Journal of Elliptic and Parabolic Equations · Journal of Elliptic and Parabolic Equations...

Journal ofElliptic and Parabolic Equations

Orthogonal Editions

JEPE Vol 2, 2016, p. 1–413

Journal of Elliptic and Parabolic Equations

Managing EditorM. Chipot (Zürich)

Email: [email protected]

Editorial Board

A. Abdulle (Lausanne)C. Alves (Campina Grande) J. A. Carrillo (London)M. Del Pino (Santiago)M. Korobkov (Novosibirsk)F. Lin (New York)S. Mardare (Rouen)H. Ninomiya (Tokyo)X. Pan (Shanghai)

K. Pileckas (Vilnius)J. Robinson (Warwick)S. Sauter (Zürich)C. Sbordone (Naples)J. K. Seo (Seoul)I. Shafrir (Haifa)C. Walker (Hannover)C. Wang (West Lafayette)J.-C. Wei (Vancouver)

THIS VOLUME IS DEDICATED TO PROFESSOR DAVID KINDERLEHRER

BIOGRAPHICAL SKETCHDavid Kinderlehrer

Alumni Professor of Mathematical SciencesProfessor of Materials Science and Engineering

Carnegie Mellon University, Pittsburgh, PAFellow, American Mathematical Society

Fellow, Society for Industrial and Applied MathematicsFellow, American Association for the Advancement of Science

Education:Massachusetts Institute of Technology, Mathematics S.B. 1963University of California, Berkeley, Mathematics Ph.D. 1968Scuola Normale Superiore, Pisa, Postdoctora1 1971 - 1973Ph.D. Advisor Hans Lewy, Postdoctoral Mentor Guido Stampacchia

Kinderlehrer has been fortunate to have a long and distinguished career in applied mathe-matics and mathematical analysis.Scientific contributions, brief highlights of high impact research:• Variational inequalities and understanding the nature of the free boundaries that arise

in constrained variational problems, including extensive collaborations with Haim Brezis,Louis Nirenberg, Joel Spruck, and the book “An Introduction to Variational Inequalitiesand their Applications” with Guido Stampacchia.• Defect structures in liquid crystals and harmonic mappings. Work with Robert Hardt

and Fang-Hua Lin in the analysis of harmonic mappings and liquid crystal theory thatestablished criteria for stable defects of mappings into manifolds. This property of defectshad not even been suspected. This gave rise to a significant amount of work in nonlinearanalysis.• Gradient Young Measures. In the Calculus of Variations, introducing new concepts of

solutions to variational problems that established a duality between variational integrandsand these solutions, with P. Pedregal, presaged by variational characterizations with MichelChipot. This became part of the contemporary analysis of shape memory alloys. In partic-ular, Kinderlehrer and Richard James made fundamental investigations of magnetic shapememory materials. (The framework was subsequently employed, just recently, twenty fiveyears later, by Sir John Ball and Richard James in their seminal work on hysteresis.)• Intracellular transport. Work helping to explain the mechanisms of molecular motor

transport, with Michel Chipot, Stuart Hastings. J. Bryce McLeod, and Michal Kowalczyk.• Mass transport theory. Work in mass transport theory showing that the gradient flow

of the Boltzmann entropy is the heat equation, thus discovering the mechanism connectingentropy and diffusion in physical systems, with Richard Jordan and Felix Otto. This is thecornerstone of a large and growing body of research in geometry and analysis worldwide, cf.comments by Ambrosio and Villani.• Evolution of microstructure and the grain boundary character distribution. Extensive

research on material texture and its evolution, the discovery of a new property, the grainboundary character distribution (GBCD). Developing new methods and enhancing simula-tion and mass transport methods to understand the statistics of observed systems, joint withmaterials scientists, colleagues, and a number of postdocs. This has been an intensive 16

year interdisciplinary project, recently involving postdocs Maria Emelianenko, YekaterinaEpshteyn, Xin Yang Lu and Richard Sharp and colleagues Katayun Barmak and ShlomoTa’asan.• He is the author/co-author of more than 160 publications, published in journals ranging

from Acta Mathematica, a top journal in mathematics, to Physical Review B, a leadingjournal in condensed matter and materials physics.Mentoring of young scientists: Kinderlehrer has had prominent Ph.D. students, especiallyIrene Fonseca (CMU), Pablo Pedregal (Universidad de Castilla-La Mancha, Spain), andAdrian Tudorascu (West Virginia). He has had many exceptional postdoctoral associates.Since 1990, these include Oscar Bruno (Cal Tech), Antonio De Simone (SISSA, Italy), GeroFriesecke (Technical University Munich), Felix Otto (Leibniz Prize recipient, founder of theHausdorff Institute, Bonn, and currently director of the Max Planck Institute for Mathe-matics in the Sciences, Leipzig), Chun Liu (Penn State), Martial Agueh (†) (U. Victoria,Canada), Michal Kowalczyk (U. Chile, Santiago), Maria Emelianenko (George Mason, NSFCareer Grant), Yekaterina Epshteyn (Utah), Russell Schwab (Michigan State), Xin YangLu (McGill), Leonard Monsaingeon (Nancy), Xiang Xu (Old Dominion).

Service to the professional community (very brief):• Coordinator (with J.L. Ericksen) 1984-85 IMA (Minnesota) program Continuum Physics

and Partial Differential Equations. This was a singularly successful program. Many iconsand leaders of the field today were participants in this program.• Scientific Director Designate and Scientific Director, Army High Performance Com-

puting Research Center, 1988-1990. Organized, wrote, and bid the scientific proposal of thislarge project at the University of Minnesota.• co-PI (with two others) of the Center for Nonlinear Analysis, an internationally rec-

ognized research center located at Carnegie Mellon University, 1991-xx, wrote and bid thesuccessful originating proposal and was the second Director. Prof. Fonseca succeeded himand remains the Director. The primary role of this Center is to fund and mentor postdocs.• co-organizer of Institute for Advanced Study/Park City summer program (2014) on

applied mathematics. This program is for graduate students. It was noted as the firstsuccessful applied math program of the 25 year Park City series.

Kinderlehrer has organized (or co-organized) many meetings or symposia for SIAM in-cluding the first mathematics and materials meeting (1994) and the first of the new seriesof mathematics and materials meetings (2010). Co-organized the SIAM Activity group onpartial differential equations. Served on organizing or scientific committees of many con-ferences. Served on several SIAM prize panels. Co-organized IPAM (UCLA) workshop.Served as AMS representative to the US National Committee on Theoretical and AppliedMechanics (2001-2009). Designed innovative undergraduate course in mathematical biologywhich has sent a number of students to graduate school, rather than to Wall Street.

Kinderlehrer serves presently on several editorial boards, notably the Archive for RationalMechanics and Analysis (since about 1989), ESAIM COCV (since about 2003), Communi-cations in Contemporary Mathematics, the Milan J. Mathematics and past service in severaldistinguished journals and book series.

JEPE Vol 2, 2016, p. 1-26

HEAT FLOW OF EXTRINSIC BIHARMONIC MAPS FROM A FOUR

DIMENSIONAL MANIFOLD WITH BOUNDARY

TAO HUANG, LEI LIU, YONG LUO, CHANGYOU WANG

Abstract. Let (M, g) be a four dimensional compact Riemannian manifold with bound-ary and (N,h) be a compact Riemannian manifold without boundary. We show the

existence of a unique, global weak solution of the heat flow of extrinsic biharmonic maps

from M to N under the Dirichlet boundary condition, which is regular with the excep-tion of at most finitely many time slices. We also discuss the behavior of solution near

the singular times. As an immediate application, we prove the existence of a smooth

extrinsic biharmonic map from M to N under any Dirichlet boundary condition.

1. Introduction

Let (M, g) be a Riemannian manifold with or without boundary and (N,h) a Riemannianmanifold without boundary and isometrically embedded in RL. For a nonnegative integer land 1 ≤ p <∞, the Sobolev space W l,p and Holder space Cl+α(0 < α < 1) are defined by:

W l,p(M,N) :=u ∈W l,p(M,RL) | u(x) ∈ N for a.e. x ∈M

,

Cl+α(M,N) :=u ∈ Cl,α(M,RL) | u(x) ∈ N ∀x ∈M

.

On W 2,2(M,N), there are two natural second order energy functionals defined by

F2(u) =

∫M

|∆u|2dvg, E2(u) =

∫M

|τ(u)|2dvg,

where ∆ is the Laplace-Beltrami operator of (M, g),

τ(u) = ∆u+A(u)(∇u,∇u)

is the tension field of u, and A(·)(·, ·) is the second fundamental form of (N,h) in RL.

2010 Mathematics Subject Classification. Primary: 35K52; Secondary: 35D30 .

Key words and phrases. Heat flow, extrinsic biharmonic maps, Dirichlet problem, Regularity.Received 07/04/2016, Accepted 13/07/2016.Huang is supported by NSF of China (No. 11601333) and NSF of Shanghai (No. 16ZR1423800). Liu

is supported by the NSF of China (No.11471299). Luo is supported by the NSF of China (No.11501421),

the Postdoctoral Science Foundation of China (No.2015M570660), and the Project-sponsored by SRF forROCS, SEM. Wang is partially supported by NSF 1522869.

1

2 TAO HUANG, LEI LIU, YONG LUO, CHANGYOU WANG

A map is called an extrinsic (or intrinsic, resp.) biharmonic map if u is a critical pointof F2 (or E2, resp.). The Euler-Lagrange equation for F2 is (cf. [3, 11, 10, 24, 25, 26])

∆2u =−L∑

i=n+1

(∆〈∇u, (dνi u)∇u〉+∇ · 〈∆u, (dνi u)∇u〉

+ 〈∇∆u, (dνi u)∇u〉)νi u

:=− f(u),

(1.1)

where νiLi=n+1 is a smooth local orthonormal frame field of the normal space of N . It iseasy to see that

|f(u)| ≤ C(|∇3u||∇u|+ |∇2u|2 + |∇u|4).(1.2)

Regularity issues for extrinsic biharmonic maps in dimensions ≥ 4 were first studied byChang etc. in [3] and for intrinsic biharmonic maps in dimension 4 by Ku [8] and alternativeproofs by Wang [23] and Strzelecki [22] when the target manifold are the standard spheres Sn.Wang extended the regularity result by [3] on biharmonic maps for general targets manifoldsN in [24, 25], where he used a Coulomb gauge frame and Riesz potentials or Lorentz spaceestimates to prove that every weakly biharmonic maps from R4 to N is smooth and everystationary biharmonic map from Rm(m ≥ 5) to N satisfies dimS ≤ m−4, i.e., the Hausdorffdimension of singular set is at most m− 4. Wang’s partial regularity result was reproved byLamm and Riviere [12] and Struwe [21] extending the lower order gauge theory techniquedeveloped in [16, 17]. See also Scheven [18] for partial regularity result for minimizingextrinsic biharmonic maps and Breiner and Lamm [2] for recent development and referencestherein.

The negative gradient flow for extrinsic biharmonic maps from a closed manifold (compactwithout boundary) was first studied by Lamm [10], where he proved the long time existenceof global smooth solution when either the dimension of M is at most 3 or under a smallinitial energy condition in dimension 4. In general, a finite time singularity may develop indimension 4 [4, 15]. Motivated by the heat flow of harmonic maps from surfaces by Struwe[20], it is natural to consider whether an extrinsic biharmonic map heat flow in dimension4 has a global weak solution, which is regular outside at most finite many singularities. Inthis direction Gastel [5] and Wang [26] independently established a global weak solution forextrinsic biharmonic map heat flow in dimension 4, which is singular at most at finite timeslices, but the problem of at most finite many singularities remains open (cf. Remark 1.2 of[26]).

In this paper we will study the extrinsic biharmonic map heat flow from a 4-dimensionalcompact manifold with boundary, i.e., we consider a solution u ∈ C4+α(M × (0, T ), N) of

∂tu+ ∆2u = −f(u)(1.3)

u(·, 0) = u0,(1.4)

u|∂M = g,(1.5)

∂νu|∂M = h,(1.6)

where u0 ∈W 2,2(M,N), g ∈ C4+α(∂M,N), and h ∈ C3+α(∂M, TgN).

HEAT FLOW OF EXTRINSIC BIHARMONIC MAPS 3

For x0 ∈M , let BMR (x0) denote the closed geodesic ball in M with center x0 and radiusR > 0, and set

E(u(t);BMR (x0)) :=

∫BMR (x0)

|∇2u(t)|2dx+( ∫

BMR (x0)

|∇u(t)|4dx) 1

2 ,

for 0 < R < 14 injM , here injM denotes the injectivity radius of M .

The main result of this work is:

Theorem 1. For dimM = 4, given any maps u0 ∈ W 2,2(M,N), g ∈ C4+α(∂M,N), andh ∈ C3+α(∂M, TgN), there exists a unique global weak solution u ∈ L∞(R+,W

2,2(M,N))of (1.3)-(1.6), with ut ∈ L2(M × R+), satisfying:(1) For any 0 < T <∞,

2

∫ T

0

∫M

|ut|2dvgdt+ F2(u(T )) ≤ F2(u0),(1.7)

and F2(u(·, t)) is monotonically non-increasing with respect to t ≥ 0.(2) There exist a positive integer K depending only on u0, g, h,M,N , and 0 < T1 < · · · <TK ≤ ∞, which is characterized by the condition

lim supt↑Tk

maxx∈M

E(u(t);BMR (x)) > ε1 for all R > 0,(1.8)

where ε1 > 0 is the constant given by Theorem 5 below, such that u ∈ C4+α,1+α4

loc

(M × (R+ \

∪Kk=1Tk), N).

(3) For each k ∈ 1, · · · ,K, there exist sequences tki ↑ Tk, xki → xk ∈M , and rki → 0 suchthat

(i) if xk ∈ M , there exists a non-constant biharmonic map ωk ∈ C∞ ∩W 2,2(R4, N)such that

uki (x) = u(xki + rki x, tki )→ ωk in C4

loc(R4).(1.9)

(ii) if xk ∈ ∂M and if lim supi→∞

dist(xki , ∂M)

rki→ ∞, then statement (i) holds. If there

exists 0 ≤ a < +∞ such that lim supi→∞

dist(xki , ∂M)

rki= a, then there exists a non-

constant biharmonic map ωk ∈ C∞ ∩W 2,2(R4a, N), with ω = constant, ∂νω = 0 on

∂R4a, such that

uki (x) = u(xki + rki x, ti)→ ωk in C4loc(R4+

a ),(1.10)

where R4a :=

(x1, x2, x3, x4) ∈ R4 | x4 ≥ −a

and R4+

a :=

(x1, x2, x3, x4) ∈R4 | x4 > −a

.

As an application of the heat flow of biharmonic maps, we obtain the following existenceresult.

Theorem 2. Let u be the global solution of (1.3)-(1.6) obtained by Theorem 1. Thenthere exists ti ↑ ∞ such that u(·, ti) converges weakly in W 2,2(M) to a biharmonic mapu∞ ∈ C4+α(M,N) with boundary data u∞|∂M = g and ∂νu∞|∂M = h.


The paper is organized as follows. In section 2, we prove a small energy regularity resultfor biharmonic maps, a main tool in our a priori estimates, and as a corollary, we obtaina gap theorem for biharmonic map under the Dirichlet boundary condition. At the endof this section, we prove several interpolation inequalities which will be used frequently inthe subsequent sections. In section 3, we give a priori estimates for the heat flow and theuniform local W 4,2 estimates in time under the assumption of small energy on a ball. Insection 4, we prove the main theorems, Theorem 1 and Theorem 2.

Throughout this paper, the letter C denotes a positive constant that depends only onM,N, u0, g, whose values may vary from lines to lines. If it depends on some other quantity,then we will point it out. For example, C(R) is a positive constant depends on R.

Additional Notations. For Ω ⊂ R4 and 0 ≤ s < t ≤ ∞, denote Ωts = Ω × [s, t],M ts = M× [s, t], and MT = M× [0, T ]. Also denote the standard Sobolev and Holder spaces

by Wm,np (MT ) and Cm+α,n+β(MT ).

We denote BR (or BR(0)) as the standard ball in R4 with radius R and center 0. Denotex′ = (x1, x2, x3) ∈ R3,

B+R :=

(x′, x4)||x′|2 + |x4|2 ≤ R2, x4 ≥ 0

, ∂0B+

R :=

(x′, x4)||x′|2 + |x4|2 ≤ R2, x4 = 0

and

V (M ts) :=

u : M × [s, t]→ N | sup

s≤σ≤t

(‖∇2u‖L2(M) + ‖∇u‖L4(M)

)+

∫Mts

(|∂tu|2 + |∇4u|2) dvgdt <∞.

2. Some basic theorems and interpolation inequalities

In this section we prove several basic theorems, including the small energy regularitytheorem and the gap theorem. At the end of this section, we derive some interpolationinequalities which will be used later.

Theorem 3. (ε1−regularity)(i) If u ∈W 4,p(B1), p > 1, is an approximated biharmonic map with bi-tension field τ2(u) ∈Lp(B1), i.e.

∆2u = −f(u) + τ2(u),

where f(u) is defined in (1.1). Then there exists a constant ε1 > 0 such that if E(u;B1) ≤ ε1,then ∥∥u− u∥∥

W 4,p(B1/2)≤ C(p,N)

(‖∇2u‖L2(B1) + ‖∇u‖L4(B1) + ‖τ2(u)‖Lp(B1)

),

where u =1

|B1|

∫B1

udx is the mean value of u over the unit ball.

(ii) If u ∈W 4,p(B+1 ), p > 1, is an approximated biharmonic map with tension field τ2(u) ∈

Lp(B+1 ) and the Dirichlet boundary value

u|∂0B+1

= g and∂u

∂~n|∂0B+

1= h,


where g ∈ C4(∂0B+1 ), h ∈ C3(∂0B+

1 ) and ~n is the outward unit normal vector of ∂0B+1 .

Then there exists a constant ε1 > 0 such that if E(u;B+1 ) ≤ ε1, then

‖u− u‖W 4,p(B+1/2

) ≤C(p,N)(‖∇2u‖L2(B+

1 )

+ ‖∇u‖L4(B+1 ) + ‖τ2(u)‖Lp(B+

1 ) + ‖g‖W 4,2(∂0B+1 ) + ‖h‖W 3,2(∂0B+

1 )

)

where u :=1

|∂0B+1/2|

∫∂0B+

1/2

u is the mean value of u over the boundary ∂0B+1 .

Proof. Here we use the idea of [14] to give the proof of boundary estimate stated in (ii), andleave the interior estimate in (i) for interested readers since it is similar to (ii) and easier toobtain.

For convenience, assume u = 0. Since u satisfies the Euler-Lagrange equation:

42u = ∇3u#∇u+∇2u#∇2u+∇2u#∇u#∇u+∇u#∇u#∇u#∇u+ τ2(u).

Here # denotes some ‘product’ for which we are only interested in the properties such as

|a#b| ≤ C|a||b|.

For 0 < σ < 1 and σ′ = 1+σ2 , let ϕ ∈ C∞0 (B+

σ′) be a cut-off function, satisfying ϕ ≡ 1 in B+σ

and |∇jϕ| ≤ 4j

(1−σ)j for j = 1, 2, 3, 4. Direct computations show that

42(ϕu) = 4(ϕ4u+ 2∇u∇ϕ+ u4ϕ)

= ϕ42u+ 4∇4u∇ϕ+ 24u4ϕ+ 4∇2u∇2ϕ+ 4∇u∇4ϕ+ u42ϕ

= (∇3u#∇u+∇2u#∇2u+∇2u#∇u#∇u+∇u#∇u#∇u#∇u+ τ2(u))ϕ

+∇3u#∇ϕ+∇2u#∇2ϕ+∇u#∇3ϕ+ u∇4ϕ

= (∇3(ϕu)#∇u+∇2(ϕu)#∇2u+∇2u#∇u#∇(ϕu) +∇u#∇u#∇u#∇(ϕu))

+∇3u#∇ϕ+∇2u#∇2ϕ+∇u#∇3ϕ+ u∇4ϕ+∇2u#∇u#∇ϕ+∇2ϕ#∇u#∇u+∇u#∇u#∇u#∇ϕ+ ϕτ2(u).

Assume first that 1 < p < 43 . Observe that

ϕu = ϕg,∂(ϕu)

∂~n= ϕh+

∂ϕ

∂~ng on ∂0B+

1 .


By the standard Lp theory (cf. [7]), we have

‖∇4(ϕu)‖Lp(B+1 ) ≤

C(‖∇u‖L4(B+

1 )‖∇3(ϕu)‖

L4p

4−p (B+1 )

+ ‖∇2u‖L2(B+1 )‖∇

2(ϕu)‖L

4p4−2p (B+

1 )

+‖∇2u‖L2(B+1 )‖∇u‖L4(B+

1 )‖∇(ϕu)‖L

4p4−3p (B+

1 )+ ‖∇u‖3

L4(B+1 )‖∇(ϕu)‖

L4p

4−3p (B+1 )

+‖∇3u‖Lp(B+

σ′ )

1− σ+‖∇2u‖Lp(B+

σ′ )

(1− σ)2+‖∇u‖Lp(B+

σ′ )

(1− σ)3

+‖u‖Lp(B+

σ′ )

(1− σ)4+‖∇2u#∇u‖Lp(B+

σ′ )

1− σ+‖∇u#∇u‖Lp(B+

σ′ )

(1− σ)2

+1

1− σ‖∇u#∇u#∇u‖Lp(B+

σ′ )+ ‖ϕτ2(u)‖Lp(B+

1 )

+‖ϕg‖W 4,p(∂0B+1 ) + ‖ϕh+

∂ϕ

∂~ng‖W 3,p(∂0B+

1 )

).

By the Sobolev embedding, if ε1 is chosen to be sufficiently small, then we get

‖∇4(ϕu)‖Lp(B+1 ) ≤

C( 1

1− σ‖∇3u‖Lp(B+

σ′ )+

1

(1− σ)2‖∇2u‖Lp(B+

σ′ )+

1

(1− σ)3‖∇u‖Lp(B+

σ′ )

+1

(1− σ)4‖u‖Lp(B+

σ′ )+

1

1− σ‖∇2u#∇u‖Lp(B+

σ′ )+

1

(1− σ)2‖∇u#∇u‖Lp(B+

σ′ )

+1

1− σ‖∇u#∇u#∇u‖Lp(B+

σ′ )+ ‖ϕτ2(u)‖Lp(B+

1 )

+‖ϕg‖W 4,p(∂0B+1 ) + ‖ϕh‖W 3,p(∂0B+

1 ) + ‖∂ϕ∂~n

g‖W 3,p(∂0B+1 )

).

Setting

Ψj(p) = sup0≤σ≤1

(1− σ)j‖∇ju‖Lp(B+σ ),

and noticing that 1− σ = 2(1− σ′), 1 < p < 43 , we have

Ψ4(p)

≤ C( 3∑j=0

Ψj(p) + ‖∇2u#∇u‖Lp(B+1 ) + ‖∇u#∇u‖Lp(B+

1 )

+ ‖∇u#∇u#∇u‖Lp(B+1 ) + ‖ϕτ2(u)‖Lp(B+

1 )

+ sup0≤σ≤1

(1− σ)4[‖ϕg‖W 4,p(∂0B+

1 ) + ‖ϕh‖W 3,p(∂0B+1 ) + ‖∂ϕ

∂~ng‖W 3,p(∂0B+

1 )

])≤ C

( 3∑j=1

Ψj(p) + ‖∇2u‖L2(B+1 ) + ‖∇u‖L4(B+

1 ) + ‖τ2(u)‖Lp(B+1 )

+ ‖g‖W 4,p(∂0B+1 ) + ‖h‖W 3,p(∂0B+

1 )

).


Using the interpolation inequality (see [14])

Ψj(p) ≤ ε4−jΨ4(p) + Cε−jΨ0(p), j = 1, 2, 3, ε > 0,

we get, by choosing sufficiently small ε > 0,

Ψ4(p) ≤C(

Ψ0(p) + ‖∇2u‖L2(B+1 ) + ‖∇u‖L4(B+

1 ) + ‖τ2(u)‖Lp(B+1 )

+ ‖g‖W 4,2(∂0B+1 ) + ‖h‖W 3,2(∂0B+

1 )

)≤C(‖∇2u‖L2(B+

1 ) + ‖∇u‖L4(B+1 ) + ‖τ2(u)‖Lp(B+

1 )

+ ‖g‖W 4,2(∂0B+1 ) + ‖h‖W 3,2(∂0B+

1 )

),

(2.1)

where we have used the Poincare inequality in the last step.If p ≥ 4

3 , we start by applying (2.1) with p = 1613 so that

‖u‖W 4, 16

13 (B+7/8

)≤ C

(‖∇2u‖L2(B+

1 ) + ‖∇u‖L4(B+1 ) + ‖τ2(u)‖Lp(B+

1 )

+‖g‖W 4,2(∂0B+1 ) + ‖h‖W 3,2(∂0B+

1 )

).

This, combined with the Sobolev embedding theorem, implies that

‖∇3u‖L

169 (B+

7/8)

+ ‖∇2u‖L

165 (B+

7/8)

+ ‖∇u‖L16(B+7/8

)

≤ C(‖∇2u‖L2(B+

1 ) + ‖∇u‖L4(B+1 ) + ‖τ2(u)‖Lp(B+

1 ) + ‖g‖W 4,2(∂0B+1 ) + ‖h‖W 3,2(∂0B+

1 )

).

With this estimate, we can bound the Lmin 85 ,p-norm of the right hand side of the Euler-

Lagrange equation of u. The interior Lp-estimate together (2.1) show that u is bounded in

W 4,min 85 ,p(B+

3/4). The lemma can be finally proved by applying the standard bootstrap-

ping method. 2

As a direct corollary of the above theorem, we can get the following gap theorem.

Theorem 4 (Gap-phenomena). Suppose either u ∈ C∞(R4, N) is a biharmonic map oru ∈ C∞(R4

+, N) is a biharmonic map with the Dirichlet boundary condition:

u|∂R4+

= constant and∂u

∂~n|∂R4

+= 0.

Then there exists a universal constant ε0 > 0 such that if either∫R4

|∆u|2dx ≤ ε20 or

∫R4

+

|∆u|2dx ≤ ε20,

then u is a constant map.

Proof. For simplicity, we only prove the upper half space case, since the proof of u ∈C∞(R4, N) is similar. By Poincare’s inequality and integration by parts, we have thatfor any R > 0, it holds

1

4R2

∫B+

2R

|∇u|2 dx ≤ C∫B+

2R

|∇2u|2 dx ≤ C∫R4

+

|∇2u|2 dx = C

∫R4

+

|∆u|2 dx ≤ Cε20.


Hence, by the standard elliptic estimates and Sobolev’s embedding, we have∫B+R

|∇2u|2 dx+( ∫

B+R

|∇u|4 dx) 1

2 ≤ C[ ∫

B+2R

|∆u|2 dx+1

R2

∫B+

2R

|∇u|2 dx]≤ Cε20.

Choosing ε0 << ε1 and applying both Theorem 3 and the Sobolev embedding, we have thatfor any R > 0, there holds

R‖∇u‖L∞(B+R) ≤ C

(‖∇2u‖L2(B+

2R) + ‖∇u‖L4(B+2R)

)≤ C.

Sending R to infinity yields that u is a constant map. 2

In the following we will prove several interpolation type inequalities, which will be usedthrough the remaining sections.

Lemma 1. For any u ∈W 4,2(M,N), we have

(2.2)

∫BMR

|∇3u|2 dx ≤ CR2

∫BMR

|∇4u|2 dx+C

R2

∫BMR

|∇2u|2 dx,

(2.3)( ∫

BMR

|∇3u|4 dx) 1

2 ≤ C( ∫

BMR

|∇4u|2 dx+C

R4

∫BMR

|∇2u|2 dx),

(2.4)

∫BMR

|∇2u|4 dx ≤ C∫BMR

|∇2u|2 dx( ∫

BMR

|∇4u|2 dx+1

R4

∫BMR

|∇2u|2 dx),∫

BMR

|∇u|8 dx ≤C∫BMR

|∇u|4 dx[ ∫

BMR

|∇2u|2 dx

×( ∫

BMR

|∇4u|2 dx+1

R4

∫BMR

|∇2u|2 dx)

+1

R4

∫BMR

|∇u|4 dx].

(2.5)

Proof. (2.2) is a standard interpolation inequality (cf. [6], page 173). By the Sobolevembedding W 1,2 → L4 on BMR we get( ∫

BMR

|∇3u|4 dx) 1

2 ≤ C( ∫

BMR

|∇4u|2 dx+1

R2

∫BMR

|∇3u|2 dx),

then (2.3) is a consequence of (2.2). By Sobolev embedding W 1, 43 → L2, we have∫BMR

|∇2u|4 dx ≤ C

R2‖|∇2u|2‖2

L43 (BMR )

+ C‖∇2u#∇3u‖2L

43 (BMR )

≤ C

R2‖∇2u‖2L2(BMR )‖∇

2u‖2L4(BMR ) + C‖∇2u‖2L2(BMR )‖∇3u‖2L4(BMR )

≤ 1

2

∫BMR

|∇2u|4 dx+C

R4‖∇2u‖4L2(BMR ) + C‖∇2u‖2L2(BMR )‖∇

3u‖2L4(BMR )

≤ 1

2

∫BMR

|∇2u|4 dx+ C

∫BMR

|∇2u|2 dx[(

∫BMR

|∇3u|4 dx)12 +

1

R4

∫BMR

|∇2u|2 dx]

≤ 1

2

∫BMR

|∇2u|4 dx+ C

∫BMR

|∇2u|2 dx( ∫

BMR

|∇4u|2 dx+1

R4

∫BMR

|∇2u|2 dx),

which implies (2.4).


By Sobolev embedding W 1,2 → L4, we have∫BMR

|∇u|8 dx ≤ C 1

R4

( ∫BMR

|∇u|4 dx)2

+ C( ∫

BMR

|∇u|2|∇2u|2 dx)2

≤ C∫BMR

|∇u|4 dx( ∫

BMR

|∇2u|4 dx+1

R4

∫BMR

|∇u|4 dx)

≤ C∫BMR

|∇u|4 dx( ∫

BMR

|∇2u|2 dx(

∫BMR

|∇4u|2 dx+1

R4

∫BMR

|∇2u|2 dx)

+1

R4

∫BMR

|∇u|4 dx),

which implies (2.5), here in the last inequality we used (2.4). 2

3. A priori estimates

In this section we will show some properties of the flow and some a priori estimates,including the monotonicity of the energy F2(u) and small energy regularity theorem ofparabolic case, which will be needed in the next section for the existence result.

From now on, we will use η as a smooth cut off function satisfying the following properties:

η ∈ C∞(M), 0 ≤ η ≤ 1, η ≡ 1 on BMR (x0), η ≡ 0 on M \BM2R(x0),

‖∇jη‖L∞ ≤C

Rj(j = 1, 2),(3.1)

where x0 ∈M and 0 < R < 14 injM .

Lemma 2. Let u ∈ V (MT ) be a solution of (1.3)-(1.6). Then for all t ∈ [0, T ), we have

F2(u(t)) + 2

∫Mt

|∂tu|2 dvgdt = F2(u0),(3.2) ∫M

|∇2u|2 dvg +( ∫

M

|∇u|4 dvg) 1

2 ≤ C(F2(u0) + ‖g‖2W 2,2(M,N)

).(3.3)

Moreover, F2(u(t)) is absolutely continuous in [0, T ) and monotonically non-increasing.

Proof. Multiplying the equation (1.3) by ∂tu and integrating by parts, we have

0 =

∫Mt

|∂tu|2 dvgdt+

∫Mt

∆2u∂tu dvgdt

=

∫Mt

|∂tu|2 dvgdt+

∫Mt

∆u∂t∆u dvgdt+

∫ t

0

∫∂M

∂ν∆u∂tu−∫ t

0

∫∂M

∆u∂t∂νu

=

∫Mt

|∂tu|2 dvgdt+

∫Mt

∂t(1

2|∆u|2) dvgdt,

where we used ∂tu|∂M = ∂t∂νu|∂M = 0. Hence (3.2) follows immediately. Moreover, it iseasy to see F2(u(t)) is absolutely continuous in [0, T ] and monotonically non-increasing.

For (3.3), we first use the L2-estimate for the Laplace operator ∆ to get∫M

|∇2u|2 dvg ≤ C(F2(u(t)) + ‖g‖2W 2,2(M,N)

)≤ C

(F2(u0) + ‖g‖2W 2,2(M,N)

).


Then, by Sobolev’s inequality we have∫M

|∇(u− g)|2 dvg ≤ C∫M

|∇2(u− g)|2dvg,

and hence ∫M

|∇u|2 dvg ≤ C( ∫

M

|∇2u|2 dvg + ‖g‖2W 2,2(M,N)

).

Observe that (3.3) is a consequence of the following Sobolev inequality( ∫M

|∇u|4 dvg) 1

2 ≤ C( ∫

M

|∇2u|2 dvg +

∫M

|∇u|2 dvg).

This completes the proof. 2

With the help of Theorem 3, we have

Lemma 3. There exists ε1 > 0 such that if u ∈ V (MT ) is a solution of (1.3)-(1.6) satisfyingE(u(t);BM2R(x0)) ≤ ε1 for some R > 0, then we have∫

BMR (x0)

|∇4u|2 dx+1

R2

∫BMR (x0)

|∇3u|2 dx ≤ C∫BM2R(x0)

|∂tu|2 dx+C

R4.(3.4)

Proof. Since u satisfies (1.3), we have that(i) if BM2R(x0) ∩ ∂M = ∅, then by taking τ2(u) = ∂tu in Theorem 3 (i) and applying astandard scaling argument, we have∫

BMR (x0)

|∇4u|2dx+1

R2

∫BMR (x0)

|∇3u|2dx ≤ C∫BM2R(x0)

|∂tu|2dx+CE(u(t);BM2R(x0))

R4

≤ C∫BM2R(x0)

|∂tu|2dx+C

R4,

(ii) if BM2R(x0) ∩ ∂M 6= ∅, then Theorem 3 (ii) implies that∫BMR (x0)

|∇4u|2dx+1

R2

∫BMR (x0)

|∇3u|2dx

≤ C∫BM2R(x0)

|∂tu|2dx+ CE(u(t);BM2R(x0)) + ‖g‖2

W 4,2(∂0BM2R(x0))+ ‖h‖2

W 3,2(∂0BM2R(x0))

R4

≤ C∫BM2R(x0)

|∂tu|2dx+C

R4.

Here ∂0BM2R(x0) = ∂BM2R(x0) ∩ ∂M . Hence the conclusion of the lemma follows. 2

From Lemma 2 and Lemma 3, we can easily obtain the following corollary.

Corollary 1. Let u ∈ V (MT ) be a solution of (1.3)-(1.6). Assume that there exists R > 0such that

sup0≤t<T

E(u(t);BM2R(x0)) ≤ ε1.


Then we have for all t ∈ [0, T ) ∫(BMR (x0))t

|∇3u|2 ≤ C +ct

R2,(3.5) ∫

(BMR (x0))t|∇4u|2 ≤ C +

ct

R4.(3.6)

Proof. Integrating (3.4) from 0 to t and applying Lemma 2 yields (3.5) and (3.6). 2

In the next step we derive an L2-estimate for ∂tu, which in turn yields an L2-estimatefor ∇4u, and then we can apply both Lp and Schauder estimates to achieve the desired Cl

estimates.

Lemma 4. Let u ∈ V (MT ) ∩σ>0 C4(MT

σ ;N) be a solution of (1.3)-(1.6). Assume thatthere exists R > 0 such that

sup0≤t<T

E(u(t);BM4R(x0)) ≤ ε1.

Then there exists 0 < δ < minT,CR4 such that for all s, t ∈ (0, T ) with s < t and|t− s| < δ, we have

sups≤t′≤t

∫M

η4|∂tu(·, t′)|2 dx ≤ C∫M

η4|∂tu(·, s)|2 dx+C

R4,(3.7)

where η is a cut off function, with support in B2R(x0), defined as in (3.1).

Proof. Differentiating equation (1.3) with respect to t, multiplying the resulting equationwith η4∂tu, and integrating over M and applying integration by parts, we get

1

2

∫Mts

η4∂t|∂tu|2 +

∫Mts

η4|∆∂tu|2 + 2

∫Mts

∇η4∇∂tu∆∂tu+

∫Mts

∆η4∂tu∆∂tu

≤C∫Mts

η4(|∇∆u||∇u||∂tu|2 + |∇2u|2|∂tu|2 + |∇u|4|∂tu|2)

:=I1 + I2 + I3.

(3.8)

Without loss of generality, we may assume that

sups≤t′≤t

∫M

η4|∂tu(·, t′)|2 =

∫M

η4|∂tu(·, t)|2.

Let’s first estimate I1. With the help of Holder’s inequality and the Sobolev embeddingW 1,2(M) → L4(M), we get

I1 ≤ C∫ t

s

(

∫BM2R(x0)

|∇u|4)14 (

∫M

η8|∂tu|4)12 (

∫BM2R(x0)

|∇∆u|4)14

≤ Cε121

∫ t

s

(

∫M

η4|∂tu|2 + |∇η|2η2|∂tu|2 + η4|∇∂tu|2)[ ∫

BM2R(x0)

|∇4u|2 +1

R2|∇3u|2

] 12


Since ∂tu|∂M = 0, by integration by part we get∫M

η4|∇∂tu|2 = −∫M

∆∂tu∂tuη4 + 4∇∂tu∂tu(η3∇η)

≤∫M

|∆∂tu∂tuη4|+1

2

∫M

η4|∇∂tu|2 + C

∫M

η2|∇η|2|∂tu|2.

Thus we have ∫M

η4|∇∂tu|2 ≤ C∫M

|∆∂tu∂tuη4|+ C

∫M

η2|∇η|2|∂tu|2.

Putting it into the estimate of I1, we have

I1 ≤ Cε121

∫ t

s

(

∫M

|∆∂tu∂tuη4|+ η4|∂tu|2 + |∇η|2η2|∂tu|2)(

∫BM2R

|∇4u|2 +1

R2|∇3u|2)

12

≤ Cε121

∫ t

s

((

∫M

η4|∆∂tu|2)12 (

∫M

η4|∂tu|2)12 +

∫M

η4|∂tu|2 +

∫M

|∇η|2η2|∂tu|2)

× (

∫BM2R

|∇4u|2 +1

R2|∇3u|2)

12

≤ Cε121

∫ t

s

[(

∫M

η4|∆∂tu|2)12 (

∫M

η4|∂tu|2)12 +

∫M

η4|∂tu|2

+ (

∫M

η4∂tu|2)12 (

∫M

|∇η|4∂tu|2)12

][

∫BM2R

|∇4u|2 +1

R2

∫BM2R

|∇3u|2]12

≤ Cε121 ( sups≤t′≤t

∫M

η4|∂tu(·, t′)|2)12 (

∫Mts

η4|∆∂tu|2 + η4|∂tu|2 +1

R4

∫Mts

|∂tu|2)12

× (C +Cδ

R4)

12 .

Therefore we obtain

I1 ≤ Cε121 (

∫M

η4|∂tu(·, t)|2 +

∫Mts

η4|∆∂tu|2 + C).(3.9)

Similarly,

I2 =

∫Mts

η4|∇2u|2|∂tu|2 ≤∫ t

s

(

∫M

η8|∂tu|4)12 (

∫BM2R(x0)

|∇2u|4)12 .

By (2.4), we get

(

∫BM2R(x0)

|∇2u|4)12 ≤ C(

∫BM2R(x0)

|∇2u|2)12 (

∫BM2R(x0)

|∇4u|2 +1

R4)

12

≤ Cε121 (

∫BM2R(X0)

|∇4u|2 +1

R4)

12 .

Then, by the same argument as in the estimates of I1, we get

I2 ≤ Cε121 (

∫M

η4|∂tu(·, t)|2 +

∫Mts

η4|∆∂tu|2 + C).(3.10)


For I3, we have

I3 =

∫ t

s

∫M

η4|∂tu|2|∇u|4 ≤∫ t

s

(

∫M

η8|∂tu|4)12 (

∫BM2R(x0)

|∇u|8)12 .

By (2.5), we get

(

∫BBM

2R(X0)

|∇u|8)12 ≤ C(

∫BM2R

|∇u|4)12 (

∫BM2R

|∇4u|2 +1

R4)

12 .

Then, by the same arguments as in the estimates of I1, I2, we obtain

I3 ≤ Cε121 (

∫M

η4|∂tu(·, t)|2 +

∫Mts

η4|∆∂tu|2 + C).(3.11)

Combining inequalities (3.8)-(3.11) yields

1

2

∫Mts

η4∂t|∂tu|2 +

∫Mts

η4|∆∂tu|2 + 2

∫Mts

∇η4∇∂tu∆∂tu+

∫Mts

∆η4∂tu∆∂tu

≤ Cε121 (

∫M

η4|∂tu(·, t)|2 +

∫Mts

η4|∆∂tu|2 + C).(3.12)

By the Cauchy-Schwartz inequality, we have

2|∫Mts

∇η4∇∂tu∆∂tu| ≤1

4

∫Mts

η4|∆∂tu|2 + 64

∫Mts

η2|∇η|2|∇∂tu|2

and ∫Mts

∆η4∂tu∆∂tu =

∫Mts

(4η3∆η + 12η2|∇η|2)∂tu∆∂tu

≥ −1

4

∫Mts

η4|∆∂tu|2 −C

R4

∫Mt

|∂tu|2

≥ −1

4

∫Mts

η4|∆∂tu|2 −C

R4.(3.13)

Furthermore, by integration by parts and noting that ∂tu|∂M = 0, we have

64

∫Mts

η2|∇η|2|∇∂tu|2 = −64

∫Mts

∇(η2|∇η|2)∇∂tu∂tu− 64

∫Mts

η2|∇η|2∆∂tu∂tu

≤∫Mts

η2|∇η|2|∇∂tu|2 +1

8

∫Mts

η4|∆∂tu|2 +C

R4

∫Mts

|∂tu|2.

Therefore we get

2

∫Mts

∇η4∇∂tu∆∂tu ≥ −1

2

∫Mts

η4|∆∂tu|2 −C

R4.(3.14)

Combining inequalities (3.12), (3.14) and (3.13) and choosing ε1 sufficiently small, we canfinally achieve (3.7). 2

Now we can derive an L2 estimate for ∇4u by the above lemma.


Lemma 5. Let u ∈ V (MT ) ∩σ>0 C4(MT

σ ;N) be a solution of (1.3)-(1.6). Assume thatthere exists R > 0 such that

sup0≤t<T

E(u(t);BM4R(x0)) ≤ ε1.

Then there exists 0 < δ < minT,CR4 such that for all t′ ∈ [ 3δ4 , T ) we have∫BMR

2

(x0)

|∇4u(·, t′)|2 ≤ C(1

δ+

1

R4).(3.15)

Proof. By Lemma 3, we have, for all t′ ∈ [ 3δ4 , T )∫BMR

2

(x0)

|∇4u|2(·, t′)dx ≤ C∫BMR (x0)

|∂tu|2(·, t′)dx+C

R4.(3.16)

Let η be a cut off function as in Lemma 4. Without loss of generality, we assume that∫M

η4|∂tu(·, s)|2dx = inft′− δ2≤s′≤t′−

δ4

∫M

η4|∂tu(·, s′)|2.

Then Lemma 4 implies

supt′− δ4≤t≤t′

∫M

η4|∂tu(·, t)|2 ≤ C∫M

η4|∂tu(·, s)|2 +C

R4

= C inft′− δ2≤s′≤t′−

δ4

∫M

η4|∂tu(·, s′)|2 +C

R4

≤ C

δ

∫ t′− δ4

t′− δ2

∫M

|∂tu|2 +C

R4≤ C(

1

δ+

1

R4).

Therefore we have ∫BMR (x0)

|∂tu|2(·, t′)dx ≤ C(1

δ+

1

R4).(3.17)

This completes the proof. 2

By Lemma 5, we have

Theorem 5. There exists ε1 > 0, depending only on M,N, u0, g, such that for 0 < T <∞,if u is a smooth solution of (1.3)-(1.6) satisfying

sup0<t≤T

∫BM4R(x0)

|∇2u(·, t)|2dx+ (

∫BM4R(x0)

|∇u(·, t)|4dx)12 ≤ ε1,(3.18)

for some R < 12 injM and x0 ∈M , then we have

(3.19) maxT2 ≤t≤T

‖u‖Ck(BMR4

(x0)) ≤ C(k,R−1, T, ‖∇2u0‖L2(M), ‖g‖Ck(∂M), ‖h‖Ck−1(∂M)

).

Proof. It follows from Lemma 5 that u is uniformly bounded in W 4,2(BMR2

(x0)) for T2 ≤

t ≤ T . It follows that ut + ∆2u ∈ Lp(BMR2

(x0) × [T2 , T ]) for any 1 < p < ∞. Therefore by

the standard parabolic Lp-theory and Schauder estimate, we can get the desired estimate. 2


To prove our main theorem, we need to establish a lower bound estimate of the timeinterval for the existence of a smooth solution of (1.3)-(1.6). First we have

Lemma 6. Let u ∈ V (MT ) be a solution of (1.3)-(1.6). Assume that there exists 0 < R < 1such that

sup0≤t<T

E(u(t);BM2R(x0)) ≤ ε1.

Then we have for all t ∈ [0, T )

(3.20) E(u(t);BMR (x0)) ≤ CE(u(0);BM2R(x0)) +Ct

R4+C√t

R2+ CR2.

Proof. Multiplying (1.3) by η4∂tu and integrating by parts, we get∫Mt

η4|∂tu|2 dxdt+1

2

∫Mt

η4∂

∂t|∆u|2 dxdt

=

∫Mt

∆u∆η4∂tu dxdt+ 2

∫Mt

∆u∇η4∇∂tu dxdt

= −∫Mt

∆u∆η4∂tu dxdt− 2

∫Mt

∇∆u∇η4∂tu dxdt

≤ 1

2

∫Mt

η4|∂tu|2 dxdt+C

R4

∫BM2R(x0)

t|∆u|2 dxdt+

C

R2

∫BM2R(x0)

t|∇3u|2 dxdt

≤ 1

2

∫Mt

η4|∂tu|2 dxdt+C

R4

∫BM2R(x0)

t|∆u|2 dxdt+ Cε

∫BM2R(x0)

t|∇4u|2 dxdt

+C

εR4

∫BM2R(x0)

t|∇2u|2 dxdt

≤ 1

2

∫Mt

η4|∂tu|2 dxdt+Ct

R4+C√t

R2by taking ε =

√t

R2.

Then we have ∫Mt

η4|∂tu|2 dxdt+

∫Mt

η4∂

∂t|∆u|2 dxdt ≤ Ct

R4+C√t

R2.(3.21)

Thus we obtain ∫BMR (x0)

|∆u|2(t) dx ≤∫BM2R(x0)

|∆u|2(0) dx+Ct

R4+C√t

R2.(3.22)

Observe that

∂t(1

2|∇u|2) = 〈∇u,∇∂tu〉 = ∇〈∇u, ∂tu〉 − 〈∆u, ∂tu〉.


Multiplying this equality by η4, integrating it over M , and applying integration by partsand (1.3), we obtain∫

Mt

∂t(1

2η4|∇u|2)dxdt =

∫Mt

η4∇〈∇u, ∂tu〉dxdt−∫Mt

η4〈∆u, ∂tu〉dxdt

= −∫Mt

∇η4〈∇u, ∂tu〉dxdt−∫Mt

η4〈∆u, ∂tu〉dxdt

≤ R2

4

∫Mt

η4|∂tu|2dxdt+C

R4

∫Mt

η2|∇u|2dxdt+C

R2

∫Mt

η4|∆u|2dxdt

≤ R2

4

∫Mt

η4|∂tu|2dxdt+C

R2

∫ t

0

(

∫B2R(x0)

|∇u|4dx)12 dt+

Ct

R2

≤ R2

4

∫Mt

η4|∂tu|2dxdt+Ct

R2

≤ R2

4

∫BM2R(x0)

|∆u|2(0)dx+ C√t+

Ct

R2.

Thus,

∫BMR (x0)

|∇u|2(t)dx ≤∫BM2R(x0)

|∇u|2(0)dx+R2

4

∫BM2R(x0)

|∆u|2(0)dx+ C√t+

Ct

R2.(3.23)

Let q ∈ C2+α(M,RN ) be a harmonic function, satisfying

∆q = 0 in M,

q = g on ∂M.

Then we have

‖q‖C2+α(M) ≤ C(M)‖g‖C2+α(M),

and hence∫BMR (x0)

|∇2(u− q)|2(t)dx+ (

∫BMR (x0)

|∇(u− q)|4(t)dx)12

≤ C∫BM3

2R(x0)

|∆(u− q)|2(t)dx+C

R2

∫BM3

2R(x0)

|∇(u− q)|2(t)dx

≤ C∫BM2R(x0)

|∆u|2(0)dx+C

R2

∫BM2R(x0)

|∇u|2(0)dx+Ct

R4+C√t

R2+

C

R2

∫BM2R(x0)

|∇q|2dx

≤ C∫BM2R(x0)

|∆u|2(0)dx+ C(

∫BM2R(x0)

|∇u|4(0)dx)12 +

Ct

R4+C√t

R2+ CR2.


This implies∫BMR (x0)

|∇2u|2(t)dx+ (

∫BMR (x0)

|∇u|4(t)dx)12

≤ C(

∫BMR (x0)

|∇2(u− q)|2(t)dx+

∫BMR (x0)

|∇2q|2(t)dx)

+ C((

∫BMR (x0)

|∇(u− q)|4(t)dx)12 + (

∫BMR (x0)

|∇q|4dx)12 )

≤ C∫BM2R(x0)

|∆u|2(0)dx+ C(

∫BM2R(x0)

|∇u|4(0)dx)12 +

Ct

R4+C√t

R2+ CR2,

which implies (3.20). This completes the proof. 2

According to Lemma 6, we have

Lemma 7. There exists 0 < ε2 ε1 <injM

2 such that if u0 ∈ C∞(M,N), g ∈ C∞(∂M,N),and h ∈ C∞(∂M, TgN) satisfies

supx∈M

(∫BM2R(x)

|∇2u0|2 dx+ (

∫BM2R(x)

|∇u0|4 dx)12

)≤ ε22,(3.24)

for some R ∈ (0, ε2). Then there exists T1 ≥ O(R4ε21) and a unique solution u ∈ C∞(M ×[0, T1], N) to (1.3)-(1.6).

Proof. Let T1 > 0 be the maximum time interval such that there exists a smooth solutionu ∈ C∞(M × [0, T1), N) of (1.3)-(1.6). Let T ′1 > 0 be the maximum time such that

(3.25) sup0≤t≤T ′1

supx∈M

E(u(t);BMR (x)) ≤ ε1.

By Theorem 5, we know T1 ≥ T ′1. By Lemma 6, we get

ε1 = E(u(T ′1);BMR (x)) ≤ CE(u(0);BM2R(x)) +CT ′1R4

+C√T ′1

R2+ CR2

≤ ε12

+CT ′1R4

+C√T ′1

R2

≤ 3ε14

+CT ′1ε1R4

.

This implies T1 ≥ T ′1 ≥ O(R4ε21). 2

4. Existence results and behavior of solutions near singularities

In this section, we show the existence of the global weak solution of the extrinsic bihar-monic map flow, which is regular with the exception of at most finitely many time slices. Wealso study the behavior of the solution near its singularities. Moreover, we get the existenceof biharmonic maps with a fixed Dirichlet boundary data. Both Theorem 1 and Theorem 2will be proved in this section.


Proof of Theorem 1. Step 1. From [19] and [1] we see that there exists a sequence ofmaps φl ∈ C4+α(M,N) such that φl = g, ∂νφl = h on ∂M , and

φl → u0 strongly in W 2,2(M,N).

Step 2. The short-time existence. Since φl → u0 inW 2,2(M,N), there exists aR ∈ (0, injM2 )such that

supl

supx∈M

(∫B2R(x)

|∇2φl|2 dx+ (

∫B2R(x)

|∇φl|4 dx)12

)≤ ε22,

where ε2 is given in Lemma 7. By the short-time existence theory in [9], there exist Tl > 0and ul ∈ C4+α,1+α

4 (M × [0, Tl), N) which solves (1.3) with the boundary-initial data (g, h).Then Lemma 7 implies that Tl ≥ O(R4ε21) and Theorem 5 implies that we have uniformly

C4+α,1+α

4

loc estimates of ul in M × (0, O(R4ε21)]. Hence we may assume that ul converges tou weakly in W 2,2(M,N), strongly in W 1,2(M,N) and in C4+α,1+α

4 (M × [ρ,O(R4ε21)], N)

for any ρ > 0. It is clear that u ∈ C4+α,1+α4

loc (M × (0, O(R4ε21)), N) is a classical solutionof (1.3). The short-time existence theory guarantees the existence of a solution to (1.3)using u(O(R4ε21)) as the new initial data so that the solution can be continued to a larger

time interval. Assume that T1 is the maximum time interval such that u ∈ C4+α,1+α4

loc (M ×(0, T1), N) solves (1.3)-(1.6). Repeating this argument, the solution can be continued untilthe first time of energy concentration excels ε1, that is, the condition

(4.1) limr↓0

limt↑T1

supx∈M

E(u(t), Br(x)) > ε1

reaches. Set

(4.2) S(T1) :=

(x, T1) | x ∈M, limr→0

lim supt↑T1

E(u(t), Br(x)) > ε1

,

which is called as the singularity set of u at time T1. It is an open question if S(T1) is afinite set.Step 3. Behavior of the solution u near its first singular time T1. By the standard blowupargument, there exist sequences t1i T1, x1i → x0 ∈M , and r1i → 0 such that

E(u(t1i ), BMr1i

(x1i )) = sup(x,t)∈M×[T1−δ2,t1i ]

r≤r1i

E(u(t), BMr (x)) =ε1C0,(4.3)

where C0 is a positive constant to be determined later. Assume that B2r1i(x0) is covered by

m balls of radius r1i constained in M and let C0 = m, then we see that

supT1−δ2≤t<T1

E(u(t);BM2r1i(x0)) ≤ ε1.

By Lemma 6, for any T1 − δ2 ≤ s ≤ t1i < T1, we have

E(u(t1i );BMr1i

(x0)) ≤ CE(u(s);BM2r1i(x0)) + C

t1i − s(r1i )

4+ C

√t1i − s

(r1i )2

+ C(r1i )2.

Set T =ε21

16C2C20

. Then we have

(4.4) E(u(s);BM2r1i(x0)) ≥ ε1

2CC0


for any s ∈ [t1i − T (r1i )4, t1i ], when i is sufficiently large.

Case 1. lim supi→∞

dist(x1i , ∂M)

r1i→∞. By passing to a subsequence, we may assume

limi→∞

dist(x1i , ∂M)

r1i→∞.

Assume t1i − δ2

4 > T1 − δ2 and define

Bi :=x ∈ R4|x1i + r1i x ∈ BMδ (x0)

,

and

vi(x, t) : = u(x1i + r1i x, t1i + (r1i )

4t), ∀ x ∈ Bi, −δ2

4(r1i )4≤ t ≤ 0.

It is easy to see that Bi → R4 as i→∞. Then vi satisfies

∂tvi + ∆2vi = −f(vi),(4.5)

along with the boundary conditionvi(x, t) = g(x1i + r1i x), if x1i + r1i x ∈ ∂M ;

∂νvi(x, t) = r1i h(x1i + r1i x), if x1i + r1i x ∈ ∂M.(4.6)

By Lemma 2, we have∫ 0

−T

∫Bi

|∂tvi|2 dxdt ≤∫ t1i

t1i−(r1i )4T

∫M

|∂tu|2 dvgdt→ 0 as i→∞,(4.7)

and

(4.8) supδ2

4(r1i)4≤t≤0

E(vi, Bi) ≤ supT1−δ2≤t≤T1

E(u) ≤ C.

By (4.3), we can see that

sup−T≤t≤0

supx∈Bi

E(vi, B1(x) ∩Bi) ≤ sup(x,t)∈M×[T1−δ2,t1i ]

r≤r1i

E(u(t), Br(x)) =ε1

2C0.

Hence, for any x ∈ R4, when i is sufficiently large, we have

(4.9) sup−T≤t≤0

E(vi, B1(x)) ≤ ε12C0

.

Combining (4.9) with Theorem 5, we have

(4.10) sup−T2 ≤t≤0

‖vi(·, t)‖C4+α(B1/2(x)) ≤ C,

which yields

(4.11) sup−T2 ≤t≤0

‖vi(·, t)‖C4+α(BR) ≤ C(R), ∀ R > 0.

From (4.7) and (4.11), we can find σi ∈ [−T2 , 0] such that as i→∞, there holds∫Bi

|∂tvi|2(x, σi) dx→ 0(4.12)


and

(4.13) ‖vi(·, σi)‖C4+αloc (R4) ≤ C.

Therefore, there exists a subsequence of vi(·, σi) and a limit map v ∈ C4(R4, N) such that

vi(·, σi)→ v in C4(BR), ∀ R > 0.(4.14)

Setting t = σi in the equation (4.5) and letting i→∞, it is easy to see that v is a biharmonicmap with

ε12CC0

≤ E(v;R4) ≤ C,

where the above inequality follows from (4.4) and (4.8). Taking t1i + (r1i )4σi as the new time

sequence, then we get that

ui(x) = vi(x, σi) = u(x1i + r1i x, t1i + (r1i )

4σi)

is the desired sequence in the theorem.

Case 2. lim supi→∞

dist(x1i , ∂M)

r1i< ∞. After taking a subsequence, we may assume that

dist(x1i ,∂M)

r1i→ a as i→∞. Then

Bi → R4a :=

(x′, x4)|x4 ≥ −a

,

where x′ := (x1, x2, x3) ∈ R3. Noting that for any x ∈x4 = −a

, x1i +r1i x→ x0. Moreover,

vi(x, t) = g(x1i + r1i x), if x1i + r1i x ∈ ∂M ;

∂νvi(x, t) = r1i h(x1i + r1i x), if x1i + r1i x ∈ ∂M.(4.15)

By Theorem 5 and (4.3), for any BR(0) ⊂ R4, R > 0, we have

(4.16) sup−T2 ≤t≤0

‖vi(·, t)‖C4+α(BR(0)∩Bi) ≤ C.

Using a similar argument as in Case 1, we can obtain v ∈ C4(R4a, N) satisfying

ε12CC0

≤ E(v;R4a) ≤ C,(4.17)

and a sequence σi ∈ [−T2 , 0] such that as i→∞, there hold

‖vi(·, σi)− v‖C4(Bi∩BR(0)) → 0,(4.18)

for any R > 0. Moreover, v is a biharmonic map with the boundary conditionv(x) = g(x0), on ∂R4

a;

∂νv(x) = 0, on ∂R4a.

(4.19)

Step 4. Global existence of weak solutions. Let (x0, T1) ∈ S(T1) be as in Step 3. Then weclaim

limr→0

lim supt↑T1

∫Br(x0)

|∆u(·, t)|2 dx ≥ ε20.(4.20)


In fact, by (4.14), (4.18) and Theorem 4, we have

ε20 ≤ limR→∞

∫BR(0)

|∆v|2dx = limR→∞

limi→∞

∫BR(0)

|∆vi|2(·, σi)dx

= limR→∞

limi→∞

∫BriR(xi)

|∆u|2(·, ti + r4i σi)dx

≤ limr→0

lim supt↑T1

∫Br(x0)

|∆u(·, t)|2dx.

Next, we claim that there is a unique weak limit u(·, T1) ∈W 2,2(M,N) such that

limt↑T1

u(·, t) = u(·, T1) weakly in W 2,2(M,N).

In fact, by Lemma 2, for any sequence ti → T1, there exists a subsequence (also denotedby ti) such that u(·, ti) → u(·, T1) weakly in W 2,2(M) as i → ∞. So, we just need to showthe weak limit u(·, T1) is independent of the choice of the time sequences. Let si → T1 beanother time sequence and the corresponding weak limit u(·, T1). Note that∫

M

|u(·, T1)− u(·, T1)|2 dx

=

∫M

〈u(·, T1)− u(·, T1), u(·, T1)− u(·, ti)〉 dx+

∫M

〈u(·, T1)− u(·, T1), u(·, ti)

− u(·, si)〉 dx+

∫M

〈u(·, T1)− u(·, T1), u(·, si)− u(·, T1)〉 dx.

(4.21)

Since ∫M

|u(·, ti)− u(·, si)|2 dx =

∫M

|∫ ti

si

∂u

∂tdt|2 dx ≤ |si − ti||

∫Mtisi

|∂u∂t|2 dxdt|,∫

MT1

|∂u∂t|2 dxdt ≤ C (see Lemma 2), u(·, ti) u(·, T1), u(·, si) u(·, T1) weakly in

W 2,2(M) by sending i→∞ in (4.21), we obtain∫M

|u(·, T1)− u(·, T1)|2 dx = 0.

Thus u(·, T1) = u(·, T1). It is easy to see that∫M

|∆u(·, T1)|2 dx ≤∫M

|∆u0|2 dx− ε20.

Now we use u(·, T1) as the initial condition and (g, h) as the boundary condition to extend theabove solution beyond T1 to obtain a weak solution u : M×(0, T2)→ N for some T2 > T1 by

piecing together the solutions at T1. Then we see that u ∈ C4+α,1+α4

loc (M×((0, T2)\T1), N).Iterating this process, we obtain a global solution defined on M × [0,∞). Let TkKk=1 beall the possible singular times. Then we have∫

M

|∆u(·, TK)|2 dx ≤ lim infti↑TK

∫M

|∆u(·, ti)|2 dx− ε20

≤∫M

|∆u0|2 dx−Kε20,


which implies

K ≤∫M|∆u0|2 dxε20

.

Hence there are at most finitely many singular time slices.Step 5. Uniqueness. The only thing left to be proven is uniqueness and we only needto prove uniqueness of the short time solution constructed above and the full uniquenessfollows by iteration. Let u, v : [0, t0) → N be two constructed smooth (for t > 0) solutionsand set w := u− v. Then

∂tw + ∆2w = f(u)− f(v)

= ∇3u#∇u+∇2u#∇2u+∇2u#∇u#∇u+∇u#∇u#∇u#∇u−(∇3v#∇v +∇2v#∇2v +∇2v#∇v#∇v +∇v#∇v#∇v#∇v

).

Multiply this equation with w and integrate over (0, s) ×M . By partial integration (w =∂νw = 0 on ∂M for any t ∈ (0, s)), we can get rid of derivatives of order > 2 (cf. [5]).Simplifying terms by using Young’s inequality we get

1

2

∫M

|w(s)|2 +

∫Ms

|∆w|2

≤C2∑k=1

∫Ms

|w|2(|∇ku|+ |∇kv|) 4k + C

2∑k=1

∫Ms

|∇w|2(|∇ku|+ |∇kv|) 2k

+ C

1∑l=0

2−l∑k=1

∫Ms

|∇2w||∇lw|(|∇ku|+ |∇kv|)2−lk

:=I4 + I5 + I6.

(4.22)

To make it more clear how the above inequality is obtained, let us give the details of theestimates of the highest order term of (f(u)− f(v))w (we denote it by ϕ(u) · ∇3u · ∇u) asfollows.∫

Ms

(ϕ(u) · ∇3u · ∇u− ϕ(v) · ∇3v · ∇v)w

=

∫Ms

[(ϕ(u)− ϕ(v)) · ∇3u · ∇u+ ϕ(v) · ∇3w · ∇u+ ϕ(v) · ∇3v · ∇w]w

=

∫Ms

∇2u#∇w#∇u#w +∇2u#∇2u∇#w2 +∇2w#∇u#∇v#w +∇2w#∇2u#w

+∇2w#∇u#∇w +∇2v#∇v#∇w#w +∇2v#∇2w#w +∇2v#∇w#∇w≤ I4 + I5 + I6,

where

ϕ(u) · ∇3u · ∇u := ϕijklAB (u) · ∇3ijku

A · ∇luB

and the last inequality follows from Young’s inequality and following property

|∇kϕ| ≤ C(N) and |ϕ(u)− ϕ(v)| ≤ C(N)(u− v).


In the following, let’s estimate the right hand side of (4.22). By Holder’s inequality, theSobolev inequality, the Poincare inequality and partial integration, we get∫

Ms

|w|2(|∇u|+ |∇v|)4

≤(

∫Ms

|w|4)12 (

∫Ms

|∇u|8 + |∇v|8)12

≤C(

∫Ms

|∇u|8 + |∇v|8)12 (

∫ s

0

(

∫M

|∇w|2 + |w|2)2)12

≤C(

∫Ms

|∇u|8 + |∇v|8)12 (

∫ s

0

(

∫M

|∇w|2)2)12

≤C(

∫Ms

|∇u|8 + |∇v|8)12 (

∫ s

0

∫M

|w|2∫M

|∇2w|2)12

≤C(

∫Ms

|∇u|8 + |∇v|8)12 ( supt∈(0,s)

∫M

|w(t)|2)12 (

∫Ms

|∇2w|2)12

≤C(

∫Ms

|∇u|8 + |∇v|8)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2),

(4.23)

and similarly we have∫Ms

|w|2(|∇2u|+ |∇2v|)2

≤ C(

∫Ms

|∇2u|4 + |∇2v|4)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2).(4.24)

Now let’s estimate I5 term by term,∫Ms

|∇w|2|∇u|2 = −∫Ms

w∆w|∇u|2 −∫Ms

w∇w∇u · ∇2u

≤ ε∫Ms

|∇2w|2 + C(ε)

∫Ms

w2|∇u|4 +1

2

∫Ms

|∇w|2|∇u|2 +

∫Ms

w2|∇2u|2.

Therefore we get∫Ms

|∇w|2|∇u|2 ≤ ε∫Ms

|∇2w|2 + C(ε)

∫Ms

w2|∇u|4 +

∫Ms

w2|∇2u|2.

Thus by (4.23) and (4.24) we have∫Ms

|∇w|2(|∇u|2 + |∇v|2)

≤ ε

∫Ms

|∇2w|2 + C(ε)((

∫Ms

|∇u|8 + |∇v|8)12 + (

∫Ms

|∇2u|4 + |∇2v|4)12 )

×( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2).


In addition,∫Ms

|∇w|2|∇2u| = −∫Ms

w∆w|∇2u| −∫Ms

w∇w · ∇|∇2u|

≤ ε∫Ms

|∇2w|2 + C(ε)

∫Ms

w2|∇2u|2 − 1

2

∫Ms

∇w2 · ∇|∇2u|

= ε

∫Ms

|∇2w|2 + C(ε)

∫Ms

w2|∇2u|2 +1

2

∫Ms

w2∆|∇2u|

≤ ε∫Ms

|∇2w|2 + C(ε)

∫Ms

w2|∇2u|2 + C(

∫Ms

w4)12 (

∫Ms

|∇4u|2)12

≤ ε∫Ms

|∇2w|2 + C(ε)(

∫Ms

|∇2u|4)12 )( sup

t∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2)

+ C(

∫Ms

|∇4u|2)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2).

Thus we get∫Ms

|∇w|2(|∇2u|+ |∇2v|)

≤ ε

∫Ms

|∇2w|2 + C(ε)(

∫Ms

|∇2u|4 + |∇2v|4)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2)

+C(

∫Ms

|∇4u|2 + |∇4v|2)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2).

In summation of the above estimates we have

I5 ≤ε∫Ms

|∇2w|2 + C(ε)(

∫Ms

|∇u|8 + |∇v|8)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2)

+ C(ε)(

∫Ms

|∇2u|4 + |∇2v|4)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2)

+ C(

∫Ms

|∇4u|2 + |∇4v|2)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2).

(4.25)

We are left to estimate the last summation I6 in (4.22). Note that by Young’s inequality

I6 ≤ ε∫Mso

|∇2w|2 + C(ε)(I4 + I5).(4.26)

Therefore from inequalities (4.22)-(4.26) we obtain∫M

|w(s)|2 +

∫Ms

|∆w(s)|2

≤ε∫Ms

|∇2w|2 + C(ε)(

∫Ms

|∇u|8 + |∇v|8)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2)

+ C(ε)(

∫Ms

|∇2u|4 + |∇2v|4)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2)

+ C(ε)(

∫Ms

|∇4u|2 + |∇4v|2)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2).


By the standard elliptic estimate, noting that w = 0 on ∂M , we have∫M|∇2w|2 ≤

C∫M|∆w|2. Choosing ε = 1

2C , we obtain∫M

|w(s)|2 +

∫Ms

|∇2w(s)|2

≤ C(

∫Ms

|∇u|8 + |∇v|8)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2)

+ C(

∫Ms

|∇2u|4 + |∇2v|4)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2)

+ C(

∫Ms

|∇4u|2 + |∇4v|2)12 ( supt∈(0,s)

∫M

|w(t)|2 +

∫Ms

|∇2w|2).

Hence for solutions u, v ∈ V (MT ) by the interpolation inequalities of Lemma 1 we see that

we can choose s small enough such that C(∫Ms |∇u|8 + |∇v|8)

12 , C(

∫Ms |∇2u|4 + |∇2v|4)

14

and C(∫Ms |∇4u|2 + |∇4v|2)

12 are all smaller than 1

4 . Without loss of generality, we assumethat

supt∈[0,s]

∫M

|w(t)|2 =

∫M

|w(s)|2.

Hence we obtain that supt∈[0,s]∫M|w(t)|2 = 0, i.e., u ≡ v on [0, s). 2

Proof of Theorem 2. By Theorem 1, we see that there exists a time sequence ti,ti → +∞ as i → +∞, such that ∂u(ti,·)

∂ti→ 0 in L2(M,N) and u(·, ti) converges weakly in

W 2,2(M,N) to a map u∞ ∈W 2,2(M,N) with Dirichlet boundary data u = g and ∂νu = h on∂M , where g ∈ C4+α(∂M,N) and h ∈ C3+α(∂M, TgN). Denote u(ti) = ui and gi = −∂ui∂ti

,

then ∆2ui + f(ui) = gi. Note that gi → 0 in L2(M,N) and hence in (W 2,2(M,N))∗, by theweak compactness theorem of Zheng [27], we see that u∞ ∈ W 2,2(M,N) is a biharmonicmap. Then we see that u ∈ C4+α,1+α

4 (M,N) by the interior regularity theorem of Wang[24] and boundary regularity theorem of Lamm and Wang [13]. 2

References

[1] P. Bousquet, A. Ponce and J. V. Schaftingen, Strong density for higher order Sobolev spaces into compactmanifolds, J. Eur. Math. Soc. 17(2015), 763-817.

[2] C. Breiner and T. Lamm, Quantitative stratification and higher regularity for biharmonic maps,

Manuscripta Math. 148(2015), 379-398.[3] S.-Y.A. Chang, L. Wang and P.C. Yang, Regularity of harmonic maps, Commun. Pure Appl. Math.

52(1999), 1099-1111.

[4] M. Coorper, Critical O(d)−equivatiant biharmonic maps, Calc. Var. Partial Differential Equations, 54(3)(2015), 2895-2919.

[5] A. Gastel, The extrinsic polyharmonic map heat flow in the critical dimension, Adv. Geom., 6(4) (2006),

501-521.[6] D. Gilbarg and N. S. Trudinger, Elliptic Partial Differential Equations of Second Order, Springer Verlag,

Heidelberg, 2001.

[7] M. Hong and H. Yin, Partial regularity of a minimizer of the relaxed energy for biharmonic maps, J.Funct. Anal. 262(2) (2012), 681-718.

[8] F. Ku, Interior and boundary regularity of intrinsic biharmonic maps to spheres, Pacific J. Math.234(2008), 43-67.


[9] T. Lamm, Biharmonischer Warmefluss, Diplomarbeit, Universitat Freiburg, 2001.

[10] T. Lamm, Heat flow for extrinsic biharmonic maps with small initial energy, Ann. Global Anal. Geom.,26(4) (2004), 369-384.

[11] T. Lamm, Biharmonic map heat flow into manifolds of nonpositive curvature, Calc. Var. Partial

Differential Equations, 22(4) (2005), 421-445.[12] T. Lamm, T. Riviere, Conservation laws for fourth order systems in four dimensions, Commun. Partial

Differ. Equ. 33(2008), 245-262.

[13] T. Lamm and C. Wang, Boundary regularity for polyharmonic maps in the critical dimension, Adv.Calc. Var. 2(2009), 1-16.

[14] L. Liu and H. Yin, Neck analysis for biharmonic maps, arxiv:1312.4600, to appear in Math. Z.

[15] L. Liu and H. Yin, On the finite time blow-up of biharmonic map flow in dimension four, Journal ofElliptic and Parabolic Equations, 1(2015), p.363-385.

[16] T. Riviere, Conservation laws for conformally invariant variational problems, Invent. Math. 168(2008),

1-22.[17] T. Riviere and M. Struwe, Partial regularity for harmonic maps and related problems, Commun. Pure

Appl. Math. 61(2008), 451-463.[18] C. Scheven, Dimension reduction for the singular set of biharmonic maps, Adv. Calc. Var. 1(2008),

53-91.

[19] R. Schoen and K. Uhlenbeck, Boundary regularity and the Dirichlet problem for harmonic maps, J.Differential Geom. 18(1983), 253-268.

[20] M. Struwe, On the evolution of harmonic mappings of Riemannian surfaces, Commun. Math. Helv. 60

(1985), 558-581.[21] M. Struwe, Partial regularity for biharmonic maps, revisited, Calc. Var. Partial Differ. Equ. 33(2008),

249-262.

[22] P. Strzelecki, On biharmonic maps and their generalizations, Calc. Var. Partial Differ. Equ. 18(2003),401-432.

[23] C. Wang, Remarks on biharmonic maps into spheres, Calc. Var. Partial Differ. Equ. 21(2004), 221-242.

[24] C. Wang, Biharmonic maps from R4 into a Riemannian manifold, Math. Z. 247(2004), 65-87.[25] C. Wang, Stationary biharmonic maps from Rm into a Riemannian manifold, Commun. Pure Appl.

Math. 57(2004), 419-444.[26] C. Wang, Heat flow of biharmonic maps in dimension four and its application, Pure. Appl. Math.

Quar., 3(2) (2007), 595-613.

[27] S. Zheng, Weak compactness of biharmonic maps, Electron. J. Diff. Equ. 190(2012), 7pp.

NYU-ECNU Institute of Mathematical Sciences at NYU Shanghai, 3663 Zhongshan Road North,

Shanghai, 200062, China

E-mail address: [email protected]

Department of Mathematics, Tsinghua University, HaiDian Road, BeiJing 100084, China; Max-planck institut fur mathematik In den naturwissenschaft, Inselstr.22, D-04103, Leipzig, Germany

E-mail address: [email protected] or [email protected]

School of mathematics and statistics, Wuhan university, Wuhan 430072, China; Max-planckinstitut fur mathematik In den naturwissenschaft, Inselstr.22, D-04103, Leipzig, Germany

E-mail address: [email protected] or [email protected]

Department of Mathematics, Purdue University, West Lafayette, IN 47907


JEPE Vol 2, 2016, p. 27-36

THE ROLE OF NON-NEGATIVE POLYNOMIALS FOR RANK-ONE

CONVEXITY AND QUASI CONVEXITY

LUIS BANDEIRA AND PABLO PEDREGAL

Abstract. We stress the relationship between the non-negativeness of polynomials and

quasi convexity and rank-one convexity. In particular, we translate the celebrated theo-rem of Hilbert ([3]) about non-negativeness of polynomials and sums of squares, into a

test for rank-one convex functions defined on 2 × 2-matrices. Even if the density for an

integral functional is a fourth-degree, homogeneous polynomial, quasi convexity cannotbe reduced to the non-negativeness of polynomials of a fixed, finite number of variables.

1. Introduction

It is well-known that the quasi convexity condition for a density φ(F) : Mm×N → R,expressed through the inequality

(1.1)

∫Q

φ(F +∇u(x)) dx ≥ φ(F)

valid for every F ∈ Mm×N , and every Q-periodic test field u : Q → Rm, is the necessaryand sufficient property for the integral functional

(1.2) I(v) =

∫Ω

φ(∇v(y)) dy

to be (sequentially) weak lower semicontinuous ([7]). This, in turn, is one of the importantingredients of the Direct Method of the Calculus of Variations to show existence of minimiz-ers ([2], [8]) for integral functionals like the one in (1.2). Q is the unit cube in RN , while Ωis a general, bounded, regular domain in RN .

This quasi convexity condition is hard to understand. Intimately related concepts, likepolyconvexity and rank-one convexity, were introduced and examined throughout the years.See [1], [2]. In particular, a main open question that remains to be answered is the equiva-lence of rank-one convexity and quasi convexity. It was shown not to be the case for m ≥ 3in [10], but still remains unsolved for the case m = 2.

2010 Mathematics Subject Classification. Primary: 49J45, 49J10; Secondary: 14P99.

Key words and phrases. Rank-one convexity, quasi convexity, non-negative polynomials.Received 20/06/2016, Accepted 20/06/2016.Research supported by National Funds through Fundacao para a Ciencia e a Tecnologia by

UID/MAT/04674/2013 - CIMA (L. Bandeira).Research supported under grant number MTM2013-47053-P of the MINECO - Spain (P. Pedregal).

27

28 LUIS BANDEIRA AND PABLO PEDREGAL

If we want to stress the dependence of the inequality (1.1) both on F and on u, we wouldwrite

Φ(F,u) ≡∫Q

[φ(F +∇u(x))− φ(F)] dx ≥ 0.

Put it in this form, we see that a certain functional depending on (F,u) should be non-negative. To clarify the dependence on the field u, we can write, taking advantage ofperiodicity,

u(x) =1

2π

∑n∈ZN

sin(2πn · x)an, an ∈ Rm,

∇u(x) =∑

n∈ZN

cos(2πn · x)an ⊗ n,

and so

Φ(F, an) =

∫Q

[φ

(F +

∑n∈ZN

cos(2πn · x)an ⊗ n

)− φ(F)

]dx.

If we further restrict the nature of φ, to be a polynomial of a certain degree, then Φ itself willbe a polynomial of the same degree on a certain number of variables (possibly infinite), thatmust be non-negative. We, hence, see that the issue of the non-negativeness of polynomialsmight have some relevance for quasi convexity.

As a matter of fact, the non-negativeness of polynomials is a very old subject but still quitealive. It is one main field of research in Real Algebraic Geometry with many applicationsin different areas within Mathematics and outside Mathematics. See for instance the nice,recent account [5]. Indeed, the issue of the non-negativeness of polynomials and rationalfunctions was the subject of Hilbert’s 17th problem ([4]). This problem was motivated byhis celebrated theorem on non-negative forms ([3]), and sum-of-squares criteria. Today, it isknown that the problem of deciding the non-negativeness of a multivariate polynomial (evenquartic) is a NP-hard problem ([9]), but there is an increasing body of knowledge about thisimportant problem in various contexts and circumstances ([5]).

Our main result, however, deals with rank-one convexity which is a necessary conditionfor quasi convexity. It is usually formulated by requiring that the sections

t 7→ φ(F + ta⊗ n)

be convex for arbitrary matrices F, and vectors a, n. If φ is smooth, rank-one convexitycan, equivalently, be formulated in the form of the so-called Legendre-Hadamard condition

∇2φ(F) : (a⊗ n)⊗ (a⊗ n) ≥ 0

again for arbitrary matrices F, and vectors a, n.Our main result can be formulated in the following terms. Consider φ : M2×2 → R, a

smooth (C2) function, and put

φ−(F) = supG−∇2φ(F) : G⊗G : det G = −1,

φ+(F) = infG∇2φ(F) : G⊗G : det G = 1.

Theorem 1. Such φ is rank-one convex if and only if

φ−(F) ≤ φ+(F)

NON-NEGATIVE POLYNOMIALS 29

for each matrix F.

By making use of Hilbert’s theorem about non-negative polynomials and sums of squares,which we briefly recall in Section 2, we are able to translate it into a test for rank-oneconvexity for smooth densities defined on M2×2. The connection between these two areasis explained in Section 3 through the investigation of quadratic forms. We then extend themain such fact to general, smooth densities (Section 4), and prove our Theorem 1 above.Finally, in Section 5, we focus on quasi convexity by assessing to what extent these ideascould lead somewhere.

2. Non-negative polynomials

The subject of non-negative polynomials has known a considerable expansion ever sincethe pioneering work of D. Hilbert ([3]). For theoretical as well as practical reasons, it isimportant to be able to decide when polynomials in several variables are non-negative. Wewill see one such situation in this contribution.

A real polynomial p(x) in n variables x = (x1, x2, . . . , xn) is said to be non-negative ifp(x) ≥ 0 for all x ∈ Rn. From a representation like

(2.1) p(x) =∑i

pi(x)2, each pi, a polynomial,

one can immediately conclude that p is indeed non-negative. Because of this reason, andlacking other criteria, the sum-of-squares test became the main focus of attention to decidethe non-negativity of polynomials. Hilbert ([3]) classified all situations in which this testis valid, i.e., those situations in which non-negativity of polynomials is equivalent to beingdecomposable as a sum of squares. To formulate such important result in more precise terms,we will talk about “forms” (like quadratic forms), as being the corresponding homogeneousrepresentation of any polynomial, by introducing an additional variable, and dividing allmonomials by a suitable power of such new variable, according to the simple rule

p(x) = xdn+1p(x/xn+1), x = (x, xn+1),

where d is the degree of p. Assume that p is a polynomial of degree d in n variables, withassociated form p. The result of Hilbert is:

Theorem 2 ([3]). Non-negative forms are the same as sums-of squares, in the followingthree cases:

(1) n = 1: polynomials of arbitrary degree in one variable;(2) d = 2: quadratic forms in any number of variables;(3) d = 4, n = 2: quartic forms in three variables, or quartic polynomials in two

variables.

In all other cases, there are non-negative forms which are not sums of squares.

Hilbert later, and motivated by his result in [3], proposed his 17th problem in the famouslist [4]:

Does every non-negative polynomial have a representation as a sum ofsquares of “rational” functions?

Equivalently, given p(x), he was asking about the existence of a polynomial q(x) so thatq(x)2p(x) is a sum of squares of polynomials. Artin proved in 1927 that this is so.


The recent development of this area is astonishing. It has become quite specialized. See[5], and references therein.

3. The fundamental fact

We pretend to find some interesting application of case (3) in Hilbert’s result, and relateit to rank-one convexity. We will restrict attention to 2× 2-matrices, and densities definedon them.

Consider a (constant) quadratic form associated with the symmetric 4× 4-matrix Q. Wewill use henceforth the identification

(3.1) F =

(F11 F12

F21 F22

)7→ F = (F11, F12, F21, F22),

so that Q is understood as a quadratic form acting on four-component vectors. Put

D =

0 0 0 10 0 −1 00 −1 0 01 0 0 0

,

and note that D : F⊗F = 2 detF if F is a 2×2-matrix understood through the identification(3.1). Given a 4 × 4-symmetric matrix Q, we refer to it as being rank-one convex if theassociated quadratic form

φ(F) = Q : F⊗ F

is a rank-one convex function. Our main lemma follows.

Lemma 1. The quadratic form Q is rank-one convex if and only if there is a number αsuch that Q = S + αD, and S is non-negative definite.

Though we use Hilbert’s theorem to prove this lemma, it was already shown in an ele-mentary way by P. Marcellini in [6].

Proof. The “if” part is immediate. Indeed, if Q = S + αD with S non-negative, then for arank-one matrix G

Q : G⊗G = S : G⊗G + αD : G⊗G = S : G⊗G + 2α detG = S : G⊗G ≥ 0.

The remarkable fact is the converse.Suppose Q is rank-one convex, i.e.

(3.2) Q : (x⊗ y)⊗ (x⊗ y) ≥ 0

for arbitrary vectors x = (x1, x2), y = (y1, y2). Condition (3.2) is a short-hand form of thetypical rank-one convex condition ∑

i,j,k,l=1,2

Qijklxiyjxkyl ≥ 0

under the identification (3.1). Because of homogeneity, we can equivalently put

Q : (x⊗ y)⊗ (x⊗ y) ≥ 0

for x = (x, 1), x = x1/x2, and likewise for y = (y, 1). In this way, this last inequality istelling us that the fourth-degree polynomial

P4(x, y) = Q : [(x, 1)⊗ (y, 1)]⊗ [(x, 1)⊗ (y, 1)]


in the two variables (x, y) is non-negative. We can also write

P4(x, y) = Q : X⊗X, X = (xy, x, y, 1).

By Hilbert’s theorem, we can find, at least, one representation of P4 as a sum of squares.But because D : X⊗X = 0, all possible representations of P4 are of the form

P4(x, y) = (Q− αD) : X⊗X

for α ∈ R. This can be checked more explicitly if we write

P4(x, y) =(xy x y 1

)Q11 Q12 Q13 Q14

Q21 Q22 Q23 Q24

Q31 Q32 Q33 Q34

Q41 Q42 Q43 Q44

xyxy1

,

perform the multiplication with some care, and take into account the symmetry of Q. Notehow the coefficient corresponding to the monomial xy is obtained through the entries Q14,Q23, Q32, Q41, but there is no further ambiguity or freedom. Hilbert’s theorem implies thenthat there should be at least one real number α, such that

(Q− αD) : X⊗X = (CX)⊗ (CX), (Q− αD) = CTC,

for a certain matrix C. Hence Q− αD is non-negative definite. This proves the claim.

The “if” part of this lemma holds for more general situations where dimensions of matricesare larger. The crucial strength, however, is the “only if” part which is only valid for 2× 2-matrices.

4. Some consequences and some examples

The following is a classical result for quadratic forms.

Theorem 3 ([2]). Let φ(F) = FTAF = A : F⊗ F be a quadratic form. Then

(1) φ is rank-one convex iff φ is quasi convex.(2) if one of the two dimensions is 2, then

φ, polyconvex ⇐⇒ φ, quasi convex ⇐⇒ φ, rank-one convex.

(3) if both dimensions are greater than 3, in general rank-one convexity does not implypolyconvexity.

Through Lemma 1, the second statement of this theorem admits some improvement inthe case in which both dimensions are 2.

Corollary 1. Every rank-one convex quadratic form on 2 × 2-matrices is the sum of aconvex quadratic form, and a multiple of the determinant (and so it is polyconvex).

This main fact can be used directly for non-quadratic, smooth functions.

Corollary 2. A smooth (C2) function φ : M2×2 → R is rank-one convex if and only ifthere is a scalar function α : M2×2 → R and a symmetric, non-negative definite matrixfield S : M2×2 →M4×4 such that

∇2φ(F) = S(F) + α(F)D.

We can further explore the condition for a matrix Q to enjoy the property that there issome real α so that Q− αD is non-negative definite.


Proposition 1. There is some real number α such that Q− αD is non-negative definite ifand only if

supG−Q : G⊗G : det G = −1 ∈ R ≤ inf

GQ : G⊗G : det G = 1 ∈ R.

Proof. The proof is easy if we use homogeneity in the condition

0 ≤ (Q− αD) : G⊗G = Q : G⊗G− 2α detG

for every matrix G.

The proof of Theorem 1 is the result of putting together Corollary 2, and this las propo-sition.

As an illustration, we reexamine two typical examples. They can be found in [2]. Thefirst one is

φ(F) = |F|4 − 2(detF)2.

It is elementary to find that

1

4∇2φ(F) : G⊗G = 2(F : G)2 + |F|2|G|2 − (DF : G)2 − 2 detF detG.

By the Cauchy-Schwarz classical inequality, we realize that

φ−(F) = supG−∇2φ(F) : G⊗G : det G = −1 ≤ −8 detF,

φ+(F) = infG∇2φ(F) : G⊗G : det G = 1 ≥ −8 detF,

and so φ is indeed rank-one convex. The second one is

(4.1) φ(F) = |F|4 − 4√3|F|2 detF.

It is also straightforward to check that

1

4∇2φ(F) = 2F⊗ F + |F|21− 2√

3detF1− 4√

3F⊗DF− 1√

3|F|2D,

where 1 stands for the identity matrix of size 4× 4.Corollary 2 enables us to change the last term to an arbitrary contribution of the form

α(F)D in order to produce a non-negative definite matrix

(4.2) 2F⊗ F + |F|21− 2√3

detF1− 4√3F⊗DF− 1√

3|F|2D + α(F)D.

If we set α(F) to the form (α− 4/√

3)|F|2, for α a constant, a few careful calculations yieldthat the eigenvalues of (4.2) are

λ =(

2± α

2

)|F|2 − 4√

3det(F)

λ = 4|F|2 − 4√

3 det(F)± 1

2

√√√√[(α+8√3

)2

+ 16

]|F|4 − 16

(α+

8√3

)|F|2 det(F).

For the choice α = −2/√

3, it turns out that these eigenvalues are non-negative: for the firstpair of eigenvalues, it is clear that the minimum is 0, and that it is attained for F = 0, whilefor the second pair, some elementary computations lead also to 0 as the minimum, attainedwhen det(F) = (

√3/4)|F|2.


5. Quasi convexity

Let φ : M2×2 → R be a density. As stated in the Introduction, the quasi convexitycondition amounts to having ∫

Q

φ(F +∇u(x)) dx ≥ φ(F)

for every matrix F, and every periodic mapping u : Q→ R2. Q is the unit cube in R2. Put

Φ(F,u) ≡∫Q

φ(F +∇u(x)) dx− φ(F).

Quasi convexity takes place if the functional Φ is always non-negative. Variable F doesnot require a particular analysis as it is a finite-dimensional variable, but u does. In fact,information on how φ behaves on sums of the form F + G, G = ∇u, might be helpful insaying something relevant.

To see this issue more clearly, let us review the quadratic case in which we take

(5.1) φ(F) = P (F) = (1/2)FTAF,

a quadratic form for a 4 × 4-, symmetric matrix A. F is identified with a four-componentvector through (3.1). We can write

P(G;F) ≡ P (F + G)− P (F) =1

2(FT + GT )A(F + G)− 1

2FTAF.

Because the variable G stands for the gradient ∇u(x), and a subsequence integration overthe unit cubeQ is to be performed, we immediately see that periodicity leads to the vanishingof the two integrals ∫

Q

∇u(x)TAF dx,∫Q

FTA∇u(x) dx.

Hence, ∫Q

P(∇u(x);F) dx =1

2

∫Q

∇u(x)TA∇u(x) dx.

If, as we did earlier,

u(x) =1

2π

∑n∈ZN

sin(2πn · x)an, an ∈ Rm,

∇u(x) =∑

n∈ZN

cos(2πn · x)an ⊗ n,(5.2)

we find∫Q

P(∇u(x);F) dx =1

2

∫Q

∑n,m∈ZN

∫Q

cos(2πn · x) cos(2πm · x) dx (an ⊗ n)TA(am ⊗m).

But the integrals ∫Q

cos(2πn · x) cos(2πm · x) dx


vanish always except when n = ±m, in which case the value is strictly positive. We concludethen that the quasi convexity condition∫

Q

P(∇u(x);F) dx ≥ 0

is equivalent to ∑n∈ZN

(an ⊗ n)TA(an ⊗ n) ≥ 0.

The arbitrariness of the full family of coefficients an implies that the non-negativeness ofthe sum can only occur when each term is non-negative

(a⊗ n)TA(a⊗ n) ≥ 0

for all a ∈ Rm, n ∈ RN . This is exactly the rank-one convex condition, and so, for aquadratic density as the one in (5.1), rank-one convexity is equivalent to quasi convexity.

We would like to explore how far this viewpoint might take us for a homogeneous, four-degree polynomial. More specifically, consider a fully symmetric, constant fourth-ordertensor T : (R2×2)2 → R, acting on matrices, and take P (X) = T(X,X,X,X) for X ∈ R2×2,a homogeneous fourth degree polynomial in the entries of X. Then, for

P(G;F) ≡ P (F + G)− P (F)

we can write

P(G;F) = T(F + G,F + G,F + G,F + G)−T(F,F,F,F),

that is

(5.3) P(G;F) = T(G,G,G,G) + 4T(G,G,G,F) + 6T(G,G,F,F) + 4T(G,F,F,F).

As indicated above, variable G stands for the gradient ∇u(x) of a smooth, Q-periodicmapping, and we are interested in examining the sign of the functional

Φ(F,u) ≡∫Q

P(∇u(x);F) dx.

Due to the periodic boundary conditions on u, the integral of the last term in (5.3) dropsout, and we are left with

Φ(F,u) =

∫Q

[T(∇u(x),∇u(x),∇u(x),∇u(x))(5.4)

4T(∇u(x),∇u(x),∇u(x),F) + 6T(∇u(x),∇u(x),F,F)] dx.

It is easy to write down necessary conditions for quasi convexity by simply selecting partic-ular groups of terms in (5.2).

(1) For three terms, we can take n3 = n1 ± n2, n1,n2, independent, ai ∈ Rm,

∇u(x) = cos(2πn1 · x)a1 ⊗ n1 + cos(2πn2 · x)a2 ⊗ n2

+ cos(2πn3 · x)a3 ⊗ n3,

and derive necessary conditions by taking this gradient to the quasi convexity in-equality.


(2) Similarly, for four terms, put n1 + n2 + n3 = n4, ai ∈ Rm, and take the gradient

∇u(x) = cos(2πn1 · x)a1 ⊗ n1 + cos(2πn2 · x)a2 ⊗ n2

+ cos(2πn3 · x)a3 ⊗ n3 + cos(2πn4 · x)a4 ⊗ n4

to the quasi convexity inequality.

In trying to say something interesting about sufficiency for quasi convexity, the crucialissue is whether there are basic families of gradients of the above form with a finite numberof terms that do not interact with each other through the corresponding trigonometricintegrals. Namely, if we put

∇u = ∇u1 +∇u2, ∇ui =∑n∈Zi

cos(2πn · x)an ⊗ n, i = 1, 2,Zi ⊂ Z2, even,

and take this decomposition to (5.4), we would have terms of three kinds, according to thethree terms for Φ(F,u). The first one is

T(∇u,∇u,F,F) =T(∇u1 +∇u2,∇u1 +∇u2,F,F)

=T(∇u1,∇u1,F,F) + T(∇u2,∇u2,F,F)

+ 2T(∇u1,∇u2,F,F).

According to the above trigonometric integrals, we would like to have that the terms in ∇u1

and ∇u2 cannot interact with each other. This is possible as long as∫Q

cos(2πn1 · x) cos(2πn2 · x) dx = 0

whenever n1 ∈ Z1, n2 ∈ Z2, and for this, it suffices to have Z1 ∩Z2 = ∅. Similarly, for cubicterms:

T(∇u,∇u,∇u,F) =T(∇u1 +∇u2,∇u1 +∇u2,∇u1 +∇u2,F)

=T(∇u1,∇u1,∇u1,F) + T(∇u2,∇u2,∇u2,F)

+ 3T(∇u1,∇u1,∇u2,F) + 3T(∇u1,∇u2,∇u2,F),

and so, we would like to have∫Q

cos(2πn1 · x) cos(2πn2 · x) cos(2πn3 · x) dx = 0

whenever n1,n2 ∈ Z1, n3 ∈ Z2, or n1,n2 ∈ Z2, n3 ∈ Z1. This condition forces to ensurethat n1,n2 ∈ Zi implies n1±n2 ∈ Zi, for each i = 1, 2, but then there is no way to separate agiven gradient into two disjoint sums of terms not interacting through the cubic terms. Thesituation is even worse for fourth-degree terms, so it seems as if there is no hope of isolatinga necessary and sufficient condition for quasi convexity for fourth-degree polynomials basedon the non-negativity of certain polynomials of a finite number of variables.

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[3] D. Hilbert, Uber die Darstellung Definiter Formen als Summe von Formenquadraten MathematischeAnnalen, 32 (1888), 342–250.


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Wiss. Gottingen Math. Phys. KL. (1900), 253–297; English transl., Bull. Amer. Math. Soc. 8 (1902),437–479; Bull. (New Series) Amer. Math. Soc. 37 (2000), 407–436.

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183–189.

[7] C. B. Morrey, Quasiconvexity and the lower semicontinuity of multiple integrals Pacific J. Math. 2(1952), 25–53.

[8] C. B. Morrey, Multiple Integrals in the Calculus of Variations, Springer 1966.

[9] J. Nie, Discriminants and nonnegative polynomials, J. Symb. Comp., 47 (2012), 167–191.

[10] V. Sverak, Rank-one convexity does not imply quasiconvexity Proc. Roy. Soc. Edinburgh Sect. 120 A

(1992), 293–300.

CIMA and Departamento de Matematica, Escola de Ciencias e Tecnologia, Universidade de

Evora, 7000-671 Evora, Portugal


Departamento de Matematicas, ETSI Industriales, Universidad de Castilla-La Mancha, 13071

Ciudad Real, SpainE-mail address: [email protected]

JEPE Vol 2, 2016, p. 37-58

EXISTENCE AND MULTIPLICITY FOR QUASI-CRITICAL FOURTH

ORDER QUASILINEAR PROBLEMS WITH GENERALIZED

VANISHING POTENTIALS

SARA BARILE

Abstract. We study the following fourth order quasilinear elliptic equation under Navier

boundary conditions in the whole space RN−∆(−∆u)

1p−1 + ρ(x)u

1p−1 = σ(x)f(u) in RN ,

u,∆u→ 0 as |x| → +∞,with weights ρ and σ which are allowed to be vanishing at infinity and in presence of quasi-critical power nonlinearities f . Existence and multiplicity results are proved by means of

Mountain Pass Theorem and its symmetric version thanks to some compact imbeddings

in σ-weighted Lebesgue spaces. These results apply to an equivalent nonlinear ellipticsystem of Lane-Emden type and in particular to a biharmonic equation under Navier

conditions in RN .

1. Introduction

In the last years we have treated the following class of fourth-order quasilinear elliptic pro-blems

−∆(−∆u)1p−1 + ρ(x)u

1p−1 = f(x, u) in Ω (resp. in RN ),

u = ∆u = 0 (resp. u,∆u→ 0) in ∂Ω (resp. as |x| → +∞),(1.1)

where Ω ⊂ RN with N ≥ 3, 1 < p ≤ 2, ρ : RN → R is a Lebesgue measurable weight andf : Ω×R→ R (resp. f : RN ×R→ R) satisfies suitable growth assumptions. For simplicity,we denote sα = sgn(s)|s|α the odd extension of the power function. Related to the growthbehavior of f we are interested in, we say that f has a ( p

p−1 − 1)-superlinear but subcritical

growth if the growth of f(x, u) is comparable with uq−1 with q > 1 and 1− 2N < 1

p + 1q < 1,

i.e. with pp−1 < q < Np

(N−2)p−N while f(x, u) has a ( pp−1 − 1)-sublinear growth if its growth

is comparable with uq−1 with q > 1 and 1p + 1

q > 1, i.e. q < pp−1 ; instead, f(x, u) possesses a

quasi-critical growth if it behaves like uq−1 with p and q belonging to the critical hyperbola

2010 Mathematics Subject Classification. 35J35; 35J50; 35J58; 35J62; 46E35.Key words and phrases. Fourth order quasilinear elliptic equations, nonlinear elliptic systems of Lane–

Emden type, vanishing potentials, variational tools, compact imbeddings.Received 31/08/2016, accepted 21/11/2016.

Research partially supported by Fondi di Ricerca di Ateneo 2014 titled “Metodi variazionali e topologicinello studio di fenomeni non lineari”.

37

38 SARA BARILE

1p + 1

q = 1 − 2N , that is q = Np

(N−2)p−N . The interest in this kind of problems arises mainly

from the need to solve nonlinear Lane-Emden elliptic systems of the type−∆u = vp−1 in Ω (resp. in RN ),

−∆v = −ρ(x)u1p−1 + f(x, u) in Ω (resp. in RN ),

u = ∆u = 0 (resp. u,∆u→ 0) in ∂Ω (resp. as |x| → +∞),

(1.2)

which are equivalent to (1.1). Indeed, when N ≥ 3 and 1 < p ≤ 2 it is well known (seee.g. [18]) that the standard functional associated to systems (1.2) is strongly indefinite ininterpolation spaces of infinite dimension. In order to overcome this problem, by arguing asin [16] (see also [17]) and by applying a decoupling technique i.e., by exploiting the “change

of variable” (−∆u)1p−1 = v which works since g(v) = vp−1 is an increasing function, it is

possible to rewrite systems (1.2) as the equivalent fourth order quasilinear elliptic problems(1.1). Clearly, if u is a weak solution of (1.1), we get a weak solution of systems (1.2)

which is the couple (u, (−∆u)1p−1 ). Existence, multiplicity and regularity results to (1.1)

(resp. (1.2)) have been stated in bounded domains Ω ⊂ RN when ρ(x) = 0 or in unboundeddomains when ρ(x) 6= 0 both in the ( p

p−1 − 1)-superlinear but subcritical case (see Barile

and Salvatore [3, 4, 5], Candela and Salvatore [14], dos Santos [21] and references therein)and in the ( p

p−1 − 1)-sublinear case (see Barile and Salvatore [6, 7, 8], Bonheure, dos Santos

and Ramos [13] and Felmer and Martinez [22] and cited papers within). In the critical case,because of the lack of compactness of the problem, non-existence of solutions has been statedin [26] and [28] by using Pohozaev type arguments.Besides the equivalence with classes of systems (1.2), problems like (1.1) are also intere-sting since, in the particular case p = 2, they reduce to biharmonic equations with Navierboundary conditions of the form

∆2u = −ρ(x)u+ f(x, u) in Ω (resp. in RN ),u = ∆u = 0 (resp. u,∆u→ 0) in ∂Ω (resp. as |x| → +∞),

which have been studied in the last years and in different cases by several authors like Alvesand do O [1], Bastos, Miyagaki and Vieira [9], Berchio and Gazzola [10], Bernıs, Garcia

Azorero and Peral [12], Chabrowski and do O [15], Demarque and Miyagaki [19], Dengand Shuai [20] and references within. Really, observe that when 1 < p ≤ 2 the operator

−∆(−∆u)1p−1 involved in (1.1) can be seen as an alternative generalization of the biharmonic

operator ∆2u with respect to the classical definition of the polyharmonic operator (−∆)mu(see Gazzola, Grunau and Sweers [23]).Specific aim of this paper is to study the following fourth order quasilinear elliptic problemunder Navier conditions in RN

−∆(−∆u)1p−1 + ρ(x)u

1p−1 = σ(x)f(u) in RN ,

u,∆u→ 0 as |x| → +∞,(1.3)

and therefore the equivalent nonlinear Lane-Emden elliptic problem−∆u = vp−1 in RN ,

−∆v = −ρ(x)u1p−1 + σ(x)f(u) in RN ,

u, v → 0 as |x| → +∞,

(1.4)

where N ≥ 5 and NN−2 < p ≤ 2, the continuous functions ρ, σ : RN → R belong to a general

class of weights such that decaying and vanishing ones are covered and f : R → R is a

QUASI-CRITICAL QUASILINEAR PROBLEMS WITH GENERALIZED VANISHING POTENTIALS 39

suitable quasi-critical power nonlinear term with critical exponent(p

p− 1

)∗∗=

Np

(N − 2)p−N∈ R

since p > NN−2 (see Section 2). The strenght of this investigation is that, to the best of

our knowledge, problem (1.3) (resp. (1.4)) has not been studied before under these generalassumptions on the vanishing functions ρ and σ and on the nonlinearities f . These classes ofpotentials and nonlinearities allow us to recover the lack of the compactness of the problem.Moreover, as we will specify later, we extend in part or complement the results establishedin the case p = 2 for the biharmonic problem

∆2u+ ρ(x)u = σ(x)f(u) in RN ,u,∆u→ 0 as |x| → +∞,(1.5)

in particular by Bastos, Miyagaki and Vieira [9] and references therein like Alves and do O[1] and Demarque and Miyagaki [19] and the ones stated by Deng and Shuai [20].Specifically, we assume the following hypotheses on the potentials ρ and σ:

(ρ1) ρ ∈ C(RN ,R) and ρ(x) > 0 for every x ∈ RN ;(σ1) σ ∈ C(RN ,R), σ(x) > 0 for every x ∈ RN and σ ∈ L∞(RN );(σ2) if Ωn ⊂ RN is a sequence of Borel sets such that meas(Ωn) ≤ R (where meas

denotes the Lebesgue measure) for all n ∈ N and some R > 0, then

limr→+∞

∫Ωn ∩ Bcr(0)

σ(x) = 0 uniformly with respect to n ∈ N.

Furthermore, one of the below conditions relating ρ and σ occurs:

(ρσ1)σ

ρ∈ L∞(RN )

or

(ρσ2) there exists m ∈(

pp−1 ,

(pp−1

)∗∗)such that

lim|x|→+∞

σ(x)

ρ(x)γ= 0 with γ =

( pp−1 )

∗∗−m

( pp−1 )

∗∗−p∈ (0, 1).

Regarding the nonlinear term f , we assume the following conditions in the origin and atinfinity:

(f1) f ∈ C(R,R) and if (ρσ1) holds, then

lim|s|→0+

f(s)

|s|1p−1

= 0

or

(f2) f ∈ C(R,R) and if (ρσ2) holds, then

lim|s|→0+

f(s)

|s|m−1= 0

with m ∈(

pp−1 ,

(pp−1

)∗∗)defined in (ρσ2);

40 SARA BARILE

(f3) f has a quasi-critical growth at infinity, namely,

lim|s|→+∞

f(s)

|s|(pp−1 )

∗∗−1= 0;

(f4) there exists µ ∈(

pp−1 ,

(pp−1

)∗∗)such that

0 < µF (s) = µ

∫ s

0

f(t)dt ≤ f(s)s for all s ∈ R \ 0;

(f5) f is odd with respect to s, i.e. f(−s) = −f(s) for every s ∈ R.

Remark 1. By making some minor technical changes throughout the paper, we can replacethe limit in assumptions (f1), (f2) and (f3) with the limit superior and in addition in (f3)we can require it is finite.

Remark 2. By assumptions (f1) and (f3), for any ε > 0 there exists a constant Cε > 0such that

(1.6) |f(s)| ≤ ε|s|1p−1 + Cε|s|(

pp−1 )

∗∗−1 for all s ∈ R

and by integration

(1.7) |F (s)| ≤ ε (p− 1)

p|s|

pp−1 +

Cε(pp−1

)∗∗ |s|( pp−1 )

∗∗

for all s ∈ R.

Similarly, by hypotheses (f2) and (f3), for any ε > 0 there exists a constant Cε > 0 suchthat

(1.8) |f(s)| ≤ ε|s|m−1 + Cε|s|(pp−1 )

∗∗−1 for all s ∈ R

and by integration

(1.9) |F (s)| ≤ ε 1

m|s|m +

Cε(pp−1

)∗∗ |s|( pp−1 )

∗∗

for all s ∈ R.

Remark 3. By hypothesis (f4), fixed any s0 > 0 we have that

(1.10) F (s) ≥ F (s0)

|s0|µ|s|µ for all s ∈ R such that |s| ≥ s0.

Setting Cs0,µ = F (s0)|s0|µ > 0, we can find constants C,C1 > 0 such that

(1.11) F (s) ≥ C |s|µ − C1 for all s ∈ R.

Therefore, since µ > pp−1 we deal with p

p−1 -superquadratic functions F .

At this point we can state our results. For the definition of the working space D2, pp−1 (RN )

(resp. D2,2(RN ) in the case p = 2) and the functional I associated to (1.3) we refer toSection 2.


Theorem 1. Let N ≥ 5 and NN−2 < p ≤ 2. Suppose that (ρ1), (σ1), (σ2), (ρσ1) (resp.

(ρσ2)) and (f1) (resp. (f2)), (f3) and (f4) hold. Then, I admits a non-trivial critical point

u ∈ D2, pp−1 (RN ) of Mountain Pass type therefore problem (1.3) admits a non-trivial weak

solution u ∈ D2, pp−1 (RN ).

Moreover, if f satisfies also (f5), then I has an unbounded sequence un of non-trivial

min-max type critical points un ∈ D2, pp−1 (RN ) thus problem (1.3) possesses an infinitely

many non-trivial weak solutions un ∈ D2, pp−1 (RN ).

Taking into account the equivalence of problem (1.3) with Lane-Emden type system (1.4),

once we have found a weak solution u ∈ D2, pp−1 (RN ) (resp. a sequence of weak solu-

tions un ⊂ D2, pp−1 (RN )) of (1.3), we get a weak solution (resp. a sequence of weak

solutions) of system (1.4) which is the couple (u, (−∆u)1p−1 ) (resp. the couples sequence

(un, (−∆un)1p−1 )). Therefore, as a direct consequence of Theorem 1 we can obtain the

following result which is also new within the framework of Lane-Emden type systems.

Theorem 2. Let N ≥ 5 and NN−2 < p ≤ 2. Suppose that (ρ1), (σ1), (σ2), (ρσ1) (resp.

(ρσ2)) and (f1) (resp. (f2)), (f3) and (f4) hold. Then, system (1.4) admits a non-trivial

weak solution (u, (−∆u)1p−1 ) with u ∈ D2, p

p−1 (RN ).

If f satisfies also (f5), problem (1.4) has an unbounded sequence (un, (−∆un)1p−1 ) then

infinitely many non-trivial weak solutions (un, (−∆un)1p−1 ) with un ∈ D2, p

p−1 (RN ).

Moreover, as observed above since in the case p = 2 problem (1.3) reduces to the biharmonicproblem (1.5), we emphasize that Theorem 1 covers the next result which holds for N ≥ 5within the framework of biharmonic equations with quasi-critical growth and generalizedvanishing potentials.

Corollary 1. Let N ≥ 5 and p = 2. Assume that (ρ1), (σ1), (σ2), (ρσ1) (resp. (ρσ2)) and(f1) (resp. (f2)), (f3) and (f4) hold (with p = 2). Then, problem (1.5) admits a non-trivialweak solution u ∈ D2,2(RN ).If in addition f satisfies (f5), problem (1.5) has an unbounded sequence un ⊂ D2,2(RN )of non-trivial weak solutions.

Note that we improve the existence of a solution in the paper by Bastos, Miyagaki and Vieira[9] since the authors establish it under a quasi-critical growth assumption on f in the originwhich is stronger with respect to our (f1) (resp. (f2)). Since they are specifically interestedin a ground state solution to (1.5), they work under a superquadraticity assumption on Fat infinity which is weaker than (f4) but under an additional monotonicity assumption onf . Moreover, we complete this paper with the multiplicity result. Consequently, we alsocomplement the results in Demarque and Miyagaki [19] for nonradial weights ρ and σ, in

Alves and do O [1] for more general weights and nonlinearities and some other referenceswithin [9]. Furthermore, we cover the existence result by Deng and Shuai [20] for µ = 1and P (x) = 0 and we establish in addition the multiplicity result. From these observa-tions, we deduce therefore that Theorem 1 allows us to extend these results to the operator

−∆(−∆u)1p−1 for exponents p such that N

N−2 < p ≤ 2 and dimensions N with N ≥ 5.

The paper is organized as follows: in Section 2 we introduce the variational formulationof the problem and we prove some useful compactness results. In Section 3 we show the

42 SARA BARILE

assumptions of Mountain Pass Theorem and its Symmetric version are satisfied thus con-cluding with the proof of Theorem 1.

2. Variational Tools

In order to prove that problem (1.3) has a variational structure, let us introduce the space

D2, pp−1 (RN ) = u ∈ L( p

p−1 )∗∗

(RN ) :

∫RN|∆u|

pp−1 dx <∞ → L( p

p−1 )∗∗

(RN )

where the critical exponent is defined as follows(p

p− 1

)∗∗=

Np

(N − 2)p−Nsince p > N

N−2 , i.e. 2 pp−1 < N.

This space is equipped with the norm

‖u‖D2,

pp−1

=

(∫RN|∆u|

pp−1 dx

) p−1p

.

In the following, we denote by | · |t the usual norm on the Lebesgue space Lt(RN ) with1 ≤ t ≤ +∞. As (ρ1) holds, we can consider the space

D2, pp−1

ρ (RN ) = u ∈ D2, pp−1 (RN ) :

∫RN

ρ(x)|u|pp−1 dx <∞

endowed with the norm

‖u‖D

2,pp−1

ρ

=

(∫RN|∆u|

pp−1 dx+

∫RN

ρ(x)|u|pp−1 dx

) p−1p

and we denote by

(D−2, p

p−1ρ (RN ), ‖ · ‖

D−2,

pp−1

ρ

)the normed dual space of D2, p

p−1ρ (RN ). From

now on, let 1 ≤ t <∞ and

Ltρ(RN ) = u ∈ Lt(RN ) :

∫RN

ρ(x)|u|t dx <∞


|u|t, ρ =(∫

RNρ(x)|u|t dx

) 1t

.

Clearly,

D2, pp−1

ρ (RN ) = D2, pp−1 (RN ) ∩ L

pp−1ρ (RN )

therefore it easily follows

(2.1) D2, pp−1

ρ (RN ) → D2, pp−1 (RN ) → L( p

p−1 )∗∗

(RN ) and D2, pp−1

ρ (RN ) → Lpp−1ρ (RN ).

Furthermore, for every w ∈ R, 1 ≤ w <∞, let us define the Lebesgue space

Lwσ (RN ) =

u ∈ Lw(RN ) :

∫RN

σ(x)|u|w dx < +∞


|u|w, σ =(∫

RNσ(x)|u|w dx

) 1w

.


Recall that a weak solution of problem (1.3) is a function u ∈ D2, pp−1

ρ (RN ) such that∫RN

(−∆u)1p−1 (−∆v) dx+

∫RN

ρ(x)|u|1p−1 v dx−

∫RN

σ(x)f(u)v dx = 0

for every v ∈ D2, pp−1

ρ (RN ).From now on, c, ci, C, Ci denote real positive constants changing line from line and

Cσ,ρ =

∣∣∣∣σρ∣∣∣∣∞

= ess sup

σ(x)

ρ(x): x ∈ RN

and Cσ = |σ|∞ = ess supσ(x) : x ∈ RN.

Now, let us recall the following classical result which will be useful in order to state thevariational formulation of problem (1.3).

Lemma 1. Let ρ : RN → R be a function satisfying (ρ1). Then, the Nemytskii operator h

associated to h(s) = s1p−1 is continuous from L

pp−1ρ (RN ) to Lpρ(RN ).

We can state now the following variational principle.

Proposition 1. Assume that (ρ1), (σ1), (ρσ1) (resp. (ρσ2)), (f1) (resp. (f2)) and (f3) hold.Then, the weak solutions of problem (1.3) are the critical points of the energy functional

defined on D2, pp−1

ρ (RN ) by

I(u) =p− 1

p

∫RN

(|∆u|

pp−1 + ρ(x)|u|

pp−1

)dx−

∫RN

σ(x)F (u) dx, u ∈ D2, pp−1

ρ (RN ).

More precisely, I ∈ C1(D2, pp−1

ρ (RN )) and its derivative dI : D2, pp−1

ρ (RN )→ D−2, pp−1

ρ (RN ) isdefined as

(2.2) dI(u)[v] =

∫RN

[(−∆u)

1p−1 (−∆v) + ρ(x)|u|

1p−1 v − σ(x)f(u)v

]dx

for all u, v ∈ D2, pp−1

ρ (RN ).

Proof. First, we prove that the functional

I(u) =p− 1

p‖u‖

pp−1

D2,

pp−1

ρ

−∫RN

σ(x)F (u) dx for any u ∈ D2, pp−1

ρ (RN )

is well defined and its Frechet derivative given in (2.2) is a continuous operator from

D2, pp−1

ρ (RN ) to D−2, pp−1

ρ (RN ). We define and study separately the two maps

Iρ(u) =p− 1

p‖u‖

pp−1

D2,

pp−1

ρ

, Iσ,F (u) =

∫RN

σ(x)F (u) dx for any u ∈ D2, pp−1

ρ (RN ).

Clearly, Iρ ∈ C1(D2, pp−1

ρ (RN )) since Iρ is continuous from D2, pp−1

ρ (RN ) to R and its Gateauxderivative at u

dIρ(u)[v] =

∫RN

(−∆u)1p−1 (−∆v) dx+

∫RN

ρ(x)u1p−1 v dx

is a linear continuous map with respect to every v ∈ D2, pp−1

ρ (RN ). Moreover, by adapting

the arguments in [3, Proof of Proposition 2.7], we prove that Iρ ∈ C1(D2, pp−1

ρ (RN )) since

(2.3) ‖dIρ(un)− dIρ(u)‖D

−2,pp−1

ρ

→ 0 if un → u in D2, pp−1

ρ (RN ).

44 SARA BARILE

Indeed, by Holder inequality and imbeddings in (2.1) we get for all v ∈ D2, pp−1

ρ (RN )

|(dIρ(un)− dIρ(u))[v]| ≤∫RN|(−∆un)

1p−1 − (−∆u)

1p−1 | |(−∆v)| dx

+

∫RN|ρ(x)(u

1p−1n − u

1p−1 )| |v| dx

≤ |(−∆un)1p−1 − (−∆u)

1p−1 |p |∆v| p

p−1

+|(ρ(x))1p (u

1p−1n − u

1p−1 )|p |(ρ(x))

p−1p v| p

p−1

≤ |(−∆un)1p−1 − (−∆u)

1p−1 |p ‖v‖

D2,

pp−1

ρ

+|(ρ(x))1p (u

1p−1n − u

1p−1 )|p ‖v‖

D2,

pp−1

ρ

.

Since −∆un → −∆u in Lpp−1 (RN ), for the continuity of the Nemytskii operator associated

to h(s) = s1p−1 from L

pp−1 (RN ) to Lp(RN ), we have that

|(−∆un)1p−1 − (−∆u)

1p−1 |p → 0.

Since un → u in Lpp−1ρ (RN ), by Lemma 1 we have that

|u1p−1n − u

1p−1 |p, ρ = |(ρ(x))

1p (u

1p−1n − u

1p−1 )|p → 0.

Hence, passing to the supremum with respect to any v ∈ D2, pp−1

ρ (RN ) with ‖v‖D

2,pp−1

ρ

≤ 1,

we obtain (2.3) holds.

Now, we have to prove that also Iσ,F ∈ C1(D2, pp−1

ρ (RN )) with

(2.4) dIσ,F (u)[v] =

∫RN

σ(x)f(u)v dx for all u, v ∈ D2, pp−1

ρ (RN ).

First suppose that (ρσ1) and (f1) hold. Then, by (σ1) and inequality (1.7) in Remark 2 it is

|Iσ,F (u)| ≤ Cσ,ρ ε(p− 1)

p

∫RN

ρ(x)|u|pp−1 dx+ Cσ Cε

1(pp−1

)∗∗ ∫RN|u|(

pp−1 )

∗∗

dx

= Cσ,ρ ε(p− 1)

p|u|

p(p−1)p

(p−1), ρ

+ Cσ Cε1(pp−1

)∗∗ |u|( pp−1 )

∗∗

( pp−1 )

∗∗

so by (2.1) we get Iσ,F (u) ∈ R for every u ∈ D2, pp−1

ρ (RN ). Similarly, by (σ1) and (1.6) inRemark 2 we get

|dIσ,F (u)[v]| ≤ Cσ,ρ ε∫RN

ρ(x)|u|1p−1 |v| dx+ Cε Cσ

∫RN|u|(

pp−1 )

∗∗−1|v| dx.

As concerns as the first integral, let us observe that by Holder inequality it is∫RN

ρ(x)|u|1p−1 |v| dx =

∫RN

(ρ(x)

1p |u|

1p−1

) (ρ(x)

p−1p |v|

)dx(2.5)

≤ |u|1p−1pp−1 , ρ

|v| pp−1 , ρ


while for the second integral we get∫RN|u|(

pp−1 )

∗∗−1|v| dx(2.6)

≤(∫

RN|u|(

pp−1 )

∗∗

dx

) ( pp−1 )

∗∗−1

( pp−1 )

∗∗ (∫RN|v|(

pp−1 )

∗∗

dx

) 1

( pp−1 )

∗∗

= |u|(pp−1 )

∗∗−1

( pp−1 )

∗∗ |v|( pp−1 )

∗∗ .

Consequently,

|dIσ,F (u)[v]| ≤ Cσ,ρ ε |u|1p−1pp−1 , ρ

|v| pp−1 , ρ

+ Cε Cσ |u|( pp−1 )

∗∗−1

( pp−1 )

∗∗ |v|( pp−1 )

∗∗

and by (2.1) it follows that dIσ,F (u)[v] ∈ R for all u, v ∈ D2, pp−1

ρ (RN ).Now, suppose that (ρσ2) and (f2) hold. Then, by (σ1) and inequality (1.9) in Remark 2,choosen a radius r > 0 we get

(2.7) |Iσ,F (u)| ≤ ε

m

∫RN

σ(x)|u|m dx+Cε(pp−1

)∗∗ ∫RN

σ(x)|u|(pp−1 )

∗∗

dx

≤ ε

m

(Cσ

∫Br(0)

|u|m dx+

∫Bcr(0)

σ(x)|u|m dx

)+

Cε(pp−1

)∗∗Cσ|u|( pp−1 )

∗∗

( pp−1 )

∗∗ .

Since m ∈(

pp−1 ,

(pp−1

)∗∗)it is L( p

p−1 )∗∗

(Br(0)) → Lm(Br(0)) from which it follows∫Br(0)

|u|m dx ≤ c

(∫Br(0)

|u|(pp−1 )

∗∗

dx

) m

( pp−1 )

∗∗

(2.8)

≤ c

(∫RN|u|(

pp−1 )

∗∗

dx

) m

( pp−1 )

∗∗

= c |u|m( pp−1 )

∗∗ .

Now, in order to estimate the integral on Bcr(0) in (2.7) let us define for any fixed x ∈ RNthe function

g(s) = ρ(x)spp−1−m + s(

pp−1 )

∗∗−m for every s > 0

whose minimum value is Cmρ(x)γ with γ defined in (ρσ2) and

Cm =

(( pp−1

)∗∗− p

p−1(pp−1

)∗∗−m

)(m− p

p−1(pp−1

)∗∗− p

p−1

) pp−1

−m

( pp−1 )

∗∗− pp−1 .

Hence,

Cm ρ(x)γ ≤ ρ(x) spp−1−m + s(

pp−1 )

∗∗−m for every x ∈ RN and s > 0.

Then, in combination with (ρσ2), in correspondence of any ε > 0 we can find a positiveradius r > 0 sufficiently large such that

σ(x)|s|m ≤ ε ρ(x)γ |s|m ≤ εC−1m

(ρ(x)|s|

pp−1 + |s|(

pp−1 )

∗∗)(2.9)

for every s ∈ R and |x| ≥ r.

46 SARA BARILE

In particular, for every u ∈ D2, pp−1

ρ (RN ) we get

(2.10)

∫Bcr(0)

σ(x)|u|m dx ≤ εC−1m

(|u|

pp−1pp−1 , ρ

+ |u|(pp−1 )

∗∗

( pp−1 )

∗∗

).

Therefore, by substituting (2.8) and (2.10) in (2.7) we get

|Iσ,F (u)| ≤ ε

m

(Cσ c |u|m( p

p−1 )∗∗ + εC−1

m

(|u|

pp−1pp−1 ,ρ

+ |u|(pp−1 )

∗∗

( pp−1 )

∗∗

))(2.11)

+Cε Cσ(pp−1

)∗∗ |u|( pp−1 )

∗∗

( pp−1 )

∗∗ .

and by (2.1) we obtain Iσ,F (u) is well defined for every u ∈ D2, pp−1

ρ (RN ).Reasoning in a similar way, we are going also to prove that dIσ,F (u)[v] is well posed for

every u, v ∈ D2, pp−1

ρ (RN ). Indeed, by (1.8) in Remark 2, (2.6) and chosen a radius r > 0 weget

(2.12) |dIσ,F (u)[v]| ≤ ε∫RN

σ(x)|u|m−1|v| dx+ Cε Cσ

∫RN|u|(

pp−1 )

∗∗−1|v| dx

≤ ε

(Cσ

∫Br(0)

|u|m−1|v| dx+

∫Bcr(0)

σ(x)|u|m−1|v| dx

)

+ Cε Cσ |u|( pp−1 )

∗∗−1

( pp−1 )

∗∗ |v|( pp−1 )

∗∗ .

Now, by Holder inequality and since m <(

pp−1

)∗∗implies

L( pp−1 )

∗∗

(Br(0)) → L

(m−1)( pp−1 )

∗∗

( pp−1 )

∗∗−1 (Br(0)),

we get ∫Br(0)

|u|m−1|v| dx(2.13)

≤

∫Br(0)

|u|(m−1)( p

p−1 )∗∗

( pp−1 )

∗∗−1 dx

( pp−1 )

∗∗−1

( pp−1 )

∗∗ (∫Br(0)

|v|(pp−1 )

∗∗

dx

) 1

( pp−1 )

∗∗

≤ c

(∫Br(0)

|u|(pp−1 )

∗∗

dx

) m−1

( pp−1 )

∗∗

|v|( pp−1 )

∗∗

≤ c |u|m−1

( pp−1 )

∗∗ |v|( pp−1 )

∗∗ .

On the other hand, by (2.9) we have

(2.14) σ(x) ≤ ε(ρ(x)|s|

pp−1−m + |s|(

pp−1 )

∗∗−m)

for every s ∈ R \ 0 and |x| ≥ r

so that

σ(x)|s|m−1 ≤ ε(ρ(x)|s|

1p−1 + |s|(

pp−1 )

∗∗−1)

for every s ∈ R and |x| ≥ r.


Therefore, by exploiting (2.5) and (2.6) we have∫Bcr(0)

σ(x)|u|m−1|v| dx(2.15)

≤ ε

(∫Bcr(0)

ρ(x)|u|1p−1 |v| dx+

∫Bcr(0)

|u|(pp−1 )

∗∗−1|v| dx

)

≤ ε(|u|

1p−1pp−1 , ρ

|v| pp−1 , ρ

+ |u|(pp−1 )

∗∗−1

( pp−1 )

∗∗ |v|( pp−1 )

∗∗

).

Consequently, by substituting (2.13) and (2.15) in (2.12) we conclude

|dIσ,F (u)[v]| ≤ ε(Cσ c |u|m−1

( pp−1 )

∗∗ |v|( pp−1 )

∗∗

+ε

(|u|

1p−1pp−1 , ρ

|v| pp−1 , ρ

+ |u|(pp−1 )

∗∗−1

( pp−1 )

∗∗ |v|( pp−1 )

∗∗

))+ Cε Cσ |u|

( pp−1 )

∗∗−1

( pp−1 )

∗∗ |v|( pp−1 )

∗∗

and by (2.1) we get dIσ,F (u)[v] is well defined for every u, v ∈ D2, pp−1

ρ (RN ).Moreover, standard arguments imply that the Gateaux derivative of Iσ,F at u is as in (2.4)

and it is linear and continuous from D2, pp−1

ρ (RN ) to R.

As concerns as the continuity of dIσ,F from D2, pp−1


ρ (RN ), i.e.

‖dϕ1(un)− dϕ1(u)‖D

2,pp−1

ρ

→ 0 if un → u in D2, pp−1

ρ ,

we can refer to and exploit Proposition 3 where the compactness of dϕ1 = dIσ,F from

D2, pp−1


ρ (RN ) is stated.

As ensured by the following result, the presence of the interacting weights ρ(x) and σ(x)

allows us to overcome the lack of compact imbeddings of the space D2, pp−1

ρ (RN ) into the

Lebesgue spaces Lt(RN ) with t ∈(

pp−1 ,

(pp−1

)∗∗). Here below we provide all the details

of the proof since we extend to D2, pp−1

ρ (RN ) the arguments employed in Bastos, Miyagakiand Vieira [9] for the space D2,2

ρ (RN ). These arguments follow the line of the compactness

results proved in Opic and Kufner [27] for D1,2ρ (RN ) in presence of vanishing potentials ρ(x)

and σ(x) which are included in our setting assumptions.

Proposition 2. Suppose that (ρ1), (σ1) and (σ2) are satisfied.

(i) If (ρσ1) holds, then

D2, pp−1

ρ (RN ) is compactly embedded in Lwσ (RN ) for any w ∈(

pp−1 ,

(pp−1

)∗∗);

(ii) if (ρσ2) holds, then

D2, pp−1

ρ (RN ) is compactly embedded in Lmσ (RN )

with m ∈(

pp−1 ,

(pp−1

)∗∗)defined in (ρσ2).

48 SARA BARILE

Proof. First, assume that (ρσ1) holds. Fixed ε > 0, since w > pp−1 there exists s0 > 0 such

that

|s|w ≤ ε |s|pp−1 for all s ∈ R, |s| ≤ s0.

Then, by (ρσ1) it is

σ(x)|s|w ≤ Cσ,ρ ρ(x) |s|w ≤ εCσ,ρ |s|pp−1 for all s ∈ R, |s| ≤ s0 and x ∈ RN .

In correspondence of ε > 0, from σ ∈ L∞(RN ) and r <(

pp−1

)∗∗we can find s1 > 0 such

that

σ(x)|s|w ≤ εCσ |s|(pp−1 )

∗∗

for all s ∈ R, |s| ≥ s1 and x ∈ RN .Since of continuity of the functions involved, there exists a constant c > 0 such that

σ(x)|s|w ≤ c σ(x)χ[s0,s1](|s|) |s|(pp−1 )

∗∗

for all s ∈ R, s0 ≤ |s| ≤ s1 and x ∈ RN

where c = maxs0≤|s|≤s1|s|w

|s|(pp−1 )

∗∗ and χ[s0,s1] denotes the characteristic function in the

interval [s0, s1].

Consequently, for any w ∈(

pp−1 ,

(pp−1

)∗∗)and ε > 0, there exist 0 < s0 < s1 and a suitable

constant C > 0 such that

σ(x) |s|w ≤ εC (ρ(x)|s|pp−1 + |s|(

pp−1 )

∗∗

) + Cσ(x)χ[s0,s1](|s|) |s|(pp−1 )

∗∗

for all s ∈ R and for every x ∈ RN .

For every u ∈ D2, pp−1

ρ (RN ) denote by

Q(u) = C

(∫RN

ρ(x)|u|pp−1 dx+

∫RN|u|(

pp−1 )

∗∗

dx

)and Ω =

x ∈ RN : s0 ≤ |u(x)| ≤ s1

. Then, for any given radius r1 > 0 we get∫

Bcr1(0)

σ(x)|u|w dx ≤ εQ(u) + C

∫Bcr1

(0)

σ(x)χ[s0,s1](|u|) |u|(pp−1 )

∗∗

dx(2.16)

= εQ(u) + C

∫Bcr1

(0)∩Ω

σ(x) |u|(pp−1 )

∗∗

dx

≤ εQ(u) + C s( pp−1 )

∗∗

1

∫Bcr1

(0)∩Ω

σ(x) dx.

Now, if un is a sequence such that un u in D2, pp−1

ρ (RN ), by (2.1) there exists a constantC1 > 0 such that

(2.17)

∫RN

ρ(x)|un|pp−1 dx ≤ C1 and

∫RN|un|(

pp−1 )

∗∗

dx ≤ C1 for all n ∈ N.

Therefore, Q(un) is bounded from above in R by 2C1, i.e.

(2.18) Q(un) ≤ 2C1 for all n ∈ N

and, denoted by Ωn = x ∈ RN : s0 ≤ |un(x)| ≤ s1, it follows that

s( pp−1 )

∗∗

0 meas(Ωn) ≤∫

Ωn

|un|(pp−1 )

∗∗

dx ≤ C1 for all n ∈ N


from which we get supn∈N|meas(Ωn)| < +∞. Consequently, from (σ2) in correspondence of

ε

C s( pp−1 )

∗∗

1

> 0 there exists a radius r2 = r2(ε) > 0 such that

(2.19)

∫Ωn ∩Bcr2 (0)

σ(x) dx <ε

s( pp−1 )

∗∗

1

for all n ∈ N.

By exploiting both the boundedness of Q(un) stated in (2.18) and (2.19) in (2.16) withu = un, choosen a suitable radius r > 0 (e.g. r = maxr1, r2) it follows∫

Bcr(0)

σ(x)|un|w dx ≤ ε(2C1 + 1) for all n ∈ N(2.20)

and by Fatou’ Lemma we get also∫Bcr(0)

σ(x)|u|w dx ≤ ε(2C1 + 1) for all n ∈ N.(2.21)

Now, since by (2.1) it is u ∈ D2, pp−1 (RN ) and D2, p

p−1 (RN ) can be also endowed with thenorm

|D2u| pp−1

=

(∫RN|D2u|

pp−1 dx

) p−1p

(see [24, page164]), we easily get D2u ∈ Lpp−1 (Br(0)). Then, by [25, Corollary, page 3] it

is Dαu ∈ Lpp−1 (Br(0)) for |α| = 0, 1 then u ∈ W 2, p

p−1 (Br(0)). By compact embeddings inbounded domains it is in particular

W 2, pp−1 (Br(0)) →→ Lw(Br(0)) for any w ∈

(pp−1 ,

(pp−1

)∗∗).

Therefore, by (σ1) it follows

(2.22) limn→+∞

∫Br(0)

σ(x)|un|w dx =

∫Br(0)

σ(x)|u|w dx

for any w ∈(

pp−1 ,

(pp−1

)∗∗). Then, from (2.20) and (2.21) for ε > 0 small enough and

(2.22) we deduce

limn→+∞

∫RN

σ(x)|un|w dx =

∫RN

σ(x)|u|w dx

from which we conclude that

un → u in Lwσ (RN ), for every w ∈(

pp−1 ,

(pp−1

)∗∗).

If (ρσ2) holds, by (2.9) we get for any ε > 0 the existence of a radius r > 0 sufficiently largesuch that∫

Bcr(0)

σ(x)|u|m dx ≤ εC−1m

(∫Bcr(0)

ρ(x)|u|pp−1 dx+

∫Bcr(0)

|u|(pp−1 )

∗∗

dx

)

for all u ∈ D2, pp−1

ρ (RN ). If un is a sequence such that un u in D2, pp−1

ρ (RN ) there existsC1 > 0 such that inequalities in (2.17) hold therefore∫

Bcr(0)

σ(x)|un|m dx ≤ 2 εC1 C−1m for all n ∈ N(2.23)

50 SARA BARILE

and by Fatou’ Lemma it is∫Bcr(0)

σ(x)|un|m dx ≤ 2 εC1 C−1m for all n ∈ N.(2.24)

Reasoning similarly as above in order to get (2.22), since m ∈(

pp−1 ,

(pp−1

)∗∗)we obtain

(2.25) limn→+∞

∫Br(0)

σ(x)|un|m dx =

∫Br(0)

σ(x)|u|m dx.

Then, from (2.23) and (2.24) for ε > 0 small enough and (2.25) it follows

limn→+∞

∫RN

σ(x)|un|m dx =

∫RN

σ(x)|u|m dx

and we obtain

un → u in Lmσ (RN ).

Thanks to Proposition 2, we can prove the following compactness result related to dIσ,F .

Proposition 3. Suppose that (ρ1), (σ1), (σ2), (ρσ1) (resp. (ρσ2)) and (f1) (resp. (f2))

and (f3) hold. Then, if un is a sequence such that un u in D2, pp−1

ρ (RN ), we get

limn→+∞

∫RN

σ(x)f(un)v dx =

∫RN

σ(x)f(u)v dx for every v ∈ D2, pp−1

ρ (RN );

consequently,

dIσ,F is compact from D2, pp−1

ρ (RN ) in D−2, pp−1

ρ (RN ).

Proof. Let un be a sequence such that un u in D2, pp−1

ρ (RN ). Fixed an arbitrary radius

r > 0 we aim to prove that for any v ∈ D2, pp−1

ρ (RN )

(2.26) limn→+∞

∫Br(0)

σ(x)f(un)v dx =

∫Br(0)

σ(x)f(u)v dx

and

(2.27) limn→+∞

∫Bcr(0)

σ(x)f(un)v dx =

∫Bcr(0)

σ(x)f(u)v dx.

Let’s start by showing (2.26). Denoted by Θn = x ∈ RN : |un(x)| ≤ 1 (resp. Θ = x ∈RN : |u(x)| ≤ 1) and χΘn (resp. χΘ) its characteristic function, we first prove that

(2.28) limn→+∞

∫Br(0)

σ(x)f(un)χΘnv dx =

∫Br(0)

σ(x)f(u)χΘv dx.

Suppose (ρσ1) holds and f satisfies (f1). Since σ(x)p−1p ≤ C

p−1p

σ,ρ ρ(x)p−1p , we get easily

σ(x)p−1p v ∈ L

pp−1 (Br(0)) and by the following relation∣∣∣σ(x)

1p f(un(x))χΘn

∣∣∣p ≤ Cσ,ρ C ρ(x)|un(x)|pp−1


we deduce the sequence σ(x)1p f(un(x))χΘn is bounded in Lp(Br(0)). Then, since∫

Br(0)


∫Br(0)

(σ(x)

1p f(un)χΘn

)(σ(x)

p−1p v)dx,

by pointwise convergence we get (2.28). Now, assume (ρσ2) holds and f satisfies (f2). By

Proposition 2 (ii) we have σ(x)1m v ∈ Lm(Br(0)) for any v ∈ D2, p

p−1ρ (Br(0)) and the sequence

σ(x)m−1m f(un(x))χΘn is bounded in L

mm−1 (Br(0)) since∣∣∣σ(x)

m−1m f(un(x))χΘn

∣∣∣ mm−1 ≤ C σ(x) |un(x)|m.

Then, pointwise convergence together with∫Br(0)


∫Br(0)

(σ(x)

m−1m f(un)χΘn

)(σ(x)

1m v)dx

allow us to show (2.28) also in this second case. Finally, it remains to prove that

(2.29) limn→+∞

∫Br(0)

σ(x)f(un)χΘcnv dx =

∫Br(0)

σ(x)f(u)χΘc v dx

where Θcn (resp. Θc) denotes the complement of Θn (resp. Θ) and χΘcn

(resp. χΘcn) its

characteristic function. In both cases, by (f3) the sequence σ(x)f(un(x))χΘcn is bounded

in L

( pp−1 )

∗∗

( pp−1 )

∗∗−1 (Br(0)) since

∣∣σ(x)f(un(x))χΘcn

∣∣ ( pp−1 )

∗∗

( pp−1 )

∗∗−1 ≤ C

( pp−1 )

∗∗

( pp−1 )

∗∗−1

σ C1|un(x)|(pp−1 )

∗∗

,

therefore (2.29) follows by pointwise convergence and since v ∈ D2, pp−1

ρ (Br(0)). By (2.28)and (2.29) we conclude (2.26) is satisfied.At this point we prove (2.27) by distinguishing here also two cases. In the first case, assume

that (ρσ1) holds. From (f1) and (f3), fixing w ∈(

pp−1 ,

(pp−1

)∗∗)and taking ε > 0, there

exists C > 0 such that

σ(x)|f(s)| ≤ εC(ρ(x)|s|

1p−1 + |s|(

pp−1 )

∗∗−1)

+ C σ(x)|s|w−1(2.30)

for all s ∈ R and x ∈ RN .Indeed, given ε > 0, from (f1) and (ρσ1) there exists s0 > 0 such that

σ(x)|f(s)| ≤ εCσ,ρ ρ(x)|s|1p−1 for all s ∈ R, |s| ≤ s0 and x ∈ RN .

Moreover, from (σ1) and by (f3) we can find s1 > 0 such that

σ(x)|f(s)| ≤ εCσ |s|(pp−1 )

∗∗−1 for all s ∈ R, |s| ≥ s1 and x ∈ RN .Since the functions involved are continuous we get the existence of a constant c > 0 suchthat

σ(x)|f(s)| ≤ C σ(x) |s|w−1 for all s ∈ R, s0 ≤ |s| ≤ s1 and x ∈ RN .

Therefore, for any fixed w ∈(

pp−1 ,

(pp−1

)∗∗), in correspondence of ε > 0 there exists a

constant C > 0 such that (2.30) holds.

Now, since un is a sequence such that un u in D2, pp−1

ρ (RN ), for any radius r > 0 we get

52 SARA BARILE

by Holder inequality, continuous embeddings in (2.1), Proposition 2 (i), (2.5) and (2.6) thefollowing∫

Bcr(0)

σ(x)|f(un)| |v| dx ≤ εC

(∫Bcr(0)

ρ(x)|un|1p−1 |v| dx+

∫Bcr(0)

|un|(pp−1 )

∗∗−1|v| dx

)

+C

∫Bcr(0)

σ(x)|un|w−1|v| dx

≤ εC

(|un|

1p−1pp−1 , ρ

|v| pp−1 , ρ

+ |un|( pp−1 )

∗∗−1

( pp−1 )

∗∗ |v|( pp−1 )

∗∗

)+C

∫Bcr(0)

(σ(x))w−1w |un|w−1 (σ(x))

1w |v| dx

≤ εC1

(|un|

1p−1pp−1 , ρ

+ |u|(pp−1 )

∗∗−1

( pp−1 )

∗∗

)‖v‖D

2,pp−1

ρ

+C1

(∫Bcr(0)

σ(x)|un|w dx

)w−1w

‖v‖D

2,pp−1

ρ

for any v ∈ D2, pp−1

ρ (RN ). Denoted by

R(un) = C1

(|un|

1p−1pp−1 , ρ

+ |u|(pp−1 )

∗∗−1

( pp−1 )

∗∗

),

it is not difficult to observe that by (2.1) we get R(un) is bounded from above by a positiveconstant C2 for all n ∈ N while since Proposition 2 (i) implies

limn→+∞

∫RN

σ(x)|un|w dx =

∫RN

σ(x)|u|w dx,

there exists a radius r1 > 0 such that(∫Bcr1

(0)

σ(x)|un|w dx

)w−1w

≤ ε for all n ∈ N.

Then, ∫Bcr1

(0)

σ(x)|f(un)| |v| dx ≤ εC1 (C2 + 1) ‖v‖D

2,pp−1

ρ

for all n ∈ N

and choosen a radius r2 ≥ r1 > by Fatou’s Lemma it is also∫Bcr2 (0)

σ(x)|f(u)| |v| dx ≤ εC1 (C2 + 1) ‖v‖D

2,pp−1

ρ

for all n ∈ N.

Consequently, for a suitable choise of the radius r > 0 we conclude (2.27) holds.Now we prove the second case; indeed, we assume (ρσ2) holds. From (2.14), given ε > 0there exists r > 0 such that

σ(x)|f(s)| ≤ εC(ρ(x)|f(s)||s|

pp−1−m + |f(s)||s|(

pp−1 )

∗∗−m)

for all s ∈ R and |x| ≥ r.

By (f2), for any ε > 0 there exists s0 > 0 such that

|f(s)| ≤ ε|s|m−1 for all s ∈ R, |s| ≤ s0


then,

σ(x)|f(s)| ≤ εC(ρ(x)|s|

1p−1 + |s|(

pp−1 )

∗∗−1)

for all s ∈ R, |s| ≤ s0 and |x| ≥ r.

Moreover, from (f3) and (σ1) for any given ε > 0 there exists s1 > s0 > 0 such that

σ(x)|f(s)|ρ(x)|s|

1p−1 + |s|(

pp−1 )

∗∗−1≤ Cσ

|f(s)||s|(

pp−1 )

∗∗−1≤ εCσ for all s ∈ R, |s| ≥ s1

which implies

σ(x)|f(s)| ≤ εC(ρ(x)|s|

1p−1 + |s|(

pp−1 )

∗∗−1)

for all s ∈ R, |s| ≥ s1 and x ∈ RN .

Therefore, there exist C > 0 and s1 > s0 > 0 satisfying

(2.31) σ(x)|f(s)| ≤ εC(ρ(x)|s|

1p−1 + |s|(

pp−1 )

∗∗−1)

for every s ∈ Ωc and |x| ≥ r

where Ωc = s ∈ R : |s| < s0 or |s| > s1.In correspondence of the radius r > 0 by (2.31) and (1.8) in Remark 2 we can write∫

Bcr(0)

σ(x)|f(un)||v| dx =

∫Bcr(0)∩Ωcn

σ(x)|f(un)||v| dx+

∫Bcr(0)∩Ωn

σ(x)|f(un)||v| dx

≤ εC∫Bcr(0)∩Ωcn

(ρ(x)|un|

1p−1 + |un|(

pp−1 )

∗∗−1)|v| dx

+ ε

∫Bcr(0)∩Ωn

σ(x)|un|m−1|v| dx+ Cε

∫Bcr(0)∩Ωn

σ(x)|un|(pp−1 )

∗∗−1|v| dx

where we denote by Ωn =x ∈ RN : s0 ≤ |un(x)| < s1

and its complement by Ωcn =

x ∈ RN : |un(x)| < s0 or |un(x)| > s1

. So, reasoning similarly as in the previous step,

by Holder inequality and the definition of R(un) we obtain∫Bcr(0)

σ(x)|f(un)||v| dx ≤ εC R(un) ‖v‖D

2,pp−1

ρ

+ ε sm−11 c

(∫Bcr(0)∩Ωn

σ(x) dx

)m−1m

‖v‖D

2,pp−1

ρ

+ Cε Cσ |un|( pp−1 )

∗∗−1

( pp−1 )

∗∗ |v|( pp−1 )

∗∗

and in particular∫Bcr(0)

σ(x)|f(un)||v| dx ≤ (εC + Cε Cσ c)R(un) ‖v‖D

2,pp−1

ρ

+ ε sm−11 c

(∫Bcr(0)∩Ωn

σ(x) dx

)m−1m

‖v‖D

2,pp−1

ρ

.

By the boundedness of R(un) for all n ∈ N and the fact that supn∈N|meas(Ωn)| < +∞ which

allows us to apply (σ2), we can exploit some arguments as in the proof of Proposition 2 thusobtaining (2.27).

54 SARA BARILE

3. Proof of Theorem 1

At this point we prove that I satisfies the Mountain Pass geometry (see [2]).

Lemma 2. Under assumptions (ρ1), (σ1), (ρσ1) (resp. (ρσ2)), (f1) (resp. (f2)), (f3) and(f4), the functional I has a mountain pass geometry, that is

(I0) I(0) = 0;

(Iδ0) there exist ρ0, δ0 > 0 such that I(u) ≥ δ0 for all u ∈ D2, pp−1

ρ (RN ) with ‖u‖D

2,pp−1

ρ

=

ρ0;

(Iu0) there exists u0 ∈ D

2, pp−1

ρ (RN ) such that ‖u0‖D

2,pp−1

ρ

> ρ0 and I(u0) ≤ 0.

Proof. (I0) By (1.7) (resp. (1.9)) in Remark 2 we get easily I(0) = 0.(Iδ0) First suppose that (ρσ1) holds. Then, by (σ1), (1.7) in Remark 2 and (2.1) in corre-spondence of ε > 0 there exists Cε > 0 such that

I(u) =p− 1

p‖u‖

pp−1

D2,

pp−1

ρ

−∫RN

σ(x)F (u) dx

≥ p− 1

p‖u‖

pp−1

D2,

pp−1

ρ

− Cσ,ρ ε(p− 1)

p|u|

p(p−1)p

(p−1),ρ− Cσ Cε

1(pp−1

)∗∗ |u|( pp−1 )

∗∗

( pp−1 )

∗∗

≥ p− 1

p(1− εCσ,ρ c) ‖u‖

pp−1

D2,

pp−1

ρ

− Cσ Cε1(pp−1

)∗∗ c1 ‖u‖( pp−1 )

∗∗

D2,

pp−1

ρ

for every u ∈ D2, pp−1

ρ (RN ). Since pp−1 <

(pp−1

)∗∗, taken ‖u‖

D2,

pp−1

ρ

= ρ with ρ and ε > 0

small enough, we get (Iδ0) holds for a suitable δ0.Now, suppose that (ρσ2) holds. Then, by (σ1), (1.9) in Remark 2 and choosen a radius r > 0,in correspondence of ε > 0 there exists Cε > 0 such that (2.7) holds and by substituting(2.8) and (2.10) in (2.7) we get (2.11) then

I(u) ≥ p− 1

p‖u‖

pp−1

D2,

pp−1

ρ

− ε

m

(Cσ c |u|m( p

p−1 )∗∗ + εC−1

m

(|u|

pp−1pp−1 ,ρ

+ |u|(pp−1 )

∗∗

( pp−1 )

∗∗

))− Cε Cσ(

pp−1

)∗∗ |u|( pp−1 )

∗∗

( pp−1 )

∗∗

≥(p− 1

p− ε2

mC−1m C1

)‖u‖

pp−1

D2,

pp−1

ρ

− ε

mCσ C2 ‖u‖m

D2,

pp−1

ρ

−

ε2

mC−1m C3 −

Cε Cσ(pp−1

)∗∗ C4

‖u‖( pp−1 )

∗∗

D2,

pp−1

ρ

where in the last inequality we have exploited (2.1). Since pp−1 < m <

(pp−1

)∗∗, taken

‖u‖D

2,pp−1

ρ

= ρ with ρ and ε > 0 small enough, we get (Iδ0) holds again for a suitable δ0.

(Iu0) Let us fix v ∈ D2, p

p−1ρ (RN )∩Cc(RN ) with v 6= 0; thus, taken t > 0, by (1.11) in Remark


3 and µ > pp−1 we get

I(t v) ≤ p− 1

ptpp−1 ‖v‖

pp−1

D2,

pp−1

ρ

− C tµ∫

supp(v)

σ(x)|v|µ dx+ C1

∫supp(v)

σ(x) dx.

Let us observe that by (σ1) it is∫supp(v)

σ(x) dx ≤ Cσ meas(supp(v)) < +∞;

moreover, since µ <(

pp−1

)∗∗, by exploiting L( p

p−1 )∗∗

(supp(v)) → Lµ(supp(v)) and (2.1) we

get∫supp(v)

σ(x)|v|µ dx ≤ Cσ c

(∫supp(v)

|v|(pp−1 )

∗∗

dx

) µ

( pp−1 )

∗∗

≤ Cσ C ‖u‖µD

2,pp−1

ρ

< +∞.

Therefore, as t → +∞ it follows I(t v) → −∞ and, taken u0 = t v with t sufficiently largewe conclude that (Iu0

) is satisfied.

Indeed, we are able to prove that the functional I satisfies the following lemma.

Lemma 3. Under assumptions (ρ1), (σ1), (ρσ1) (resp. (ρσ2)), (f1) (resp. (f2)), (f3) and(f4), the functional I satisfies

(I ′U0) for every finite-dimensional subspace U0 ⊂ D2, pp−1

ρ (RN ) there esists a R = R(U0)such that I(u) ≤ 0 for every u ∈ U0 \BR(U0).

Proof. Now, fix u ∈ D2, pp−1

ρ (RN ) and s0 > 0 with |s0| ≤ 1. Denote Ωu,s0 = x ∈ RN :|u(x)| ≥ s0. By (σ1), (1.10) in Remark 3 and µ > p

p−1 , we have that

I(u) ≤ p− 1

p‖u‖

pp−1

D2,

pp−1

ρ

− C∫

Ωu,s0

σ(x)|u|µ dx

=p− 1

p‖u‖

pp−1

D2,

pp−1

ρ

− C∫RN

σ(x)|u|µ dx+ C

∫RN\Ωu,s0

σ(x)|u|µ dx

≤ p− 1

p‖u‖

pp−1

D2,

pp−1

ρ

− C∫RN

σ(x)|u|µ dx+ C Cσ

∫RN\Ωu,s0

|u|µ dx

≤ p− 1

p‖u‖

pp−1

D2,

pp−1

ρ

− C∫RN

σ(x)|u|µ dx+ C Cσ

∫RN\Ωu,s0

|u|pp−1 dx

≤ p− 1

p‖u‖

pp−1

D2,

pp−1

ρ

− C∫RN

σ(x)|u|µ dx+ C Cσ |u|pp−1pp−1

.

Then, by (2.1) and (ρ1) it follows

I(u) ≤(p− 1

p+ C Cσ c

)‖u‖

pp−1

D2,

pp−1

ρ

− C∫RN

σ(x)|u|µ dx.(3.1)

Let U0 be a finite dimensional subspace ofD2, pp−1

ρ (RN ). Clearly the term( ∫

RN σ(x)|u|µ dx) 1µ

is a norm in D2, pp−1

ρ (RN ), hence by (3.1) and the equivalence of all norms in U0, there exists

56 SARA BARILE

a positive constant R = R(U0) such that

I(u) ≤ 0 if u ∈ U0, ‖u‖D

2,pp−1

ρ

≥ R

and (I ′U0) is proved.

Since by Lemma 2 the functional I has the Mountain pass geometry, hence there exists a

Palais-Smale sequence un ⊂ D2, pp−1

ρ (RN ) at level cMP , namely such that

I(un)→ cMP(3.2)

and

dI(un)→ 0 in D−2, pp−1

ρ (RN ),(3.3)

where cMP is the minimax level of the Mountain Pass Theorem (see [2]) applied to I, thatis

cMP = infη∈Γ

maxt∈[0,1]

I(η(t))

with

Γ =η ∈ C([0, 1],D2, p

p−1ρ (RN )) : η(0) = 0 and I(η(t)) ≤ 0

.

At this point, by exploiting Proposition 3, we can prove that I satisfies (PS) condition atlevel cMP .

Lemma 4. Suppose that ρ and σ satisfy (ρ1), (σ1), (σ2), (ρσ1) (resp. (ρσ2)) and f verifies

(f1) (resp. (f2)), (f3) and (f4). Then, every Palais-Smale sequence un in D2, pp−1

ρ (RN ) at

level cMP converges in D2, pp−1

ρ (RN ), up to subsequences.

Proof. Let un be a (PS) sequence at level cMP , namely satisfying (3.2) and (3.3). Thenby (2.2), (σ1) and (f4) it follows

c (1 + ‖un‖D

2,pp−1

ρ

) ≥ µI(un)− dI(un)[un]

=

(µp− 1

p− 1

)‖un‖

pp−1

D2,

pp−1

ρ

+

∫RN

σ(x) (f(x, un)un − µF (x, un)) dx

≥(µp− 1

p− 1

)‖un‖

pp−1

D2,

pp−1

ρ

,

for a suitable positive constant c. Hence, un is bounded in D2, pp−1

ρ (RN )). So, there exists

u ∈ D2, pp−1

ρ (RN )) such that, up to subsequences, un u weakly in D2, pp−1

ρ (RN ) and, fromProposition 3,

(3.4) dIσ,F (un)→ dIσ,F (u) in D−2, pp−1

ρ (RN ).

Thus, by (3.3) we get

on(1) = (dI(un)− dI(u))[un − u]

= (dIρ(un)− dIρ(u))[un − u]− (dIσ,F (un)− dIσ,F (u))[un − u]

as n→ +∞ which together with (3.4) implies

(dIρ(un)− dIρ(u))[un − u]→ 0 if n→ +∞,


i.e., ∫RN

((−∆un)1p−1 − (−∆u)

1p−1 )(−∆un − (−∆u))dx(3.5)

+

∫RN

ρ(x)(u1p−1n − u

1p−1 )(un − u)dx→ 0.

Now, let us remark that, as 1p−1 ≥ 1, it is not difficult to prove that a constant C > 0 exists

such that(s

1p−1 − t

1p−1 ) (s− t) ≥ C|s− t|

pp−1 for all s, t ∈ R;

whence the conclusion follows from (3.5) and (ρ1).

At this point, we can prove Theorem 1.

Proof of Theorem 1. By Proposition 1 we get the functional I ∈ C1(D2, pp−1

ρ (RN )) and byLemma 2 it has the Mountain Pass geometry. Moreover, Lemma 4 implies that I satisfies

the Palais–Smale condition in D2, pp−1

ρ (RN ). Whence, the classical Mountain Pass Theorem

applies (see [2, Theorem 2.1]) and a critical point u of I in D2, pp−1

ρ (RN ) hence a weak solution

u ∈ D2, pp−1

ρ (RN ) of problem (1.3) exists.Furthermore, if also condition (f5) holds, the functional I is even. Then by (I0) and (Iδ0)in Lemma 2, Lemma 3 and Lemma 4 we get the hypotheses of the Symmetric versionof Mountain Pass Theorem are satisfied (see [2, Corollary 2.9]) and I has an unbounded

sequence un ⊂ D2, pp−1

ρ (RN ) of critical points.

References

[1] C.O. Alves and J.M. do O, Positive solutions of a fourth order semilinear problem involving critical

growth, Adv. Nonlinear Stud. 2 (4) (2002), 437-458.

[2] A. Ambrosetti and P.H. Rabinowitz, Dual variational methods in critical point theory and applications,J. Functional Anal. 14 (1973), 349-381.

[3] S. Barile and A. Salvatore, Weighted elliptic systems of Lane-Emden type in unbounded domains,

Mediterr. J. Math. 9 (3) (2012), 409-422.[4] S. Barile and A. Salvatore, Some Multiplicity and Regularity Results for Perturbed Elliptic Systems,

Dynamic Systems and Applications, Dynamic, Atlanta, GA, 6 (2012), 58-64.

[5] S. Barile and A. Salvatore, Existence and Multiplicity Results for some Elliptic Systems in UnboundedCylinders, Milan J. Math. 81 (1) (2013), 99-120.

[6] S. Barile and A. Salvatore, Existence and Multiplicity Results for some Lane-Emden Elliptic Systems:Subquadratic case, Adv. Nonlinear Anal. 4 (1) (2015), 25-35.

[7] S. Barile and A. Salvatore, Some results on weighted subquadratic Lane-Emden Elliptic Systems in

unbounded domains, Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl. 27 (1) (2016), 89-103.[8] S. Barile and A. Salvatore, Some new results on subquadratic Lane-Emden Elliptic Systems with weights

in unbounded domains, to appear on Mediterr. J. Math.

[9] W.D. Bastos, O.H. Miyagaki and R.S. Vieira, Solution to a biharmonic equation with vanishing potential,Illinois J. Math. 57 (3) (2013), 839-854.

[10] E. Berchio and F. Gazzola, Some remarks on biharmonic elliptic problems with positive, increasing and

convex nonlinearities, Electron. J. Differential Equations 34 (2005), 1–20.[11] H. Berestycki and P.L. Lions, Nonlinear scalar field equations I. Existence of a ground state, Arch.

Ration. Mech. Anal. 82 (4) (1983), 313-345.

[12] F. Bernis, J. Garcıa Azorero and I. Peral, Existence and multiplicity of nontrivial solutions in semilinearcritical problems of fourth order, Adv. Differential Equations 1 (2) (1996), 219-240.

[13] D. Bonheure, E. Moreira dos Santos and M. Ramos, Ground state and non-ground state solutions ofsome strongly coupled elliptic systems, Trans. Amer. Math. Soc. 364 (1) (2012), 447-491.

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[14] A.M. Candela and A. Salvatore, Elliptic systems in unbounded domains, Complex Var. Elliptic Equ.

56 (12) (2011), 1143-1153.

[15] J. Chabrowski and J.M. do O, On some fourth order semilinear elliptic problems in RN , Nonlinear

Anal. 49 (2002), 861-884.[16] Ph. Clement and E. Mitidieri, On a class of nonlinear elliptic systems, Nonlinear Evolution Equations

and their Applications, Research Institute for Mathematical Science - Kyoto 1009 (1997), 132-140.

[17] D.G. de Figueiredo and P. Felmer, On superquadratic elliptic systems, Trans. Amer. Math. Soc. 343(1) (1994), 99-116.

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Math. 1 (4) (2004), 417-431.[19] R. Demarque and O.H. Miyagaki, Radial solutions of inhomogeneous fourth order elliptic equations

and weighted Sobolev embeddings, Adv. Nonlinear Anal. 4 (2) (2014), 135-151.[20] Y. Deng and W. Shuai, Non-trivial solutions for a semilinear biharmonic problem with critical growth

and potentials vanishing at infinity, Proc. Roy. Soc. Edinburgh Sect. A 145 (2) (2015), 281-299.

[21] E.M. dos Santos, Multiplicity of solutions for a fourth-order quasilinear nonhomogeneous equation, J.Math. Anal. Appl. 342 (1) (2008), 277-297.

[22] P. Felmer and S. Martınez, Existence and uniqueness of positive solutions to certain differential systems,

Adv. Differential Equations 3 (4) (1998), 575-593.[23] F. Gazzola, H.C. Grunau and G. Sweers, Polyharmonic boundary value problems. Positivity preserving

and nonlinear higher order elliptic equations in bounded domains, Lecture Notes in Mathematics, 1991,

Springer-Verlag, Berlin 2010.[24] P.L. Lions, The concentration-compactness principle in the calculus of variations. The Limit case. I.,

Rev. Mat. Iberoamericana 1 (1) (1985), 145-201.

[25] W.G. Maz’ja, Sobolev Spaces, Springer, Berlin, 1985.[26] E. Mitidieri, A Rellich type identity and applications, Comm. Partial Differential Equations 18 (1-2)

(1993), 125-151.[27] B. Opic and A. Kufner, Hardy-Type Inequalities, Pitman Research Notes in Mathematics Series 219,

Longman Scientific and Technical, Harlow, 1990.

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Dipartimento di Matematica, Universita degli Studi di Bari Aldo Moro, Via E. Orabona 4,

70125 Bari, Italy


JEPE Vol 2, 2016, p. 59-71

SYMMETRIC–CONVEX FUNCTIONALS OF LINEAR GROWTH

FRANZ GMEINEDER

Abstract. We discuss existence and regularity theorems for convex functionals of lineargrowth that depend on the symmetric rather than the full gradients. Due to the failure

Korn’s Inequality in the L1–setup, the full weak gradients of minima do not need to

exist, and the paper aims for presenting methods that help to overcome these issues asto partial regularity and higher integrability of minimisers.

1. Introduction

The purpose of the present paper is to survey and to announce existence and regularityresults for minima of autonomous variational integrals which depend on the symmetric ratherthan the full gradients. More precisely, let Ω be an open and bounded Lipschitz subset ofRn and consider the variational principle

to minimise F[v] :=

∫Ω

f(ε(v)) dx over a Dirichlet class D ,(1.1)

where ε(v) := 12 (Dv+DTv) is the symmetric part of the weak gradient of a function v : Ω→

Rn, and f ∈ C(Rn×nsym ) is a convex function of linear growth. By the latter, we understandthat there exist c1, c2 > 0 such that

c1|Z| 6 f(Z) 6 c2(1 + |Z|) for all Z ∈ Rn×nsym .(1.2)

Under these conditions imposed on f , F is well–defined on the space LD(Ω) consisting ofall v ∈ L1(Ω;Rn) whose weak symmetric gradients belong to L1(Ω;Rn×nsym ). This space isequipped with the canonical norm ‖v‖LD := ‖v‖L1 + ‖ε(v)‖L1 , and we define LD0(Ω) to bethe ‖ · ‖LD–closure of C1

c(Ω;Rn). Consequently, we put D := u0 + LD0(Ω) for some fixedu0 ∈ LD(Ω) and easily conclude that F is coercive on D with respect to the LD–norm.

It is important to note that the aforementioned coerciveness fails when LD is replaced byW1,1. The reason for this is the lack of Korn’s Inequality, a fundamental obstruction whichwe briefly describe now. Given 1 < p < ∞, Korn’s Inequality asserts that there exists aconstant C > 0 such that ∫

Ω

|Dv|p dx 6 C

∫Ω

|ε(v)|p dx(1.3)

2010 Mathematics Subject Classification. Primary: 49J06; Secondary: 35J06.Key words and phrases. Functionals of Linear Growth, Regularity Theory, Functions of Bounded

Deformation.Received 02/09/2016, accepted 26/10/2016.

59

60 FRANZ GMEINEDER

holds for all v ∈ C1c(Ω;Rn). In consequence, for convex and continuous integrands g : Rn×nsym →

R which satisfy c1|Z|p 6 g(Z) 6 c2(1 + |Z|p) for all Z ∈ Rn×nsym and two constants c1, c2 > 0,

(1.3) implies that the variational integral G[v] :=∫

Ωg(ε(v)) dx not only is well–defined on

W1,p(Ω;Rn) but also coercive on suitable Dirichlet subclasses of W1,p(Ω;Rn) with respectto the usual W1,p–norm. As shall be explained in more detail in section 2 below, the reasonfor (1.3) to hold is that the map Φ: ε(v) 7→ Dv (where v ∈ C1

c(Ω;Rn) is tacitly identifiedwith its trivial extension to the entire Rn) is a singular integral of convolution type. Thus,by standard results for such operators, Φ is of strong–(p, p) type if and only if 1 < p < ∞.In turn, if p = 1, inequality (1.3) fails to hold true, a fact which is often referred to asOrnstein’s Non–Inequality. Even stronger statements are available, some of which shall bediscussed in section 2.

Since LD(Ω) is a non–reflexive space, the second chief obstruction is that minimisingsequences (vk) ⊂ D might not possess weakly convergent subsequences even though they areuniformly bounded with respect to the LD–norm. To overcome this lack of compactness, it isreasonable to define the space BD(Ω) of functions of bounded deformation as the collection ofall v ∈ L1(Ω;Rn) such that the distributional symmetric gradient ε(v) can be represented bya Rn×nsym –valued Radon measure of finite total variation on Ω, in symbols ε(v) ∈M(Ω;Rn×nsym );see [39, 1] for a detailled treatment of these spaces. In particular, by Ornstein’s Non–Inequality, there exist elements v ∈ BD(Ω) such that Dv /∈ M(Ω;Rn×n) and hence BD(Ω)contains BV(Ω;Rn) as a proper subspace. In many respects, the properties of BD–functionsare reminiscent of those of BV–functions, and we shall discuss similarities and discrepanciesbetween the two function spaces as the paper evolves. Before passing to criteria that ensurethe regularity of minima, we briefly revisit the treatment of the boundary value problemin BD which appears in a similar vein as that in BV as set up in the fundamental work ofGiaquinta, Modica and Soucek [22].

1.1. Relaxation and Generalised Minima. As an easy consequence of the Banach–Alaoglu and Rellich–Kondrachov Theorems, uniformly bounded sequences in LD(Ω) possesssubsequences that converge to some v ∈ BD(Ω) in the weak*–sense. By this we understand

that for some (vj(k)) ⊂ (vj) there holds vj(k) → v strongly in L1(Ω;Rn) and ε(vj(k))∗ ε(v)

in the sense of weak*–convergence of Radon measures on Ω as k → ∞. In this situation,the weak*–limit map can be shown to exist, however, to establish a reasonable notion ofminimality for v, the functional F must be extended to BD(Ω) first. To keep the presentationsimple, we stick to the L1–Lebesgue–Serrin extension given by

F[v] := inf

lim infk→∞

F[vk] : (vk) ⊂ D , vk → v in L1(Ω;Rn) as k →∞, v ∈ BD(Ω).

Note that this type of relaxation is reasonable indeed: If (vk) ⊂ D is bounded with respectto the LD–norm and converges to some v ∈ L1(Ω;Rn) strongly in L1(Ω;Rn), then v ∈BD(Ω) by lower semicontinuity of the total deformation |ε(·)|(Ω) with respect to strongL1–convergence. It needs to be noted that the functionals F admit the explicit integralrepresentation

F[v] =

∫Ω

f(E v) dx+

∫Ω

f∞(

dEv

d|Esv|

)d|Esu|+

∫∂Ω

f∞(Tr(v − u0) ν∂Ω) dHn−1

REGULARITY FOR FUNCTIONALS OF LINEAR GROWTH 61

for v ∈ BD(Ω), where

ε(v) = Eacv + Esv =dEv

dL nL n +

dEsv

d|Esv||Esv| = E vL n +

dEsv

d|Esv||Esv|

is the Radon–Nikodym decomposition of ε(v) into its absolutely continuous and singularparts with respect to Lebesgue measure L n; moreover, f∞ : Rn×nsym → R is the recessionfunction of f defined as

f∞(Z) := limt0

tf (Z/t) , Z ∈ Rn×nsym ,

and captures the behaviour of the integrand at infinity, that is to say, where the Lebesguedensity of ε(v) with respect to L n becomes singular. Note that by [1, 28], the density dEv

dLn

can be shown to equal the symmetric part E v of the approximate gradient of v L n–a.e..The trace terms in fact are sensible, as by [39, 3], BD–functions attain boundary values inthe L1–sense. Noting that for a, b ∈ Rn, a b := 1

2 (abT + baT) is the symmetric tensorproduct and ν∂Ω the outer unit normal to ∂Ω, the boundary integral term appearing in theintegral representation admits the interpretation of a penalisation term that leads to largervalues of the functional provided the L1–distance of Tr(v) from Tr(u0) is increased. Theproof of the above representation follows along the lines of [7] in the full gradient case, andfor more background information, the reader is referred to [26]. For completeness, we makethe following

Definition 1 (BD–Minima). An element u ∈ BD(Ω) is called a BD–minimiser if and onlyif F[u] 6 F[v] for all v ∈ BD(Ω).

On a sidenote, let us remark that convexity of f substantially simplifies the proof of theintegral representation for the relaxed functional. In fact, in the convex setup, it is possibleto use the Goffman–Serrin relaxation machinery [27], whereas in the quasiconvex situationmore subtle arguments need to be invoked; see the work of Rindler [36] for more detail.

Another notion of minimisers has been employed by Bildhauer & Fuchs [9, 7] in thesetting of linear growth functionals on BV, whose adaption to the present situation readsas follows:

Definition 2 (Generalised Minima). Let Ω be an open and bounded Lipschitz subset ofRn and fix a boundary datum u0 ∈ LD(Ω). The set of generalised minima of F givenby (1.1) consists of all those u ∈ BD(Ω) for which there exists an F–minimising sequence(uk) ⊂ Du0 := u0 + LD0(Ω) such that uk → u strongly in L1(Ω;Rn) as k →∞. The set ofall generalised minima is denoted GM(F).

Now, if f ∈ C(Rn×nsym ) is convex – that is to say, f is symmetric–convex – then u ∈ BD(Ω)can be shown to be a BD–minimiser if and only if it is a generalised minimiser, and in thiscase there holds

F[u] = minF(BD(Ω)) = inf F[D ].(1.4)

The crucial point thus is to establish existence of a BD–minimiser. This is, however, eas-ily achieved by employing the direct method and by use of Reshetnyak–type theorems onthe lower semicontinuity of functionals of measures with respect to the weak*– and stricttopologies; see [35, 7]. Hence a satisfactory existence theory is established, and the foremostaim of the present paper is to survey the reguarity properties of generalised minima.

62 FRANZ GMEINEDER

1.2. Organisation of the Paper and Description of Results. Having settled existenceof generalised minima in the previous section, the paper focusses on regularity results in allof what follows. In section 2, we revisit Korn’s Inequality and Ornstein’s Non–Inequalityand strengthen the sketchy arguments outlined above to understand the chief obstructionsfor regularity results in the symmetric–convex case. In section 3, we report on recent devel-opments regarding the Holder and Sobolev regularity of generalised minima of symmetric–convex functionals subject to strong convexity conditions imposed on the variational inte-grands f . Firstly describing a partial regularity result due to the author [25] in the spiritof Anzellotti & Giaquinta [2], we then turn to conditions on the variational integrands to

produce generalised minima of class BVloc or even W1,ploc for some 1 < p < ∞, the latter

being joint work with Jan Kristensen [26]. To our best knowledge, these results are the firstof their kind and extend the regularity theory on the Dirichlet problem on BV to that onBD; see [7, 5]. To conclude with, in section 4 we introduce the spaces WA,1 and BVA assuitable generalisatons of BV and BD and highlight open questions that would lead to asatisfactory existence and regularity theory in this fairly general setup.

Acknowledgment

The author is indebted to the University of Zurich for financial support to attend the9th European Conference on Elliptic and Parabolic Problems held in Gaeta in May 2016.Moreover, he gratefully acknowledges the comments of an anonymous referee which helpedto improve the exposition of the material.

Notation

The symmetric n × n–matrices are denoted Rn×nsym , and we use the symbol 〈·, ·〉 for the

euclidean inner product on finite dimensional spaces. Lastly, L n and Hn−1 denote the n–dimensional Lebesgue– or (n− 1)–dimensional Hausdorff measures, respectively, and we use(u)U := −

∫Uudx := (L n(U))−1

∫Uudx for the mean value of a locally integrable function

u : U → RN whenever this is well–defined.

2. Korn’s Inequality and Ornstein’s Non–Inequality

Before we embark on the regularity of generalised minima as addressed in the introduction,we briefly wish to comment on Korn’s Inequality in slightly more detail. To this end, let A[D]be a linear, homogeneous first order and constant coefficient differential operator betweenthe two finite–dimensional real vector spaces V and W , i.e., A[D] can be written in the form

A[D] =∑|α|=1

Aα∂α,(2.1)

where Aα : V →W are fixed linear mappings. We associate with A[D] its symbol map

A[ξ] :=∑|α|=1

ξαAα, ξ = (ξ1, ..., ξn) ∈ Rn,(2.2)

and call A[D] elliptic if and only if A[ξ] : V →W is injective for any ξ 6= 0. Given an ellipticoperator A[D] and u ∈ C∞c (Rn;Rn), we can thus retrieve u from A[D]u by means of the


operator

u(x) = G[A[D]u](x) = cnF−1ξ 7→x((A∗[ξ] A[ξ])−1A∗[ξ]A[D]u) =: Φ(A[D]u)(x), x ∈ Rn,

where cn > 0 is a constant and A∗[ξ] is the adjoint symbol of A[ξ] being defined in theobvious manner. Since (A∗[ξ] A[ξ])−1A∗[ξ] is homogeneous of degree −1, it is easily seenthat G is a Riesz potential operator of order 1, in particular, we have the bound

|G[v](x)| .∫Rn

|v(y)||x− y|n−1

dy for all x ∈ Rn.

Finally, differentiating G[A[D]u] immediately yields that Du can be written as a singular in-tegral of convolution type acting on A[D]u; see [38]. Therefore, the operator Φ: A[D]u 7→ Du

given by Φ(A[D]u) := D(G[A[D]u]

)extends to a bounded linear operator Φ : Lp(Rn;W )→

Lp(Rn;Rn × V ) provided 1 < p < ∞. Hence, given 1 < p < ∞ and an open subset Ω ofRn, Korn’s inequality ‖Du‖Lp(Ω;Rn×V ) 6 C‖A[D]u‖Lp(Ω;W ) for all u ∈ C∞c (Ω;V ) with afinite constant C = C(A, p) > 0 follows from the aforementioned boundedness of singularintegrals by extending elements of C∞c (Ω;V ) to Rn by zero.

Korn’s Inequality in the form as given above can be generalised to various other settings;see [32] and [10, 11] for more recent developments in the context of Orlicz functions. Itsfailure in the case p = 1 has witnessed a variety of notable contributions beyond Ornstein’soriginal article [34]; see, among others, [14, 29]. In particular, as can be seen best throughTheorem 1.3 of Kirchheim and Kristensen’s study [29], there are only trivial L1–estimatesin the following sense:

Theorem 1 ([29], Theorem 1.3). Let V,W,X be three finite–dimensional vector spaces andconsider two k–th order linear and homogeneous differential operators A1[D] and A2[D] ofthe form

A1[D] =∑|α|=k

A1α(x)∂α and A2[D] =

∑|α|=k

A2α(x)∂α,

with locally integrable coefficients A1α ∈ L1

loc(Rn; L (V ;W )) and A2α ∈ L1

loc(Rn; L (V ;X)) forall |α| = k, respectively, the following are equivalent:

(1) There exists a constant C > 0 such that ‖A2[D]ϕ‖L1(Rn;W ) 6 c‖A1[D]ϕ‖L1(Rn;W )

holds for all ϕ ∈ C∞c (Rn;V ).(2) There exists C ∈ L∞(Rn; L (W ;X)) with ‖C‖L∞(Rn;L (W ;X)) 6 c such that A2

α(x) =

C(x)A1α(x) for L n–a.e. x ∈ Rn and each α ∈ Nn0 with |α| = k.

Here, triviality of L1–estimates means that if a Korn–type inequality holds (1), thenthe coefficients are the same up to multiplication with an L∞–function. It is important tonote that both the symmetric gradient operator A1[D]u = ε(u) or the trace–free symmetricgradient operator A1[D]u = ε(u)− 1

n div(u)1n×n with the (n×n)–unit matrix 1n×n ∈ Rn×ndo not verify (2) with A2[D] = Du and hence they do not admit Korn–type inequalities inthe L1–setup.

3. Partial and Sobolev Regularity

Besides existence of generalised minima as outlined in the introduction, it is natural toinvestigate their regularity properties, a program which has been launched in the BV–settingin [22, 31]. Here, we focus on Holder– and Sobolev regularity and shall describe the main

64 FRANZ GMEINEDER

obstructions that come along both with the linear growth hypothesis and Ornstein’s Non–Inequality. In particular, the latter motivates to study conditions imposed on the variationalintegrand f that guarantee existence of the full gradients of generalised minima as elementsof M(Ω;Rn×n) or L1(Ω;Rn×n).

3.1. Holder Regularity. To begin with, let us note that by the genuine vectorial nature ofthe functional F, BD–minima cannot be shown to share everywhere C1,α–regularity unlessstrong structural conditions are imposed on the variational integrands f . This is in the spiritof the famous counterexamples due to De Giorgi [15] and Giusti & Miranda [24] (see also[33] for an excellent overview) which demonstrate that in the case N > 1, functionals of theform

G[v] :=

∫Ω

g(∇v) dx, v : Ω→ RN

do not necessarily produce minimisers of class C1,αloc (Ω;RN ) even if suitable ellipticity, bound-

edness and measurability assumptions are imposed on the integrands g. The correct substi-tute is then given by the notion of partial regularity : Given u ∈ BD(Ω), we define its regularset

Ωu := x ∈ Ω: ∇u is Holder continuous in a neighbourhood of x,(3.1)

and note that the definition of Ωu depends on the full weak gradients ∇u indeed. We callΩu the regular set of u and its relative complement Σu := Ω \Ωu the singular set. Adoptingthese notions, we say that u ∈ BD(Ω) is partially regular if and only if Ωu is open andL n(Σu) = 0.

Referring the reader to [4, 23, 33] for a comprehensive overview of techniques to establishpartial regularity of minima of elliptic variational integrals, we note that most approaches tothe partial regularity rely on the higher integrability of gradients. Such higher integrabilityresults in turn often stem from Caccioppoli–type inequalities in conjunction with Gehring’slemma. Indeed, to sketch the prototype form of such an argument, let 1 < p < 2 and assumethat u ∈W1,p(Ω;RN ) satisfies a Caccioppoli–type inequality of the form

−∫

B(z,r)

|Du|p dx 6 C−∫

B(z,2r)

∣∣∣∣u− (u)B(z,2r)

r

∣∣∣∣p dx(3.2)

for all z ∈ Ω and 0 < r < dist(z, ∂Ω)/2. Now, applying the Sobolev–Poincare–inequality tothe right side, we deduce that −

∫B(z,r)

|Du|p can be locally estimated against −∫

B(z,2r)|Du|q dx

for some 1 < q < p, and in this sense Du satisfies a reverse Holder inequality. By Gehring’sLemma, we then obtain that Du belongs to some Lp+εloc for some ε > 0, and the readerwill notice that the above argument remains unchanged when D is replaced by ε and themean values on the right side of (3.2) by suitable rigid deformations, that is, elements ofthe nullspace of ε.

If p = 1, then Gehring’s Lemma self–improves the integrability of Du or ε(u), respectively,if and only if we can choose q < p = 1. Thus a suitable Sobolev–Poincare inequality wouldbe required, estimating the L1–norm of a function against the Lq–norm of its gradient orsymmetric gradient, respectively. However, in the full generality as needed here, this is ruledout by a counterexample due to Buckley & Koskela [13]. In turn, one is lead to the so–called weak reverse Holder classes WRH whose elements satisfy suitable Sobolev–Poincareinequalities for q < p = 1 by definition in a natural way; see [13] for more background


information. Coming back to the higher integrability addressed above, it is not clear thatthe gradients of minima belong to such weak reverse Holder classes at the relevant stageof the proof; hence different methods are required in the linear growth setting. In thesymmetric–convex case as described in the introduction, such estimates can be achieved byuse difference quotient–type methods, but in turn require strong ellipticity assumptions onthe integrands; see section 3.2 below.

A direct approach to the partial regularity that is particularly designed for convex func-tionals and applies to functionals of linear growth, too, is that of Anzellotti & Giaquinta [2].As usual, this particular method also relies on decay estimates for suitable excess functionalstoo. To describe the decisive feature of this method, let us remark that unlike other, perhapsmore standard approaches like blow–up proofs, Anzellotti & Giaquinta derive the requireddecay estimates through comparing minima with suitable mollifications thereof. These mol-lifications in turn are shown to be close to solutions of elliptic second order PDE and thusenjoy good decay estimates which then are shown to carry over to the generalised minimathemselves. Relying on mollifications and, consequently, Jensen’s inequality, the method iswell–suited for convex problems, whereas it is not clear how to generalise it to quasiconvexintegrands, for instance. With the case of full gradients being treated in [2], the respectivegeneralisation to the symmetric gradient case will be given in [25] by the following

Theorem 2. Let f ∈ C2(Rn×nsym ;R≥0) be convex and of linear growth. Suppose that u ∈BD(Ω) is a BD–minimiser of F given by (1.1). If (x, z) ∈ Ω× Rn×nsym is such that

limR0

[−∫

B(x,R)

|E u− z|dx+|Esu|(B(x,R))

L n(B(x,R))

]= 0

and f ′′(z) is positive definite, then u ∈ C1,α(U ;Rn) for a suitable neighbourhood U of x forall 0 < α < 1.

Assuming the theorem, the standard Lebesgue differentiation theorem for Radon measuresyields the claimed partial regularity: There exists an open subset Ωu of Ω with L n(Ω\Ωu) =0 such that for any x ∈ Ωu there exists r > 0 with u ∈ C1,α(B(x, r);Rn) for every 0 < α < 1.We wish to conclude with the following

Remark 1 (Non–Autonomous Integrands). Since F given by (1.1) is autonomous, it ispossible to overcome the higher integrability of the symmetric gradients in the proof of theabove theorem. If the integrand in addition is x–dependent, then the higher integrabilityseems to be a necessary to conclude the result in this non–autonomous case too. Going backto the discussion at the beginning of the section, such a result is therefore unlikely to beestablished by means of the method as described above.

3.2. Sobolev Regularity. Next we turn to Sobolev regularity of generalised minima andhereafter aim for conditions on the integrands f under which generalised minima genuinelybelong to BVloc(Ω;Rn) or W1,p

loc(Ω;Rn) for some 1 < p < ∞. To obtain such results, weshall work with a strong convexity adapted from that of Bildhauer & Fuchs [9] in the fullgradient case:

66 FRANZ GMEINEDER

Definition 3 (µ–ellipticity). Let µ > 1. A C2–integrand f : Rn×nsym → R≥0 is called µ–ellipticif and only if there exist 0 < λ 6 Λ <∞ such that

λ|A|2

(1 + |B|2)µ2

6 〈f ′′(B)A,A〉 6 Λ|A|2

(1 + |B|2)12

(3.3)

holds for all A,B ∈ Rn×nsym . We further say that the variational integral F is µ–ellipticprovided its integrand f is.

Prototypical examples are given by the area–type integrands mp(ξ) := (1+ |ξ|p)1p for ξ ∈

Rn×nsym , 1 < p <∞; so, for instance, the area integrand m2 is 3–elliptic, whereas mp coincideswith a µ = p + 1–elliptic integrand away from the unit ball; also see [9] for more detail. Itis important to remark that µ = 1 is explicitely excluded in definition 3; indeed, 1–ellipticintegrands correspond to L logL–growth. For such integrands, ε(u) ∈ L logLloc(Ω;Rn×nsym )

already implies Du ∈ L1loc(Ω;Rn×n) and hence the full gradients are known to exist and

belong to L1 locally. In this setup of L logL–growth, the corresponding regularity theoryhas been established by Fuchs, Seregin and collaborators [17, 19, 18] among others; see theextensive monograph [20] for more information.

3.2.1. Results on the Dirichlet Problem on BV. To explain our method, it is convenient tofirstly report on the available higher integrability results for µ–elliptic functionals in the fullgradient case

F [u] :=

∫Ω

f(∇v) dx, v ∈ D := u0 + W1,10 (Ω;RN ),(3.4)

where u0 ∈ W1,1(Ω;RN ) is a given boundary datum. In analogy with Definition 2, wesay that v ∈ BV(Ω;RN ) is a generalised minimiser for F if and only if there exists anF–minimising sequence (vk) ⊂ D such that vk → v strongly in L1(Ω;RN ) as k →∞.

Using a vanishing viscosity approach, Bildhauer [8] provided the first higher integrabilityresults for gradients of generalised minima. Precisely, assuming u0 ∈ W1,2(Ω;RN ) for theboundary data, the functional F is stabilised by adding small Laplacians, i.e., we consider

Fδ[v] := F [v] +δ

2

∫Ω

|∇v|2 dx on D := u0 + W1,20 (Ω;RN )

and finally aim for sending δ 0. Denoting the unique minimiser of F over D by uδ, it iseasy to prove that (uδ) is a minimising sequence for F . Bildhauer, in turn building on ideasof Seregin [37], then was able to show that if (uδ) satisfies the local boundedness assumption

for all K b Ω there exists C(K) > 0 with sup0<δ<1

‖uδ‖L∞(K;RN ) 6 C(K),(3.5)

then the weak*–limit u of (uδ) belongs to W1,ploc(Ω;RN ) for some p = p(µ) > 1 provided

1 < µ < 3, and to W1,L logLloc (Ω;RN ) provided µ = 3. Apart from the strong assumptions

made on the particular minimising sequence, the boundary data and the functional itself,the result merely applies to one particular such generalised minimiser. The reason for thisis the possible non–uniqueness of generalised minima which, in turn, is a consequence ofthe recession parts in the relaxed variational integral. Indeed, even if f ∈ C2(RN×n) isstrictly convex, the recession function f∞ : RN×n → R is positively 1–homogeneous andthus never strictly convex. In consequence, if a minimiser does not have vanishing singularpart with respect to Lebesgue measure, uniqueness in general fails as f and f∞ act on


two mutually singular parts of the gradients. To achieve uniqueness, one must thereforegenuinely rule out the singular parts of minima. This has been achieved recently by Beck &Schmidt [5] by sophisticated use of the Ekeland variational principle in the negative Sobolevspace W−1,1 (see Prop. 1 below) for the borderline case µ = 3. Referring the readerfor the precise outline to [5], the general streamline is this: Starting from an arbitrary

minimising sequence (uk) ⊂ u0 + W1,10 (Ω;RN ), the Ekeland variational principle yields

another minimising sequence (vk) that is W−1,1–close to (uk) and has the same weak*–limit. This new sequence (vk) is then shown to be a sequence of almost minimisers to

suitably stabilised functionals and thus can be proved to belong to W2,2loc. At this point it is

possible to adapt Bildhauer’s approach and hence, by arbitrariness of (uk), uniqueness andthe aforementioned higher regularity results follow at once. However, it needs to be stressedthat Beck & Schmidt’s method of proof also relies on a version of the local boundednessassumption. Further, assuming Uhlenbeck, i.e., radial structure of the integrands, strongerresults such as C1,α–regularity of generalised minima of the F can be achieved; see [7, 8, 6].

3.2.2. Results on the Dirichlet Problem on BD. Going back to functionals F as given by(1.1), the main difficulty lies in the appearance of the full difference quotients when aim-ing for higher integrability estimates and testing the Euler–Lagrange equation of a suit-ably stabilised functional with the canonical choice ϕ := ∆−s,h(ρ2∆+

s,hv), where ∆±s,hv(x) :=1h (v(x±hes)−v(x)) are the forward or backward difference quotients, respectively. By Orn-

stein’s Non–Inequality, ∆+s,hv cannot even be bounded locally in L1 for v ∈ BD in general,

and thus the suitable device hence is to work with finite differences instead of differencequotients and to establish estimates for carefully chosen Besov–norms of the symmetricgradients.

We pass on to a more precise description of the method which is, to some extent, inspiredby [5]. Let (vk) ⊂ D be a minimising sequence for the µ–elliptic functional F given by (1.1),with µ to be determined later on. Then we consider for a suitable sequence (αk) ⊂ R>0

with αk 0 as k →∞ the stabilised functionals

Fk[w] :=

∫Ω

f(ε(w)) dx+ αk

∫Ω

(1 + |ε(w)|p) dx =:

∫Ω

fk(ε(w)) dx

with p ≥ n on appropriately modified Dirichlet classes Dk to keep track of the fact thatthe leading part of Fk is the p–th Dirichlet energy; indeed, as p > 1, minima of Fk belongto W1,p and thus possess full gradients in Lp by Korn’s Inequality. Extending each Fkto (W1,∞

0 (Ω;Rn))∗ by infinity on (W1,∞0 (Ω;Rn))∗ \ Dk, we obtain a lower semicontinuous

functional on (W1,∞0 (Ω;Rn))∗ which is continuous with respect to the norm topology on

(W1,∞(Ω;Rn))∗. To continue, we recall the following instrumental

Proposition 1 (Ekeland’s Variational Principle). Let (X, d) be a complete metric space andassume that f : X → [0,∞] is lower semicontinuous with inf F (X) <∞. If for some ε > 0and x ∈ X there holds F [u] 6 inf F (X) + ε, then there exists v ∈ X such that d(x, v) 6

√ε

and

F [v] 6 F [w] +√εd(v, w) for all w ∈ X.

For a proof and a discussion of this result, see [23], Thm. 5.6.. Using suitable approxi-

mations and Proposition 1 in the Banach space (W1,∞0 )∗, we obtain a sequence (uk), each

68 FRANZ GMEINEDER

of whose members is an almost minimiser of F, is (W1,∞0 )∗–close to vk and, most crucially,

each uk satisfies the perturbed Euler–Lagrange equation∣∣∣∣∫Ω

〈f ′k(ε(uk)), ε(ϕ)〉dx∣∣∣∣ 6 1

k‖ϕ‖(W1,∞

0 (Ω;Rn)∗ +(small perturbation

)for all ϕ ∈ W1,p

0 (Ω;Rn). Given x0 ∈ Ω, 0 < r < R < dist(x0, ∂Ω), we consider for anarbitrary standard unit vector es, s = 1, ..., n, the test functions ϕ := τ−s,h(ρ2τ+

s,huk) with

τ±s,h = h∆±s,h. Essentially following, e.g., [23], section 10.1, and employing µ–ellipticity of f ,we end up with a coercive inequality∫

Ω

|ρτ+s,hε(uk)|2

(1 + |ε(uk)|)µdx .

∣∣∣∣∫Ω

〈f ′k(ε(uk)), ρ τ+s,huk〉dx

∣∣∣∣+1

k‖τ−s,h(ρ2τ+

s,huk)‖(W1,∞0 )∗ ,(3.6)

with the constants implicit in ’.’ being uniformly bounded in k. Let us briefly explain howto handle the two terms on the right side: As to the first term, we note that f ′k(ε(uk))converges in a suitable sense to the solution of the dual problem associated with (1.1) (inthe sense of convex duality, see [16]). Since the dual solution σ ∈ L∞(Ω;Rn×n) itself belongs

to W1,2loc(Ω;Rn×n), this regularity can be shown to inherit to f ′k(ε(uk)) uniformly in k. If

we wish to fruitfully use this estimate, we need to suitably bound ‖ρτ+s,huk‖L2 uniformly

in k too. For general n, the fractional Sobolev–type embeddings BDloc → Ws,n/(n−1+s)loc ,

0 < s < 1, with the fractional Sobolev spaces Wθ,r, 0 < θ < 1 and r ≥ 1, are optimal, andn/(n − 1 + s) = 2 is achieved if and only if n = 2 and s = 0. In this case, however, weloose all smoothness information and hence may invoke a condition that is slightly weakerthan a local boundedness assumption in the spirit of (3.5); namely, we require the so–calledlocal BMO–assumption, meaning that for each relatively compact K ⊂ Ω, the seminorms‖uk‖BMO(K;Rn) are bounded independently of k. As shall be demonstrated in [26] by meansof so–called Dorronsoro–type estimates which have been fruitfully used in [30] by Kristensen& Mingione in a different context, we have

BDloc ∩BMOloc →W1p−ε,ploc(3.7)

for all 1 6 p < ∞ and suitably small ε > 0; the limiting case ε = 0 is not even reachedin general even if BDloc is replaced by the considerbaly smaller space W1,1

loc, a fact whichhas been pointed out by Bourgain, Brezis & Mironescu in [12], Remark 3. The upshot ofthis in comparison with the aforementioned embedding without the BMO–side constraint isthat although the BMO–condition is not reflected by the first derivatives, it improves bothfractional differentiability and the corresponding integrability at a uniform rate. Puttingp = 2 in (3.7), it is possible to estimate the first term on the right side of (3.6) by Ch3/2−ε

for any suitably small ε > 0. The second term on the right side of (3.6) can be estimated in

the same way, using ‖∆+s,hv‖(W1,∞

0 )∗ . ‖v‖L1 together with BDloc →Wθ,1loc for any 0 < θ < 1.

Going back to (3.6), we then obtain the uniform bound∫Ω

∣∣∣∣∣ρτ+s,hε(uk)

h34−

ε2

∣∣∣∣∣2

ωk dx :=

∫Ω

∣∣∣∣∣ρτ+s,huk

h34−

ε2

∣∣∣∣∣2

1

(1 + |ε(uk)|)µdx 6 C

This is a weighted Nikolskiı–type estimate for ε(uk). At this stage of the proof, it possibleto deduce that the weights ωk uniformly belong to certain Muckenhoupt classes Ap for some1 < p <∞ and hence, using embedding results for weighted Nikolskiı spaces and the theory


of singular integrals on Muckenhoupt weighted Lebesgue spaces, it is possible to deducefor a non–empty range of µ ∈ (1, 2) that the symmetric gradients ε(uk) are uniformlybounded in certain non–weighted Lploc–spaces with p > 1. By arbitariness of the intially

chosen minimising sequence, this implies u ∈ W1,ploc(Ω;Rn) by Korn’s Inequality and thus

establishes the higher integrability of generalised minima of (1.1) subject to the above localBMO–assumption. In summary, the strategy leads to the following theorem which shall beestablished in [26]:

Theorem 3. Let f ∈ C2(Rn×nsym ) be a convex integrand of linear growth and Ω an openand bounded Lipschitz subset of Rn. For every n ∈ N with n ≥ 2 there exists a number1 < µ(n) < 2 and an exponent p ≥ 1 such that the following holds: If f is µ–elliptic withµ 6 µ(n), then any generalised minimiser u ∈ BD(Ω) ∩ BMOloc(Ω;Rn) of the functional F

given by (1.1) belongs to W1,ploc(Ω;Rn) for some p ≥ 1.

Let us remark that it is plausible for the preceding result to hold true for a fairly largerrange of µ than described in Theorem 3, however, this seems hard to be achieved by useof the above method. In particular, a Sobolev regularity result regarding the limiting caseµ = 3 (an instance of which is the area integrand f(Z) :=

√1 + |Z|2) would be desirable.

Finally, the strategy outlined above allows to weaken even the local BMO–assumption inview of uniform local Lp–bounds on the single members of minimising sequences, and shallbe addressed in a future publication.

4. A–Convex Functionals

It is a natural extension of the problems and results outlined in the previous sections toreplace the symmetric gradient operator by an elliptic differential operator of the form (2.1).In turn, within the framework of section 2, we aim for existence and regularity propertiesfor minima functionals of the form

F[v] :=

∫Ω

f(A[D]u) dx

over Dirichlet classes D := u0+WA,10 (Ω), where f : W → R is of linear growth, thus verifying

(1.2) with the obvious modifications. The spaces WA,1(Ω) are defined similarly to W1,1 or

LD, namely, we say that a measurable map v : Ω→ V belongs to WA,1(Ω) if and only if v ∈L1(Ω;V ) and the weak differential expression A[D]u belongs to L1(Ω;W ). Equipped with

the canonical norm, one defines WA,10 (Ω) as the closure of C1

c(Ω;V ) with respect to the normtopology. Similarly, we define the space of functions of bounded A[D]–variation BVA(Ω) toconsist of all v ∈ L1(Ω;V ) for which the distributional differential expression A[D]v can berepresented by a finite W–valued Radon measure. Based on the linear growth assumption,the direct method in conjunction with the obvious changes of section 1.1 would lead to asatisfactory existence theory within the class of functions bounded A[D]–variation providedthe trace operator for such function spaces would be well–understood. This, however, is notthe case by now: For instance, even if A[D] is an elliptic operator in the sense of section

2, elements of BVA or WA,1 do not neccesarily possess traces in L1(∂Ω;V ) for arbitraryLipschitz domains Ω ⊂ Rn:

Remark 2. For n = 2, let the trace–free symmetric gradient operator defined by εD(v) :=ε(v)− 1

2 div(v)12×2 with the (2× 2)–unit matrix. In this situation, εD is an elliptic, linear,

70 FRANZ GMEINEDER

homogeneous first order differential operator on R2 from V = R2 to W = R2×2. IdentifyingR2 ∼= C, it is easy to see that ker(εD) contains all holomorphic functions. To argue that

elements of WεD,1(B(0, 1);R2)('WεD,1(D;C)) with the unit disk D ⊂ C do not have tracesin L1(∂ B(0, 1);R2)(' L1(∂D;C)), consider f : D 3 z 7→ 1/(z−1) ∈ C. Then f ∈ L1(D;C), is

holomorphic and thus belongs to WεD,1(D;C) whereas it is easy to see that∫∂D |f(z)|dz =

∞.

The previous example is due to Fuchs & Repin [21], and motivates the characterisation

of all A[D] such that the corresponding spaces WA,1(D;C) possess trace space L1(∂Ω;V )at least for the large class of bounded Lipschitz subsets Ω of Rn. For such operators, theSobolev regularity result, Theorem 3 is easily shown to hold true as its proof does not use thespecific structure of the symmetric gradient operator. In this respect, it is important to notethat the techniques available in the literature – so for instance Strang & Temam’s approach[39] in the case of BD – which are taylored for the symmetric gradient case, do not apply tothe setting of arbitrary elliptic differential operators of the form (2.1) without substantialmodifications. We hope to succesfully tackle this problem in a future publication.

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Mathematical Institute, University of Oxford, Andrew Wiles Building, OX2 6HG Oxford,United Kingdom


URL: https://www.maths.ox.ac.uk/people/franz.gmeineder

JEPE Vol 2, 2016, p. 73-85

ON THE REGULARIZING EFFECT OF SOME ABSORPTION AND

SINGULAR LOWER ORDER TERMS IN CLASSICAL DIRICHLET

PROBLEMS WITH L1 DATA

LINDA MARIA DE CAVE AND FRANCESCANTONIO OLIVA

Abstract. We are interested in existence and regularity results concerning the solutionto the following problem

−∆u+ us =f(x)

uγin Ω,

u > 0 in Ω,

u = 0 on ∂Ω,

where Ω is an open and bounded subset of RN , 0 < γ ≤ 1, s ≥ 1 and f is a nonnegative

function that belongs to some Lebesgue space.

1. Introduction

In this paper we investigate the interaction between two regularizing terms in the followingsemilinear problem

(1)

−∆u+ us =

f(x)

uγin Ω,

u > 0 in Ω,

u = 0 on ∂Ω,

where Ω is an open and bounded subset of RN , N > 2, 0 < γ ≤ 1, s ≥ 1 and f is just anonnegative L1(Ω) function.The motivations in the study of the above problem mainly arise by two papers, [3] and [4].In [3] the authors investigate the regularizing effect of the term us on the solution to thefollowing classical problem

(2)

−∆u = f(x) in Ω,

u = 0 on ∂Ω,

when the term us is added in the left hand side of (2), namely when we consider the following

(3)

−∆u+ us = f(x) in Ω,

u = 0 on ∂Ω.

2010 Mathematics Subject Classification. Primary: 35J61; Secondary: 35J75.

Key words and phrases. Semilinear elliptic equations, Singular elliptic equations, Regularizing effects.Received 04/09/2016, accepted 04/11/2016.

73

74 LINDA MARIA DE CAVE AND FRANCESCANTONIO OLIVA

We refer to the term us in (3) as the absorption term.On the other side, in [4], the authors analyze the following singular problem

(4)

−∆u =

f(x)

uγin Ω,

u > 0 in Ω,

u = 0 on ∂Ω,

discovering, among other things, a regularizing effect of the singular termf(x)

uγ, to which

we refer as the singular sourcing term, on the solution to the classical problem (2), whenthis term replaces the non-singular right hand side.

Here we want to focus on a possibly double regularization effect on the solution to (2)when we add both the absorption term on the left hand side and the singular sourcing termon the right hand side.

For the sake of completeness, we start recalling some literature regarding the above men-tioned problems.Firstly we consider the following

(5)

−∆v = f(x) in Ω,

v = 0 on ∂Ω,

where f ∈ Lr(Ω), r ≥ 1.If r = 1, by Theorem 1 of [2] (see also Theorem 1 in [1]), we can ensure the existence of at

least a distributional solution v to (5) such that v ∈W 1,q0 (Ω for every q <

N

N − 1. Moreover,

by the classical Calderon-Zygmund theory for infinite energy solution, if 1 < r < (2∗)′ it

is guaranteed the existence of at least a distributional solution v ∈ W 1,r∗

0 (Ω) to (5) (seeTheorem 3 of [2]). If instead m ≥ (2∗)′, by the classical variational results, problem (5) hasa finite energy solution.Thus, the regularization effects mentioned above will be with respect to the Sobolev regu-larity of the solution v to the classical problem (5).Now we turn our attention recalling some results contained in the papers [4] and [3].In [4] the authors study the following problem

(6)

−∆z =f(x)

zγin Ω,

z = 0 on ∂Ω,

and they prove the existence of a solution z to (6) such that:

(i) if γ = 1 and 0 ≤ f ∈ L1(Ω), then z ∈ H10 (Ω),

(ii) if γ < 1 and 0 ≤ f ∈ L( 2∗1−γ )′(Ω), then z ∈ H1

0 (Ω),

(iii) if γ < 1 and 0 ≤ f ∈ Lr(Ω) with r < ( 2∗

1−γ )′, then z ∈W 1,q0 (Ω) with q = Nr(γ+1)

N−r(1−γ) .

ON THE REGULARIZING EFFECT... 75

In all the above cases, the singular sourcing term has a regularizing effect on the Sobolevregularity of the solution z, compared to the one of the distributional solution to problem(5). Indeed, for example, it is immediate to see that, if γ = 1, N > 2 and f ∈ L1(Ω), then(6) has a finite energy solution, while, if γ = 0, i.e. in the case of problem (5), the solution

belongs only to W 1,q0 (Ω) for every q <

N

N − 1.

On the other hand, in [3], the authors perturb problem (5) with an absorption term. Theyconsider the following

(7)

−∆w + ws = f(x) in Ω,

w = 0 on ∂Ω,

and, in [3, Theorem 5], it is proved that, if f ∈ L1(Ω) and s >N

N − 2, then there exists w

solution to (7) belonging to W 1,q0 (Ω) for all q <

2s

(s+ 1). Since

N

N − 1<

2s

s+ 1⇐⇒ s >

N

N − 2,

also in this case there is a regularization effect of the absorption term with respect to theSobolev regularity of v solution to (5).It is worth recalling also the paper [5], where the author strengthens the assumption on thedatum f and studies the regularity of a solution w to (7) depending simultaneously on theregularity of f and on the exponent s. To be more precise, in [5] it is proved that, if s ≥ 1 and

f is such that |f | log(1 + |f |) ∈ L1(Ω), then w has the limiting regularity w ∈ W1, 2s

(s+1)

0 (Ω)

and that, if f ∈ Lr(Ω) with r ∈(1, (2∗)′

), then w ∈ W

1, 2sr(s+1)

0 (Ω) if s ∈(

1(2r−1) ,

1(r−1)

),

while w ∈ H10 (Ω) if s > 1

r−1 .

As already said, we want to understand the mutual behavior of the two regularizing ef-fects. In particular, we want to exploit if the combination of the two regularizations leadsto an even more regular solution or, alternatively, if the regularization given by the sin-gular sourcing term to the solution is too strong to expect some effects when we add alsoan absorption term. As we will see in the next sections, both the addition of a singularsourcing term to a problem with only an absorption term and the addition of an absorptionterm to a problem with only a singular sourcing term, will improve the regularity of thesolution. Anyway we underline that, when we add a singular sourcing term to a problemwith an absorption term, we improve the regularity of the solution anyhow we choose thesingularity exponent γ > 0, while, when we add an absorption term to a problem with asingular sourcing term, in order to improve the regularity we have to consider an exponents ≥ 1 large enough, namely we are not able to improve the regularity of the solution for eachs ≥ 1. This tells us that a singular sourcing term gives rise to a more powerful regularizationwith respect to the one generated by an absorption term.


1.1. Notations. For a fixed k > 0, we define the truncation functions Tk : R → R andGk : R→ R as follows

Tk(s) := max(−k,min(s, k)),

Gk(s) :=(|s| − k)+ sign(s).

We will denote with R∗ the set R \0, with R+ the set t ∈ R s.t. t > 0, with r∗ the

Sobolev conjugate of 1 ≤ r < N, given byNr

N− r, and with r′ =

r

r − 1the Holder conjugate

of 1 < r < ∞ (if r = 1 we define r′ = ∞, if r = ∞ we define r′ = 1). Moreover, if nootherwise specified, we will denote by c several positive constants whose value may changefrom line to line and, sometimes, on the same line. These values will only depend on thedata (for instance c can depend on Ω, γ, s, N) but they will never depend on the indexesof the sequences we will introduce.

2. Existence of a regularized solution

We want to prove existence and regularity results for problem (1) in case Ω is an open

bounded subset of RN (N > 2), 0 < γ ≤ 1, s ≥ 1 and 0 ≤ f ∈ Lr(Ω) with r ≥ 1. We arelooking for a distributional solution u that belongs to a smaller Sobolev space compared tothe ones where v, z, w solutions to, respectively, (5), (6) and (7) belong.For the sake of completeness we state what we mean for distributional solution (or simplysolution) to (1).

Definition 1. A function u ∈ W 1,10 (Ω) is a distributional solution to problem (1) in case

γ ≤ 1, s ≥ 1 and f ∈ Lr(Ω) with r ≥ 1 if

(8) ∀ω ⊂⊂ Ω exists cω > 0 s.t. u ≥ cω a.e. in ω,

us ∈ L1(Ω),

and ∫Ω

∇u · ∇ϕ+

∫Ω

usϕ =

∫Ω

fϕ

uγ∀ϕ ∈ C1

c (Ω).

Our main result is the following.

Theorem 1. Let γ ≤ 1, s ≥ 1 and 0 ≤ f ∈ Lr(Ω) with r ≥ 1. Then there exists a distribu-tional solution u to (1). Moreover u belongs to H1

0 (Ω) ∩ Ls+1(Ω) if

(i) γ = 1, f ∈ L1(Ω) or

(ii) γ < 1, f ∈ Lr(Ω) for some r > 1 and s ≥ 1−rγr−1 or

(iii) γ < 1, f ∈ Ls+1s+γ (Ω)

while if

(iv) γ < 1 and f ∈ L1(Ω), then u ∈W 1,2(s+γ)s+1

0 (Ω) ∩ Ls+γ(Ω).


2.1. Approximating problems. In order to prove Theorem 1, we will work by approxi-mation, namely by introducing the following

(9)

−∆un,k + Tk(|un,k|s−1un,k) =fn(x)(

|un,k|+ 1n

)γ in Ω,

un,k = 0 on Ω,

where n, k ∈ N, 0 ≤ fn(x) := Tn(f(x)) ∈ L∞(Ω), γ ≤ 1 and s ≥ 1.Thanks to [6, Theoreme 2], we know that there exists un,k ∈ H1

0 (Ω) weak solution to (9) foreach n, k ∈ N fixed. Moreover un,k ∈ L∞(Ω) for all n, k ∈ N since, if m ≥ 1 is fixed, takingGm(un,k) ∈ H1

0 (Ω) as test function in (9) and using that Gm(un,k) and Tk(|un,k|s−1un,k)have the same sign of un,k, we immediately find that∫

Ω

|∇Gm(un,k)|2 ≤∫

Ω

fnGm(un,k),

and so we can proceed as in [7] to end up with un,k ∈ L∞(Ω). Moreover the previousL∞ estimate is independent from k ∈ N. Now taking un,k as a test function in the weakformulation of (9), we find that un,k is bounded in H1

0 (Ω) with respect to k for n ∈ N fixed.Since un,k is bounded in L∞(Ω) independently on k, for each n ∈ N fixed we choose kn largeenough to obtain the following scheme of approximation

(10)

−∆un + |un|s−1un =fn(x)(|un|+ 1

n

)γ in Ω,

un = 0 on ∂Ω,

where un ∈ H10 (Ω) ∩ L∞(Ω) is given by un,kn .

As concerns the sign of un, taking u−n := min(un, 0) ∈ H10 (Ω) ∩ L∞(Ω) as test function in

(10), we find ∫Ω

|∇u−n |2 +

∫Ω

|un|s−1(u−n )2 =

∫Ω

fn

(|un|+ 1n )γ

u−n ≤ 0,

and so that un ≥ 0 almost everywhere in Ω.Now we prove some local positivity property that will guarantee that the limit of the ap-proximations (10) satisfies (8).

Proposition 1. For each n ∈ N fixed, the nonnegative un ∈ H10 (Ω) ∩ L∞(Ω) weak solution

to (10) is nondecreasing in n ∈ N and it results

(11) ∀ω ⊂⊂ Ω ∃ cω > 0 (independent of n ∈ N) s.t. un ≥ cω in ω ∀n ∈ N .

Proof. We can prove that the sequence un is nondecreasing in n ∈ N proceeding preciselyas in [4, Lemma 2.2], namely taking (un − un+1)+ := max(un − un+1, 0) ∈ H1

0 (Ω) ∩ L∞(Ω)as test function in the difference between the problem solved by un and the one solved byun+1, so we will omit the details. To prove (11), we will instead use that

(12) un ≥ u1 ∀n ∈ N a.e. in Ω

and we will apply the strong maximum principle to u1 ∈ H10 (Ω) ∩ L∞(Ω), that solves−∆u1 + us1 =

f1(x)

(u1 + 1)γ ≥

f1(x)(‖u1‖L∞(Ω) + 1

)γ ≥ 0 in Ω

u1 = 0 on ∂Ω.


Indeed, since u1,∆u1 ∈ L1loc(Ω), u1 ≥ 0 almost everywhere in Ω, ∆u1 ≤ us1 and∫

0

(ts+1

)− 12 =∞ ⇐⇒ s ≥ 1,

we can apply [8, Theorem 1] and deduce that

∀ω ⊂⊂ Ω ∃ cω > 0 s.t. u1 ≥ cω in ω.

Then (11) follows from (12).

2.2. A priori estimates. Now we need some compactness results on the sequence of ap-proximating solutions un, at least up to subsequences.

Proposition 2. Let n ∈ N and un ∈ H10 (Ω) ∩ L∞(Ω) be a solution to (10) where s ≥ 1.

(i) If one of the following holdsγ = 1, f ∈ L1(Ω) ,

γ < 1, f ∈ Lr(Ω) for some r > 1 and s ≥ 1−rγr−1 ,

γ < 1, f ∈ Ls+1s+γ (Ω) ,

then un is bounded in H10 (Ω) ∩ Ls+1(Ω).

(ii) If instead γ < 1 and f ∈ L1(Ω), then un is bounded in W1,

2(s+γ)s+1

0 (Ω) ∩ Ls+γ(Ω).

Proof. The case (i). Let us take un ∈ H10 (Ω) ∩ L∞(Ω) as test function in (10). We obtain

(13)

∫Ω

|∇un|2 +

∫Ω

us+1n ≤

∫Ω

fnu1−γn .

If γ = 1, we immediately find that un is bounded in H10 (Ω) and in Ls+1(Ω).

If γ < 1, we apply Young’s inequality with weights (ε, c(ε)) and exponents (r, r′) on the righthand side of the previous, obtaining∫

Ω

|∇un|2 +

∫Ω

us+1n ≤ 1

c(ε)

∫Ω

frn + ε

∫Ω

u(1−γ)r′

n ≤ 1

c(ε)

∫Ω

fr + εc

∫Ω

us+1n .

If ε is small enough, we deduce the following estimate∫Ω

|∇un|2 + c(Ω, ε)

∫Ω

us+1n ≤ 1

c(ε)

∫Ω

fr ≤ c.

If γ < 1 and f ∈ Ls+1s+γ (Ω), we apply Young’s inequality with weights (ε, c(ε)) and exponents(

s+ 1

s+ γ,s+ 1

1− γ

)on the right hand side of (13). Proceeding as before, we can easily prove the last assertion.The case (ii). Let us take (un + ε)γ − εγ ∈ H1

0 (Ω) ∩ L∞(Ω) as test function in (10), where0 < ε < 1

n . We obtain

γ

∫Ω

|∇un|2(un + ε)γ−1 +

∫Ω

usn((un + ε)γ − εγ) ≤∫

Ω

fn

and so, in particular, we deduce that∫Ω

usn((un + ε)γ − εγ) ≤∫

Ω

f,


that, letting ε→ 0, implies ∫Ω

us+γn ≤∫

Ω

f.

Moreover we deduce ∫Ω

|∇un|2

(un + ε)1−γ ≤ c.

Now, if q < 2, applying Holder inequality with exponents2

qand

2

2− q, we find

∫Ω

|∇un|q =

∫Ω

|∇un|q

(un + ε)(1−γ) q2(un + ε)(1−γ) q2 ≤ c

(∫Ω

(un + ε)(1−γ)q2−q

)1− q2.

Finally we choose q such that

(1− γ)q

2− q= s+ γ.

It is easy to verify that

q =2(s+ γ)

(s+ 1)< 2,

and this gives us the result.

2.3. Proof of Theorem 1.

Proof of Theorem 1. We have to pass to the limit the approximation (10).Thanks to the a priori estimates of Proposition 2, by weak convergence we can pass to thelimit in the left hand side of the distributional formulations of the approximating problemswith C1

c (Ω) test functions. For what concerns the right hand side, using (11) we find∣∣∣∣ fnϕ

(un + 1n )γ

∣∣∣∣ ≤ ∣∣∣∣ fϕ

cγsuppϕ

∣∣∣∣ ∀ϕ ∈ C1c (Ω).

Then, thanks to Lebesgue Theorem, we can pass to the limit also in the right hand side ofthe distributional formulation of (10). This concludes the proof.

2.4. Some comments on the regularizing effect. First of all, it is easy to verify that,if

(14) γ < 1 and s >N + 2

N − 2

then

s+ 1

s+ γ<

(2∗

1− γ

)′.

Since f ∈ L( 2∗1−γ )

′

(Ω) is the weaker assumption on the datum in order to find a priori esti-mates in H1

0 (Ω) for the sequence of approximating solutions to (4) (see [4, Theorem 5.1]),it follows that, if we add the term us, with s satisfying (14), in the left hand side of (4),we find a priori estimates in H1

0 (Ω) for the sequence of approximating solutions also for lessregular data.


Furthermore, if f ∈ L1(Ω) and γ < 1, the Sobolev space in which the sequence of approxi-

mating solutions to (4) is bounded is given by W1,

N(γ+1)N−(1−γ)

0 (Ω) (see [4, Theorem 5.6]). It iseasy to verify that, if

(15) γ < 1, and s >N + 2γ

N − 2

thenN(γ + 1)

N − (1− γ)<

2(s+ γ)

s+ 1.

So we have another regularizing effect of the lower order term us, with s such that (15)holds, on the a priori estimates for the approximating solutions.

Finally we recall that, if f ∈ L1(Ω) and s >N

N − 2, then the sequence of approximating

solutions to (3) is bounded in W 1,q0 (Ω) for all q ∈

[1,

2s

(s+ 1)

)(see [3, Theorem 5]). Since

2s

(s+ 1)<

2(s+ γ)

(s+ 1)⇐⇒ γ > 0,

we immediately obtain the, if we perturb the right hand side of (3) through the singular term1

uγwith γ > 0, we find more regular a priori estimates on the sequence of approximating

solutions.

3. More general singular and absorption terms: combined effect on theexistence and regularity of the solution

Here we intend to analyze a more general case, namely the following

(16)

−∆u+ g(u) = h(u)f(x) in Ω,

u > 0 in Ω,

u = 0 on ∂Ω.

The absorption term g : R→ R is a nondecreasing and continuous function on R satisfyingthe assumptions listed below:

(g1) g(t)t ≥ 0 for all t ∈ R,

(g2)

∫0

(g(t)t)− 1

2 =∞,

(g3) g(t) ≥ ts for all t ≥ 0 and for some s ≥ 1.

The singular sourcing term h : R∗ → R+ is a nonincreasing function on R∗, continuous andbounded on R∗ such that

(h1) limt→0

h(t) = +∞,


and such that

(h2) there exist c > 0, δ′ > δ > 0 and γ, θ ∈ (0, 1] s.t.

h(t) ≤ c

tθif t < δ

h(t) ≤ c

tγif t > δ′.

The example we have in mind for the function g is

g(t) =

0 if t ≤ 0

ts if 0 ≤ t ≤ 1

ts+1 if t ≥ 1,

while the function h is a general singular nonincreasing function with two different behaviorsnear zero and near infinite.This time the definition of distributional solution is as follows.

Definition 2. Let g : R → R be a nondecreasing and continuous function satisfying (g1),(g2) and (g3) and let h : R∗ → R+ be a nonincreasing function on R∗, continuous and

bounded on R∗, satisfying (h1) and (h2). Under these assumptions a function u ∈W 1,10 (Ω)

is a distributional solution to problem (16), with f ∈ Lr(Ω) and r ≥ 1, if

(17) ∀ω ⊂⊂ Ω exists cω > 0 s.t. u ≥ cω a.e. in ω,

g(u) ∈ L1(Ω),

and

(18)

∫Ω

∇u · ∇ϕ+

∫Ω

g(u)ϕ =

∫Ω

h(u)fϕ ∀ϕ ∈ C1c (Ω).

Remark 1. We underline that property (17) implies in particular that h(u)f ∈ L1loc(Ω), as

it was in the case with monotone singularity given by h(u) =1

uγ(γ > 0). Consequently,

the right hand side of (18) is well defined.

We intend to prove the following result.

Theorem 2. Let g : R → R be a nondecreasing and continuous function satisfying (g1),(g2) and (g3) and let h : R∗ → R+ be a nonincreasing function on R∗, continuous andbounded on R∗, satisfying (h1) and (h2). If f ∈ Lr(Ω) with r ≥ 1, then there exists adistributional solution u to (16). Moreover u belongs to H1

0 (Ω) ∩ Ls+1(Ω) if

(i) θ, γ = 1, f ∈ L1(Ω) or

(ii) θ, γ < 1, f ∈ Lr(Ω) for some r > 1 and s ≥ 1−rγr−1 or

(iii) θ, γ < 1, f ∈ Ls+1s+γ (Ω),

while if

(iv) θ < γ < 1 and f ∈ L1(Ω), then u ∈W 1,2(s+γ)s+1

0 (Ω) ∩ Ls+γ(Ω).


Since Theorem 2 can be seen as an extension of Theorem 1, for shortness we will omit somedetails in the proof, referring to the corresponding ones in the proof of Theorem 1.

Proof of Theorem 2. As before, we introduce the following scheme of approximation forproblem (16)

(19)

−∆un,k + Tk(g(un,k)) = hn(un,k)fn(x) in Ω,

un,k = 0 on ∂Ω,

where n, k ∈ N, 0 ≤ fn(x) := Tn(f(x)) ∈ L∞(Ω) and hn(x) := Tn(h(x)).Thanks to [6, Theoreme 2], we know that there exists un,k ∈ H1

0 (Ω) weak solution to (19) foreach n, k ∈ N fixed. As in Theorem 1, we can show that un,k ∈ L∞(Ω) for all n, k ∈ N andthat this L∞-estimate is independent on k and, taking un,k as a test function in the weakformulation of the previous, we find once again that un,k is bounded in H1

0 (Ω) with respectto k for n ∈ N fixed. Since un,k is bounded in L∞(Ω) independently on k, as already donefor problem (9) we choose k large enough to obtain the following scheme of approximation

(20)

−∆un + g(un) = hn(un)fn(x) in Ω,

un = 0 on ∂Ω,

where un ∈ H10 (Ω) ∩ L∞(Ω) is given by a certain un,k for a k large enough.

Moreover, analogously to what done in Proposition 1, we are able to prove that the sequenceun is nonnegative and nondecreasing in n ∈ N.In order to prove the local positivity of un, we will use, once again, the strong maximumprinciple applied to u1 ∈ H1

0 (Ω) ∩ L∞(Ω), that solves−∆u1 + g(u1) = h1(u1)f1 ≥ h1(||u1||L∞(Ω))f1 ≥ 0 in Ω,

u1 = 0 on ∂Ω.

Thus, thanks to (g2), we can apply [8, Theorem 1] and deduce that

∀ω ⊂⊂ Ω ∃ cω > 0 s.t. u1 ≥ cω in ω.

From now on we divide the proof in the different cases with respect to the parameters.

Proof of (i). In this case, it is sufficient to take un as a test function in the weak for-mulation of (20) obtaining∫

Ω

|∇un|2 +

∫Ω

us+1n ≤

∫Ω

|∇un|2 +

∫Ω

g(un)un

≤ c∫un≤δ

fn +

∫δ<un<δ′

hn(un)fnun + c

∫un≥δ′

fn

≤ 2c

∫Ω

f + δ′||h||L∞(δ,δ′)

∫Ω

f.

Thus, un is bounded in H10 (Ω)∩Ls+1(Ω) and we have no problems in passing to the limit in

the approximation (20) with C1c (Ω) test functions, using Lebesgue Theorem for the singular

sourcing term and monotone convergence Theorem for the absorption term.

Proof of (ii). If θ, γ < 1, we take un as test function and we apply Young’s inequality


with weights (ε, c(ε)) and exponents (r, r′) on the last term of the right hand side of theweak formulation, obtaining∫

Ω

|∇un|2 +

∫Ω

us+1n ≤

∫Ω

|∇un|2 +

∫Ω

g(un)un

≤ c∫un≤δ

fnu1−θn + δ′

∫δ<un<δ′

hn(un)fn + c

∫un≥δ′

fnu1−γn

≤ cδ1−θ∫

Ω

f + δ′||h||L∞((δ,δ′))

∫Ω

f +1

c(ε)

∫Ω

frn + cε

∫Ω

u(1−γ)r′

n

≤ cδ1−θ∫

Ω

f + δ′||h||L∞((δ,δ′))

∫Ω

f +1

c(ε)

∫Ω

fr + cε

∫Ω

us+1n .

Thus, if ε is small enough, we deduce

(21)

∫Ω

|∇un|2 + c

∫Ω

us+1n ≤ c.

As already done in case (i), we can pass to the limit the approximation scheme (20) in orderto prove the existence of a solution u ∈ H1

0 (Ω) ∩ Ls+1(Ω) to problem (16).

Proof of (iii). For this case we can proceed in a similarly to what done for (ii), obtain-ing∫

Ω

|∇un|2 +

∫Ω

us+1n ≤

∫Ω

|∇un|2 +

∫Ω

g(un)un

≤ c∫un≤δ

fnu1−θn + δ′

∫δ<un<δ′

hn(un)fn + c

∫un≥δ′

fnu1−γn

≤ cδ1−θ∫

Ω

f + δ′||h||L∞((δ,δ′))

∫Ω

f +1

c(ε)

∫Ω

fs+1s+γn + cε

∫Ω

us+1n .

This allows us to deduce (21) once again and to obtain a solution u ∈ H10 (Ω) ∩ Ls+1(Ω) to

(16).

Proof of (iv). This time we take (un + ε)γ − εγ ∈ H10 (Ω) ∩ L∞(Ω) as test function in

(20), where 0 < ε < 1n . We obtain

γ

∫Ω

|∇un|2(un + ε)γ−1 +

∫Ω

usn((un + ε)γ − εγ)

≤ γ∫

Ω

|∇un|2(un + ε)γ−1 +

∫Ω

g(un)((un + ε)γ − εγ)

≤ c∫un≤δ

fn(un + ε)γ

uθn+

∫δ≤un≤δ′

fnhn(un)(un + ε)γ + c

∫un≥δ′

fn(un + ε)γ

uγn,

and so we have in particular∫Ω

usn((un + ε)γ − εγ) ≤ c∫un≤δ

fn(un + ε)γ

uθn+

∫δ≤un≤δ′

fnhn(un)(un + ε)γ

+ c

∫un≥δ′

fn(un + ε)γ

uγn,


that, letting ε→ 0, implies ∫Ω

us+γn ≤ cδγ−θ + c.

Moreover, we can reason as in proof case (ii) of Proposition 2 obtaining that un is bounded

in W1,

2(s+γ)s+1

0 (Ω) with respect to n. From here we conclude as in case (i) of the currentTheorem. This concludes the proof.

3.1. A final remark. Here we want to explain the motivations that brought us to investi-gate only the case with singularity indexes θ, γ ≤ 1.First of all, we had in mind the idea of understanding the regularization effect of a singu-lar sourcing term joined to an absorption term and, the regularization effect of a singularsourcing term with an exponent of singularity far from the origin greater than one, is itselfvery strong, also without the addition of an absorption term.Indeed, we consider the following

(22)

−∆u = h(u)f in Ω

u = 0 on ∂Ω,

where the singularity h : R∗ → R+ is assumed to be of class Cb(R∗) along with the followinggrowth conditions

∃ c, δ > 0, θ ≤ 1 s.t. h(s) ≤ csθ

if s < δ

∃ c > 0, δ′ > δ, γ ≥ 1 s.t. h(s) ≤ csγ if s > δ′.

Formally, if we take u itself as a test function in (22) we obtain∫Ω

|∇u|2 ≤ c∫u≤δ

fu1−θ +

∫δ<u<δ′

h(u)fu+ c

∫u≥δ′

fu1−γ

≤ cδ1−θ∫u≤δ

f + δ′||h||L∞((δ,δ′))

∫δ<u<δ′

f + cδ′1−γ∫u≥δ′

f,

that gives an H1-estimate for the solution u. Clearly, the above argument can be tightenedpassing to the limit in a suitable approximation scheme. For the motivation explained above,it was not interesting for us the case γ > 1.For what concerns the case θ > 1, the regularizing effect of the singular sourcing term is onlylocal for problem (22), as well explained for the model case in [4] where they find solutionsu ∈ H1

loc(Ω), and we have chosen to omit the study of any local regularization effect obtainedby adding an absorption term.Therefore, for exponents greater than one, it seemed to be not of interest the combinationof the two regularization effects, the one of absorption term with the one of the singularsourcing term.

References

[1] L. Boccardo and T. Gallouet, Nonlinear Elliptic and Parabolic Equations involving Measures Data, J.Funct. Anal., 87 (1989), pp. 149-169.

[2] L. Boccardo and T. Gallouet, Nonlinear Elliptic Equations with Right Hand Side Measures, Comm.

Partial Differential Equations, 17 (1992), pp. 641-655.

[3] L. Boccardo, T. Gallouet and J. L. Vazquez, Nonlinear elliptic equations in RN without growth restric-

tions on the data, J. Differential Equations, 105 (1993), pp. 334-363.


[4] L. Boccardo and L. Orsina, Semilinear elliptic equations with singular nonlinearities, Calc. Var. Partial

Differential Equations, 37 (2009), pp. 363-380.[5] G. R. Cirmi, Regularity of the solutions to nonlinear elliptic equations with a lower-order term, Non-

linear Anal., 25 (1995), pp. 569-580.

[6] J. Leray and J.-L. Lions, Quelques resultats de Visik sur les problemes elliptiques non lineaires par lesmethodes de Minty-Browder, Bull. Soc. Math. France, 93 (1965), pp. 97-107.

[7] G. Stampacchia, Equations elliptiques du second ordre a coefficients discontinus, Les Presses de

l’Universite de Montreal (1966).[8] J. L. Vazquez, A Strong Maximum Principle for Some Quasilinear Elliptic Equations, Appl. Math.

Optim., 12 (1984), pp. 191-202.

Linda Maria De Cave, Section de Mathematiques, Ecole Polytechnique Federale de Lausanne,

1015 Lausanne, Switzerland

E-mail address: [email protected], [email protected]

Francescantonio Oliva, Dipartimento di Scienze di Base e Applicate per l’ Ingegneria, Sapienza

Universita di Roma, Via Scarpa 16, 00161 Roma, ItalyE-mail address: [email protected]

JEPE Vol 2, 2016, p. 87-104

EXISTENCE OF WEAK SOLUTIONS OF AN UNSTEADY

THERMISTOR SYSTEM WITH p-LAPLACIAN TYPE EQUATION

JOACHIM NAUMANN

Abstract. In this paper, we consider an unsteady thermistor system, where the usual

Ohm law is replaced by a non-linear monotone constitutive relation between current andelectric field. This relation is modeled by a p-Laplacian type equation for the electrostatic

potential ϕ. We prove the existence of weak solutions of this system of PDEs under mixed

boundary conditions for ϕ, and a Robin boundary condition and an initial condition forthe temperature u.

1. Introduction

Let Ω ⊂ Rn (n = 2 or n = 3) be a bounded domain with Lipschitz boundary ∂Ω, and setQT = Ω× ] 0, T [ (0 < T < +∞).

Let J and q denote the electric current field density and the heat flux, respectively, of athermistor occupying the domain Ω under unsteady operating conditions. Then the balanceequations for the electric current and the heat flow within the thermistor material are thefollowing two PDEs

∇ · J = 0,∂u

∂t+∇ · q = f(x, t, u,∇ϕ) in QT ,

where ϕ = ϕ(x, t) and u = u(x, t) represent the electrostatic potential and the temperature,respectively (see, e.g., [29, Chap. 8]).

We make the following constitutive assumptions on J and q

J = σ(u, |E|

)E Ohm’s law, q = −κ(u)∇u Fourier’s law,

where

E = −∇ϕ density of the electric field,

σ = σ(u, |E|

)electrical conductivity,

κ = κ(u) thermal conductivity.

2010 Mathematics Subject Classification. 35J92, 35K20, 35Q79, 80A20.Key words and phrases. Thermistor system, Robin boundary condition, p-Laplacian, saturation of cur-

rent, self-heating.Received 05/09/2016, accepted 27/10/2016.

87

88 JOACHIM NAUMANN

With these notations the above system of PDEs takes the form

−∇ ·(σ(u, |∇ϕ|

)∇ϕ)

= 0 in QT ,(1.1)

∂u

∂t−∇ ·

(κ(u)∇u

)= f(x, t, u,∇ϕ) in QT .(1.2)

The function f = f(x, t, u,∇ϕ) represents a heat source that will be specified below (see(1.13) and (H3), Section 2).

We supplement system (1.1)–(1.2) by boundary conditions for ϕ and u, and an initialcondition for u. Without any further reference, throughout the paper we assume

∂Ω = ΓD ∪ ΓN disjoint, ΓD non-empty, open.

Define

ΣD = ΓD× ]0, T [ , ΣN = ΓN× ] 0, T [ .

We then consider the conditions

ϕ = ϕD on ΣD, J · n = 0 on ΣN ,(1.3)

q · n = g(u− h) on ∂Ω× ] 0, T [ ,(1.4)

u = u0 in Ω× 0(1.5)

(n = unit outward normal to ∂Ω). The first condition in (1.3) means that there is an appliedvoltage ϕD along ΣD, whereas the second condition characterizes electrical insulation of thethermistor along ΣN . The Robin boundary condition (1.4)1) means that the flux of heatthrough ∂Ω× ] 0, T [ is proportional to the temperature difference u − h, where g denotesthe thermal conductivity of the surface ∂Ω of the thermistor, and h represents the ambienttemperature (cf. [10], [15], [22], [29, Chap. 8] and [32] (nonlinear boundary conditions)).

We present two prototypes for the electrical conductivity σ. To this end, let σ0 : R →

R+2) be a continuous function such that

0 < σ∗ ≤ σ(u) ≤ σ∗ <∞ ∀ u ∈ R (σ∗, σ∗ = const).

We then consider the following functions

(1.6) σ(u, τ) = σ0(u)(δ + τ2)(p−2)/2, (u, τ) ∈ R× R+ (δ = const > 0, 1 < p < +∞)

and

(1.7) σ(u, τ) = σ0(u)τp−2, (u, τ) ∈ R× R+ (2 ≤ p < +∞).

The electrical conductivities which correspond to these functions σ = σ(u, τ) read

(1.8) σ(u, |E|

)= σ0(u)

(δ + |E|2

)(p−2)/2

and

(1.9) σ(u, |E|

)= σ0(u)|E|p−2,

1) This boundary condition is also called “Newton’s cooling law” or “third boundary condition”.2) R+ = [0,+∞ [ .

THERMISTOR SYSTEM WITH p-LAPLACIAN 89

respectively (E = electrical field density). Here, the factor σ0(u) characterizes the thermaldependence of the electrical conductivity of the thermistor material. Observing that E =−∇ϕ, equ. (1.1) takes the form of p-Laplacian equations

−∇ ·(σ0(u)

(δ + |∇ϕ|2

)(p−2)/2∇ϕ)

= 0,

resp.−∇ ·

(σ0(u)|∇ϕ|p−2∇ϕ

)= 0.

Let p = 2. Then both (1.8) and (1.9) lead to J = σ0(u)E. If the right hand side in (1.2)is of the form f = σ0(u)|∇ϕ|2 = J · E (Joule heat), (cf. (1.13) below), then (1.1)–(1.2)represents the “classical” thermistor system (see [1], [9], [15], [33]). This system has beenstudied in [18]–[20] with a degeneration of the coefficients σ0(u) and κ(u) (cf. also [10] fora similar degeneration of σ0(u)).

Remark 1. (The case 1 < p ≤ 2.) Let be σ = σ(u, τ) as in (1.6). Then Ohm’s law reads

(1.10) J = σ0(u)(δ + |E|2

)(p−2)/2E

(cf. (1.8)). To make things clearer, let I = |J | and V = |E| denote the current andvoltage, respectively, in an electrical conductor. Equ. (1.10) then gives the current-voltagecharacteristic

(1.11) I = σ0(u)(δ + V 2)(p−2)/2V.

If p = 2, then this current-voltage characteristic turns into the well-known linear (i.e.,Ohmic) characteristic I = σ0(u)V . If p is “sufficiently near to 1”, then (1.11) can be used asan approximation of current-voltage characteristics for transistors (see, e.g., [23], [31, Chap.6.2.2]).

The characteristic (1.11) continues to make sense if p = 1, i.e.,

(1.12) I =σ0(u)

(δ + V 2)1/2V.

This current-voltage characteristic is widely used to describe the effect of saturation ofcurrent in certain transistors under high electric fields (see, e.g., [27, Chap. 2.5] for details).The following figure gives an illustration of the relationship between the limit case p = 1and the effect of saturation of current.

Voltage V

Cur

rent

I

VS

I0

Fig. Current-voltage characteristic I vs. V (I0 = σ0(u))

90 JOACHIM NAUMANN

Broken line: I = I0(δ+V 2)(2−p)/2 V , 1 < p < 2 (cf. (1.11));

dotted line: I = I0(δ+V 2)1/2

V (cf. (1.12), i.e., asymptotic saturation of current I I0

when V increases;

bold-faced line: experimental data I vs. V of MOSFETs, i.e., linear slope I = σ0(u)V for

voltages V << VS (cf. (1.11) with p = 2), and saturation of current I = I0 for voltages

V ≥ VS (see, e.g., [23], [31, p. 304, fig 9]).

Finally, we notice that for the case δ = 0 and p = 1, Ohm’s law (1.10) and the current-voltage characteristic (1.11) have to be replaced by

J ∈ Br0(0) if E = 0, J =r0

|E|E if E 6= 0,

0 ≤ I ≤ r0 if V = 0, I = r0 if V > 0,

respectively, where Br0(0) = ξ ∈ Rn; |ξ| ≤ r0, r0 = r0(u) (cf. [21]).

Remark 2. (The case 2 ≤ p < +∞.) In [11], the author considers the steady case of (1.1)with σ = σ

(|∇ϕ|

), where

limτ→+∞

σ(τ)

τp−2= a > 0, p ≥ 2

(cf. (1.7)). Electrical conductors obeying the constitutive law J = −σ(|∇ϕ|

)∇ϕ are called

varistors (= varying resistors).Equ. (1.1) with this constitutive law is then studied under the boundary conditions

ϕ = 0 on Γ′D, ϕ = Φ on Γ′′D,∂ϕ

∂n= 0 on ΓN (ΓD = Γ′D ∪ Γ′′D disjoint),

where Φ is an unknown constant (cf. (1.3)). The constant Φ is related to ∇ϕ by a nonlocalboundary condition on Γ′′D which models a current limiting device (see, e.g., [15] for moredetails).

A second topic of [11] concerns the steady case of (1.1)–(1.2) with J = −σ(u)∇ϕ andf = σ(u)|∇ϕ|2 under analogous boundary conditions as above.

Similar studies of the steady case of (1.1)–(1.2) with J = −σ(u, ϕ)∇ϕ and f = σ(u, ϕ)|∇ϕ|2can be found in [12].

Another type of non-Ohmic current-voltage characteristics is

I =(σ0(x, u)V p(x)−2

)V, 2 ≤ p(x) < +∞ (x ∈ Ω),

where p = p(x) is a jump function (cf. (1.7) and (1.9)). The experimental findings whichlead to this characteristic, are presented in [14]. This characteristic is used to model bothOhmic and non-Ohmic behavior of the device material (i.e., x ∈ Ω; p(x) = 2 and x ∈Ω; 2 < pi(x) < +∞, respectively, (i = 1, . . . ,m)) (see also [24] for more details).

We present a prototype for the heat source term f in (1.2) which motivates hypotheses(H3) in Section 2.

Let be σ = σ(u, τ) as in (1.6) or (1.7). For (x, u, ξ) ∈ Ω× R× Rn we consider functionsf such that

(1.13)

f(x, u, ξ) = α(x, u, ξ)σ

(u, |ξ|

)|ξ|2,

α : Ω× R× Rn −→ R+ is Caratheodory,

0 ≤ α(x, u, ξ) ≤ α0 = const ∀ (x, u, ξ) ∈ Ω× R× Rn (α0 = const).


If α ≡ 1, then

f(x, u,∇ϕ) = σ(u, |∇ϕ|

)(−∇ϕ) · (−∇ϕ) = J ·E.

Let be α of the form

α(x, u, ξ) = α(x, u,−ξ)or

α(x, u, ξ) = α(x, u,−σ

(u, |ξ|

)ξ),

where α : Ω × R × Rn is a Caratheodory function such that 0 ≤ α ≤ 1 everywhere. Then(1.2) models a self-heating process with source term

f = αJ ·E,

where the factor

α = α(x, u,E) or α = α(x, u,J)

characterizes a loss of Joule heat (cf. [24] for more details).

The existence of weak solutions to the steady case of (1.1)–(1.4) has been proved for thefirst time in [24] for 2 < p < +∞ and in [17] for 2 ≤ p(x) < +∞ (n = 2 in both papers).Extensions of these results for measurable exponents p = p(x) such that 1 < p1 ≤ p(x) ≤p2 < +∞ (p1, p2 = const), and any dimension n have been recently presented in [7], [8].

In [28], we proved the existence of a weak solution of (1.1)–(1.5) when the functionτ 7→ σ(u, τ) is strictly monotone and f satisfies hypothesis (H3) below (see Section 2) whichincludes (1.13) as a special case. The aim of the present paper is to prove an analogousexistence result when τ 7→ σ(u, τ) is merely monotone whereas the function f , however, hasto satisfy a structure condition of type (1.13).

2. Weak formulation of (1.1)–(1.5)

We introduce the notations which will be used in what follows.By W 1,p(Ω) (1 ≤ p < +∞) we denote the usual Sobolev space. Define

W 1,pΓD

(Ω) =v ∈W 1,p(Ω); v = 0 a.e. on ΓD

.

This space is a closed subspace of W 1,p(Ω). Throughout the paper, we consider W 1,pΓD

(Ω)equipped with the norm

|v|W 1,p =

∫Ω

|∇v|pdx

1/p

.

Let X denote a real normed space with norm | · |X and let X∗ be its dual space. By〈x∗, x〉X we denote the dual pairing between x∗ ∈ X∗ and x ∈ X. The symbol Lp(0, T,X)(1 ≤ p ≤ +∞) stands for the vector space of all strongly measurable mappings u : ] 0, T [→ Xsuch that the function t 7→

∣∣u(t)∣∣X

is in Lp(0, T ) (cf. [4, Chap. III, §3; Chap. IV, §3], [5,

App.], [13, Chap. 1]). For 1 ≤ p < +∞, the spaces Lp(0, T ;Lp(Ω)

)and Lp(QT ) are linearly

isometric. Therefore, in what follows we identify these spaces.Let H be a real Hilbert space with scalar product (·, ·)H such that X ⊂ H densely and

continuously. Identifying H with its dual space H∗ via Riesz’ Representation Theorem, weobtain the continuous embedding H ⊂ X∗ and

(2.1) 〈h, x〉X = (h, x)H ∀ h ∈ H, ∀ x ∈ X.

92 JOACHIM NAUMANN

Given any u ∈ L1(0, T ;X) we identify this function with a function in L1(0, T ;X∗) anddenote it again by u. If there exists U ∈ L1(0, T ;X∗) such that

T∫0

u(t)α′(t)dtinX∗= −

T∫0

U(t)α(t)dt ∀ α ∈ C∞c ( ] 0, T [ ),

then U will be called derivative of u in the sense of distributions from ] 0, T [ into X∗ anddenoted by u′ (see [5, App.], [13, Chap. 21]).

Let 1 < p < +∞ be fixed. We make the following assumptions on the coefficients σ, κand the right hand side f in (1.1)–(1.2):

(H1)

σ : R× R+ → R+ is continuous,

c1τp − c2 ≤ σ(u, τ)τ2, 0 ≤ σ(u, τ) ≤ c3(1 + τ2)(p−2)/2

∀ (u, τ) ∈ R× R+, where c1, c3 = const > 0 and c2 = const ≥ 0;

(H2)

κ : R→ R+ is continuous,

0 < κ0 ≤ κ(u) ≤ κ1 ∀ u ∈ R, where κ0, κ1 = const,

and

(H3)

f : QT × R× Rn → R+ is Caratheodory,

0 ≤ f(x, t, u, ξ) ≤ c4(1 + |ξ|p

)∀ (x, t, u, ξ) ∈ QT × R× Rn, where c4 = const > 0.

It is readily seen that (H1) and (H3) are satisfied by the prototypes for σ and f we haveconsidered in Section 1.

Definition. Assume (H1)–(H3) and suppose that the data in (1.3)–(1.5) satisfy

ϕD ∈ Lp(0, T ;W 1,p(Ω)

);(2.2)

g = const, h = const;(2.3)

u0 ∈ L1(Ω).(2.4)

The pair

(ϕ, u) ∈ Lp(0, T ;W 1,p(Ω)

)× Lq

(0, T ;W 1,q(Ω)

) (1 < q <

n+ 2

n+ 1

)


is called weak solution of (1.1)–(1.5) if∫QT

σ(u, |∇ϕ|

)∇ϕ · ∇ζ dxdt = 0 ∀ ζ ∈ Lp

(0, T ;W 1,p

ΓD(Ω));(2.5)

ϕ = ϕD a.e. on ΣD;(2.6)

∃ u′ ∈ L1(0, T ;

(W 1,q′(Ω)

)∗);(2.7)

T∫0

⟨u′(t), v(t)

⟩W 1,q′dt+

∫QT

κ(u)∇u · ∇v dxdt+ g

T∫0

∫∂Ω

(u− h)v dxSdt

=

∫QT

f(x, t, u,∇ϕ)v dxdt ∀ v ∈ L∞(0, T ;W 1,q′(Ω)

);

(2.8)

u(0) = u0 in(W 1,q′(Ω)

)∗.(2.9)

From (H1) and (H3) it follows that f(·, ·, u,∇ϕ)∈L1(QT ). Therefore, u∈Lq(0, T ;W 1,q(Ω)

)(1 < q < n+2

n+1

)is standard for weak solutions of parabolic equations with right hand side in

L1 (see, e.g., the papers cited in [28]).

We notice that v ∈ L∞(0, T,W 1,q′(Ω)

)can be identified with a function in L∞(QT )

(cf. [28]). Hence, the integral on the right hand side of the variational identity in (2.8) iswell-defined.

To make precise the meaning of (2.9), let 2nn+2 < q < n+2

n+1 . Then nqn−q > 2 and q′ > n+ 2.

Identifying L2(Ω) with its dual, we obtain

W 1,q′(Ω) ⊂ W 1,q(Ω) ⊂ L2(Ω) ⊂(W 1,q′(Ω))∗.(2.10)

continuously compactly continuously

Therefore, u can be identified with an element in Lq(0, T ;

(W 1,q′(Ω)

)∗). Together with (2.7)

this implies the existence of a function u ∈ C([0, T ];

(W 1,q′(Ω)

)∗)such that

u(t) = u(t) for a.e. t ∈ [0, T ]

(see, e.g., [13, p. 45, Th. 2.2.1]).

On the other hand, there exists a uniquely determined u0 ∈(W 1,q′(Ω)

)∗such that

(2.11) 〈u0, z〉W 1,q′ =

∫Ω

u0z dx ∀ z ∈W 1,q′(Ω).

Thus, (2.9) has to be understood in the sense

u(0) = u0 in(W 1,q′(Ω)

)∗.

Remark 3. Let (ϕ, u) be a sufficiently regular solution of (1.1)–(1.5). We multiply (1.1)and (1.2) by smooth test functions ζ and v, respectively, satisfying the conditions

ζ = 0 on ΣD, v(·, T ) = 0 in Ω.

94 JOACHIM NAUMANN

Then we integrate the div-terms by parts over Ω and the term ∂u∂t v by parts over the interval

[0, T ]. It follows

−∫QT

u∂v

∂tdxdt+

∫QT


T∫0

∫∂Ω

(u− h)v dxSdt

=

∫Ω

u0v(·, 0)dx+

∫QT

f(x, t, u,∇ϕ)v dxdt.(2.12)

This variational formulation of initial/boundary-value problems for parabolic equations isfrequently used in the literature.

We notice that from a variational identity of type (2.12) it follows the existence of adistributional time derivative of u (see the arguments concerning (4.25) and (4.26) below).

Remark 4. Let (ϕ, u) be a weak solution of (1.1)–(1.5). From (2.8) it follows that, for any

z ∈W 1,q′(Ω),⟨u′(t), z

⟩W 1,q′ +

∫Ω

κ(u(x, t)

)∇u(x, t) · ∇z(x)dx+ g

∫∂Ω

(u(x, t)− h

)z(x)dxS

=

∫Ω

f(x, t, u(x, t),∇ϕ(x, t)

)z(x)dx(2.13)

for a.e. t ∈ [0, T ], where the null set in [0, T ] of those t for which (2.13) fails, does notdepend on z. We integrate (2.13) (with s in place of t) over the interval [0, t] (0 ≤ t ≤ T )and integrate the first term on the left hand side by parts. Using the above notation u and(2.11), we obtain

⟨u(t), z

⟩W 1,q′+

t∫0

∫Ω

κ(u(x, s)

)∇u(x, s) · ∇z(x)dxds+ g

t∫0

∫∂Ω

(u(x, s)−h

)z(x)dxSds

=

∫Ω

u0(x)z(x)dx+

t∫0

∫Ω

f(x, s, u(x, s),∇ϕ(x, s)

)z(x)dxds.(2.14)

Let be p = 2 and let be f(x, t, u, ξ) = σ0(u)|ξ|2((

(x, t), u, ξ)∈ QT ×R×Rn; cf. (1.13)

).

Taking z ≡ 1 in (2.14), we obtain

⟨u(t), 1

⟩W 1,q′ + g

t∫0

∫∂Ω

(u(x, s)− h

)dxSds =

∫Ω

u0(x)dx+

t∫0

∫Ω

J ·E dxds, t ∈ ]0, T ].

3. Existence of weak solutions

Our existence result for weak solutions of (1.1)–(1.5) is the following

Theorem. Assume (H1) and (H2). Suppose further that

(3.1)(σ(u, |ξ|

)ξ − σ

(u, |η|

)η)· (ξ − η) ≥ 0 ∀ u ∈ R, ∀ ξ, η ∈ Rn,


and

(3.2)

f(x, t, u, ξ) = α(x, t, u)σ

(u, |ξ|

)|ξ|2 ∀

((x, t), u, ξ

)∈ QT × R× Rn,

where α : QT × R→ R+ is Caratheodory,

0 ≤ α(x, t, u) ≤ α0 = const ∀((x, t), u

)∈ QT × R,

σ = σ(u, τ) as in (H1).

Let ϕD and u0 satisfy (2.2) and (2.4), respectively, and suppose that

(3.3) g = const > 0, h = const.

Then there exists a pair

(ϕ, u) ∈ Lp(0, T ;W 1,p(Ω)

)×( ⋂

1<q<(n+2)/(n+1)

Lp(0, T ;W 1,q(Ω)

))such that

(2.5) and (2.6) are satisfied,

∃ u′ ∈⋂

n+2<r<+∞L1(0, T ;

(W 1,r(Ω)

)∗),(3.4)

and for any n+ 2 < s < +∞ there holds

T∫0

〈u′, v〉W 1,sdt+

∫QT


T∫0

∫∂Ω

(u− h)v dxSdt

=

∫QT

f(x, t, u,∇ϕ)v dxdt ∀ v ∈ L∞(0, T ;W 1,s(Ω)

),

(3.5)

u(0) = u0 in(W 1,s(Ω)

)∗.(3.6)

Moreover, u satisfies ‖u‖L∞(L1) + λ

∫QT

|∇u|2(1 + |u|

)1+λdxdt

≤ c(1 + ‖u0‖L1 +

∥∥ |∇ϕD|∥∥pLp

), 0 < λ < 1 3)

(3.7)

u ∈⋂

1<r<(n+2)/n

Lr(0, T ;Lr(Ω)

).(3.8)

The proof of this theorem is a further development of the approximation method we usedin [28]. In this paper, the function τ 7→ σ(u, τ) is assumed to satisfy the condition of strictmonotonicity(

σ(u, |ξ|

)ξ − σ

(u|η|

)η)· (ξ − η) > 0 ∀ u ∈ R, ∀ ξ, η ∈ Rn, ξ 6= η.

This condition allows to prove that the sequence (∇ϕε)ε>0 converges a.e. in QT as ε → 0,where (ϕε, uε)ε>0 is an approximate solution of the problem under consideration. Therefore,the discussion in [28] includes the large class of source functions f characterized by (H3).

3) For notational simplicity, in what follows, for indexes we write Lp(X) in place of Lp(0, T ;X). If thereis no danger of confusion, we briefly write Lp in place of Lp(E) (E ⊂ Rm).

96 JOACHIM NAUMANN

However, due to (3.1), in the present paper we have to work only with the weak con-vergence of the sequence (ϕε)ε>0 in Lq

(0, T ;W 1,q(Ω)

)as ε → 0, which in turn makes the

structure condition (3.2) necessary for the passage to the limit ε→ 0.

4. Proof of the theorem

We begin by introducing two notations. For ε > 0, define

fε(x, t, u, ξ) =f(x, t, u, ξ)

1 + εf(x, t, u, ξ),((x, t), u, ξ

)∈ QT × R× Rn.

To our knowledge, this approximation has been introduced for the first time by Bensoussan-Frehse [2] for the study of nonlinear elliptic systems in stochastic game theory. Detailedproofs of [2] are presented in [3]. Later on the above approximation has been widely usedfor the study of nonlinear elliptic and parabolic problems with right hand side in L1.

The function fε is Caratheodory and satisfies the inequalities

0 ≤ fε(x, t, u, ξ) ≤1

ε∀((x, t), u, ξ

)∈ QT × R× Rn.

Let (u0,ε)ε>0 be a sequence of functions in L2(Ω) such that u0,ε → u0 strongly in L1(Ω)as ε→ 0.

We divide the proof of the theorem into five steps.

1 Existence of approximate solutions. We have

Lemma 1. For every ε > 0 there exists a pair

(ϕε, uε) ∈ Lp(0, T ;W 1,p(Ω)

)× L2

(0, T ;W 1,2(Ω)

)such that

ε

∫QT

|∇ϕε|p−2∇ϕε · ∇ζ dxdt+

∫QT

σ(uε, |∇ϕε|

)∇ϕε · ∇ζ dxdt

= 0 ∀ ζ ∈ Lp(0, T ;W 1,p

ΓD(Ω))

4) ;

(4.1)

ϕε = ϕD a.e. on ΣD;(4.2)

∃ u′ε ∈ L2(0, T ;

(W 1,2(Ω)

)∗);(4.3)

T∫0

〈u′ε, v〉W 1,2dt+

∫QT

κ(uε)∇uε · ∇v dxdt+ g

T∫0

∫∂Ω

(uε − h)v dxSdt

=

∫QT

fε(x, t, uε,∇ϕε)v dxdt ∀ v ∈ L2(0, T ;W 1,2(Ω)

);

(4.4)

uε(0) = u0,ε in L2(Ω).(4.5)

4) If 1 < p < 2, for z ∈W 1,p(Ω) we define∣∣∇z(x)

∣∣p−2∇z(x) = 0 a.e. in x ∈ Ω;∇z(x) = 0.


Proof. To begin with, we notice that, for all ξ, η ∈ Rn,(|ξ|p−2ξ − |η|p−2η

)· (ξ − η)

≥

p− 1(

1 + |ξ|+ |η|)2−p |ξ − η|2 if 1 < p ≤ 2,

min1

2,

1

2p−2

|ξ − η|p if 2 < p < +∞

(4.6)

(cf. [25, pp. 71, 74], [28]).For ε > 0 and (u, τ) ∈ R× R+, define

σε(u, 0) = σ(u, 0) if τ = 0,

σε(u, τ) = ετp−2 + σ(u, τ) if 0 < τ < +∞.Thus, by (3.1) and (4.6),(

σε(u, |ξ|

)ξ − σε

(u, |η|

)η)· (ξ − η) ≥ ε

(|ξ|p−2ξ − |η|p−2η

)· (ξ − η) > 0

for all u ∈ R and all ξ, η ∈ Rn, ξ 6= η.The assertion of Lemma 1 now follows from [28, Lemma 1] with σε in place of σ.

2 A-priori estimates. We have

Lemma 2. Let be (ϕε, uε) as in Lemma 1. Then, for all 0 < ε ≤ 1,

(4.7) ε∥∥ |∇ϕε|∥∥pLp + ‖ϕε‖pLp(W 1,p) ≤ c

(1 +

∥∥ |∇ϕD|∥∥pLp

)5) ;

(4.8)

‖uε‖L∞(L1) + λ

∫QT

|∇uε|2(1 + |uε|

)1+λdxdt

≤ c(1 + ‖u0,ε‖L1 +

∥∥ |∇ϕD|∥∥pLp

), 0 < λ < 1;

‖uε‖Lq(W 1,q) ≤ c ∀ 1 < q <n+ 2

n+ 1,(4.9)

‖uε‖Lr(Lr) ≤ c ∀ 1 < r <n+ 2

n,(4.10)

‖u′ε‖L1((W 1,q′ )∗) ≤ c ∀ 1 < q <n+ 2

n+ 1.(4.11)

Proof. By (4.2), the function ϕε − ϕD is in Lp(0, T ;W 1,p

ΓD(Ω)). Inserting this function into

(4.1), we find

ε

∫QT

|∇ϕε|pdxdt+

∫QT

σ(uε, |∇ϕε|

)|∇ϕε|2dxdt

= ε

∫QT

|∇ϕε|p−2∇ϕε · ∇ϕD dxdt+

∫QT

σ(uε, |∇ϕε|

)∇ϕε · ∇ϕD dxdt.

5) Without any further reference, in what follows, by c we denote constants which may change theirnumerical value from line to line, but do not depend on ε.

98 JOACHIM NAUMANN

From this, (4.7) easily follows by combining (H1) and Holder’s inequality.Estimates (4.8)–(4.11) can be proved by following line by line the proof of [28, Lemma 2].

3 Convergence of subsequences. Let be (ϕε, uε) as in Lemma 1. From (4.7) and (4.9),(4.10) we conclude that there exists a subsequence of (ϕε, uε)ε>0 (not relabelled) such that

(4.12) ϕε −→ ϕ weakly in Lp(0, T ;W 1,p(Ω)

)and

(4.13)

uε → u weakly in Lq

(0, T ;W 1,q(Ω)

) (1 < q <

n+ 2

n+ 1

)and weakly in Lr

(0, T ;Lr(Ω)

) (1 < r <

n+ 2

n

)as ε→ 0. Then (4.2) and (4.12) yield ϕ = ϕD a.e. on ΣD, i.e., ϕ satisfies (2.6).

Next, fix any 1 < q < n+2n+1 . Taking into account the embeddings (2.10), from (4.9) and

(4.11) we obtain by the aid of a well-known compactness result [6, Prop. 1] or [30, Cor. 4]the existence of a subsequence of (uε)ε>0 (not relabelled) such that uε → u strongly inLq(0, T ;L2(Ω)

), and therefore

(4.14) uε −→ u a.e. in QT as ε −→ 0.

We prove estimate (3.7). To begin with, we find an 0 < ε0 ≤ 1 such that

‖u0,ε‖L1 ≤ 1 + ‖u0‖L1 ∀ 0 < ε ≤ ε0.

Then, given any ψ ∈ L∞(0, T ), ψ ≥ 0 a.e. in [0, T ], from (4.8) it follows that

(4.15)

∫QT

∣∣uε(x, t)ψ(t)∣∣dxdt ≤ C0

T∫0

ψ(t)dt ∀ 0 < ε ≤ ε0

whereC0 := c

(1 + ‖u0‖L1 +

∥∥ |∇ϕD|∥∥pLp

).

Taking the lim infε→0

in (4.15), we find

∫QT

∣∣u(x, t)ψ(t)∣∣dxdt ≤ C0

T∫0

ψ(t)dt.

Hence, ∫Ω

∣∣u(x, t)|dx ≤ C0 for a.e. t ∈ [0, T ].

Next, from (4.8) and (4.14) we infer (by passing to a subsequence if necessary) that

∇uε(1 + |uε|

)(1+λ)/2−→ ∇u(

1 + |u|)(1+λ)/2

weakly in[L2(QT )

]nas ε→ 0. Then taking the lim inf

ε→0in (4.8) gives

λ

∫QT

|∇u|2(1 + |u|

)1+λdxdt ≤ C0.


Summarizing, from (4.12)–(4.14) we deduced the existence of a pair

(ϕ, u) ∈ Lp(0, T ;W 1,p(Ω)

)×( ⋂

1<q<(n+2)/(n+1)

Lq(0, T ;W 1,q(Ω)

))which satisfies (2.6) and (3.7), (3.8). It remains to prove that (ϕ, u) satisfies the variationalidentity in (2.5) and that (3.4)–(3.6) hold true. This can be easily done by the aid ofLemma 3 and 4 we are going to prove next.

4 Passage to the limit ε→ 0. We have

Lemma 3. Let be (ϕε, uε) as in Lemma 1, and let be (ϕ, u) as in (4.12), (4.13). Then

(4.16)

∫QT

σ(u, |∇ϕ|

)∇ϕ · ∇ζ dxdt = 0 ∀ ζ ∈ Lp

(0, T ;W 1,p

ΓD(Ω))

i.e., (ϕ, u) satisfies (2.5);

σ(uε, |∇ϕε|

)∇ϕε −→ σ

(u, |∇ϕ|

)∇ϕ weakly in

[Lp′(QT )

]nas ε −→ 0;(4.17)

σ(uε, |∇ϕε|

)|∇ϕε|2 −→ σ

(u, |∇ϕ|

)|∇ϕ|2 weakly in L1(QT ) as ε −→ 0.(4.18)

Proof of (4.16) (cf. the “monotonicity trick” in [26, pp. 161, 172], [34, p. 474]). The function

ϕε − ϕD is in Lp(0, T ;W 1,p

ΓD(Ω))

(see (4.2)). Thus, given any ψ ∈ Lp(0, T ;W 1,p

ΓD(Ω)), the

function ζ = ϕε − ϕD − ψ is admissible in (4.1). By the monotonicity condition (3.1)(ξ = ∇ϕε and η = ∇(ψ + ϕD)),

0 = ε

∫QT

|∇ϕε|p−2∇ϕε · ∇(ϕε − (ψ + ϕD)

)dxdt

+

∫QT

σ(uε, |∇ϕε|

)∇ϕε · ∇

(ϕε − (ψ + ϕD)

)dxdt

≥ −ε∫QT

|∇ϕε|p−2∇ϕε · ∇(ψ + ϕD)dxdt

+

∫QT

σ(uε,∣∣∇(ψ + ϕD)

∣∣)∇(ψ + ϕD) · ∇(ϕε − (ψ + ϕD)

)dxdt.

The passage to the limit ε→ 0 gives

(4.19) 0 ≥∫QT

σ(u,∣∣∇(ψ + ϕD)

∣∣)∇(ψ + ϕD) · ∇(ϕ− (ψ + ϕD)

)dxdt

(cf. (4.7), (4.12) and (4.14)).

Let ζ ∈ Lp(0, T ;W 1,p

ΓD(Ω)). For any λ > 0, we insert ψ = ϕ− ϕD ∓ λζ into (4.19), divide

then by λ and carry through the passage to the limit λ→ 0. It follows∫QT

σ(u, |∇ϕ|

)∇ϕ · ∇ζ dxdt = 0.

100 JOACHIM NAUMANN

Proof of (4.17). From (H1) and (4.7) it follows that there exists a subsequence of (∇ϕε)ε>0

(not relabelled) such that

σ(uε, |∇ϕε|

)∇ϕε −→ F weakly in

[Lp′(QT )

]nas ε −→ 0.

The function ζ = ϕ− ϕD being admissible in (4.1), we find∫QT

F · ∇(ϕ− ϕD)dxdt = limε→0

∫QT

σ(uε, |∇ϕε|

)∇ϕε · ∇(ϕ− ϕD)dxdt = 0.

Thus, using (4.1) with ζ = ϕε − ϕD, it follows∫QT

F · ∇ϕdxdt =

∫QT

F · ∇ϕD dxdt

= limε→0

∫QT

σ(uε, |∇ϕε|

)∇ϕε · ∇ϕD dxdt

≥ lim infε→0

∫QT

σ(uε, |∇ϕε|

)|∇ϕε|2dxdt.(4.20)

Claim (4.17) is now easily seen by the aid of the “monotonicity trick” with respect to

the dual pairing([Lp(QT )

]n,[Lp′(QT )

]n). Indeed, let G ∈

[Lp(QT )

]n. Using (3.1) with

ξ = G, η = ∇ϕε, we find by the aid of (4.12), (4.20) and Lebesgue’s Dominated ConvergenceTheorem ∫

QT

σ(u, |G|

)G · (G−∇ϕ)dxdt ≥

∫QT

F · (G−∇ϕ)dxdt.

Hence, given H ∈[Lp(QT )

]nand λ > 0, we take G = ∇ϕ±λH, divide by λ > 0 and carry

through the passage to the limit λ→ 0 to obtain∫QT

σ(u, |∇ϕ|

)∇ϕ ·H dxdt =

∫QT

F ·H dxdt.

Whence (4.17).

Proof of (4.18). Define

gε =(σ(uε, |∇ϕε|

)∇ϕε − σ

(uε, |∇ϕ|

)∇ϕ)· ∇(ϕε − ϕ) a.e. in QT .

By the aid of (4.17), (4.16) and uε → u a.e. in QT (see (4.14)) one easily obtains

limε→0

∫QT

gε dxdt = 0.

By (3.1), gε ≥ 0 a.e. in QT . Thus

(4.21) limε→0

∫QT

gεz dxdt = 0 ∀ z ∈ L∞(QT ).

We next multiply each term of the equation

σ(uε, |∇ϕε|

)|∇ϕε|2 = gε + σ

(uε, |∇ϕε|

)∇ϕε · ∇ϕ+ σ

(uε, |∇ϕ|

)∇ϕ · ∇(ϕε − ϕ)


by z ∈ L∞(QT ) and integrate over QT . Then (4.18) follows from (4.21), (4.17) and (4.14),(4.12).

The next lemma is fundamental to the passage to the limit ε→ 0 in (4.4).

Lemma 4. Let be (ϕε, uε) as in Lemma 1, and let be (ϕ, u) as in (4.12), (4.13). Then, forany z ∈ L∞(QT ),

(4.22) limε→0

∫QT

fε(x, t, uε,∇ϕε)z dxdt =

∫QT

f(x, t, u,∇ϕ)z dxdt.

Proof. For notational simplicity, we write (·, ·) in place of the variables (x, t).The structure condition (3.2) and the definition of fε yield∫

QT

f(·, ·, uε,∇ϕε)1 + εf(·, ·, uε,∇ϕε)

z dxdt−∫QT

f(·, ·, u,∇ϕ)z dxdt = J1,ε + J2,ε + J3,ε

where

J1,ε =

∫QT

AεBεdxdt,

Aε = zα(·, ·, uε)( 1

1 + εα(·, ·, uε)σ(uε, |∇ϕε|

)|∇ϕε|2

− 1)

Bε = σ(uε, |∇ϕε|

)|∇ϕε|2,

and

J2,ε =

∫QT

z(α(·, ·, uε)− α(·, ·, u)

)Bε dxdt,

J3,ε =

∫QT

zα(·, ·, u)(Bε − σ

(u, |∇ϕ|

)|∇ϕ|2

)dxdt.

Observing that 0 ≤ α ≤ α0 = const a.e. in QT (see (3.2)), we find

(4.23) |Aε| ≤ α0‖z‖L∞ a.e. in QT , ∀ ε > 0.

On the other hand, from∫QT

α(·, ·, uε)σ(uε, |∇ϕε|

)|∇ϕε|2dxdt ≤ c ∀ ε > 0

it follows (by going to a subsequence if necessary) that

εα(·, ·, uε)σ(uε, |∇ϕε|

)|∇ϕε|2 −→ 0 a.e. in QT as ε −→ 0.

Hence,

(4.24) Aε −→ 0 a.e. in QT as ε −→ 0.

From (4.23), (4.24) and Bε → σ(u, |∇ϕ|

)|∇ϕ|2 weakly in L1(QT ) (see (4.18)) we conclude

with the help of Egorov’s theorem and the absolute continuity of the integral that

J1,ε =

∫QT

AεBε dxdt −→ 0 as ε −→ 0

102 JOACHIM NAUMANN

(see, e.g., [16, p. 54, Prop. 1 (i)]). Analogously,

Jk,ε −→ 0 as ε −→ 0 (k = 2, 3).

Whence (4.22).

5 Proof of (3.4)–(3.6). Let n+2 < r < +∞ (i.e., setting q = r′, then 1 < q < n+2n+1 , q′ = r,

and vice versa).Let be z ∈ W 1,r(Ω) and ψ ∈ C1

([0, T ]

), ψ(T ) = 0. We set v(x, t) = z(x)ψ(t) for a.e.

(x, t) ∈ QT . An integration by parts gives

T∫0

〈u′ε, v〉W 1,2dt = −⟨uε(0), z

⟩W 1,2ψ(0)−

T∫0

〈zψ′, uε〉W 1,2dt

= −∫Ω

uε(·, 0)z dxψ(0)−∫QT

uεzψ′dxdt [by (2.1)]

(see [13, p. 54, Prop. 2.5.2 with p = q = 2, r = 1 therein]).With the help of (4.13), (4.14) and (4.22) the passage to the limit ε → 0 in (4.4) (with

v = zψ therein) is easily done. We find

−∫QT

uzψ′dxdt+

∫QT

κ(u)∇u · ∇zψ dxdt+ g

T∫0

∫∂Ω

(u− h)zψ dxSdt

=

∫Ω

u0z dxψ(0) +

∫QT

f(x, t, u,∇ϕ)zψ dxdt(4.25)

(recall uε(·, 0) = u0,ε → uε strongly in L1(Ω)). Following line by line the arguments in [28],from (4.25) we deduce the existence of the distributional derivative

u′ ∈ L1(0, T ;

(W 1,r(Ω)

)∗)(cf. [5, p. 154, Prop. A6]), i.e., (3.4) holds. Moreover, we have

(4.26)

T∫0

⟨u′(t), zψ(t)

⟩W 1,rdt+

⟨u(0), z

⟩W 1,rψ(0) = −

∫QT

uzψ′dxdt [by (2.1)],

where u ∈ C([0, T ];

(W 1,r(Ω)

)∗)is as in Section 2 (see [13, p. 54, Prop. 2.5.2 with p = 1,

q = +∞, r = 1 therein]). We insert (4.26) into (4.25) and obtain

T∫0

⟨u′(t), zψ(t)

⟩W 1,rdt+

⟨u(0), z

⟩W 1,rψ(0)

+

∫QT

κ(u)∇u · ∇zψ dxdt+ g

T∫0

∫∂Ω

(u− h)zψ dxSdt

=

∫Ω

u0z dxψ(0) +

∫QT

f(x, t, u,∇ϕ)zψ dxdt(4.27)


for all z ∈W 1,r(Ω) and all ψ ∈ C1([0, T ]

), ψ(T ) = 0.

To prove (3.5), we take ψ ∈ C1c

(] 0, T [

)in (4.27). A routine argument yields⟨

u′(t), z⟩W 1,r +

∫Ω

κ(u)∇u · ∇z dx+ g

∫∂Ω

(u− h)z dxS

=

∫Ω

f(x, t, u,∇ϕ)z dx(4.28)

for all z ∈W 1,r(Ω) and a.e. t ∈ [0, T ], where the null set in [0, T ] of those t for which (4.28)fails, does not depend on z. Now, given v ∈ L∞

(0, T ;W 1,s(Ω)

)(n+2 < s < +∞), we insert

z = v(·, t) into (4.28) (with r = s therein) and integrate over the interval [0, T ]. Whence(3.5).

Equ. (3.6) in(W 1,s(Ω)

)∗is now easily seen. Indeed, let z ∈W 1,s(Ω) (n+ 2 < s < +∞),

and let ψ ∈ C1([0, T ]

), ψ(0) = 1 and ψ(T ) = 0. We multiply (4.28) by ψ(t) and integrate

over [0, T ]. Combining (4.27) and (4.28), we obtain⟨u(0), z

⟩W 1,s =

∫Ω

u0z dx,

i.e., (3.6) holds (cf. (2.11) with q′ = s therein).The proof of the theorem is complete.

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Mathematics Department, Humboldt University Berlin, Unter den Linden 6, D-10099 BerlinE-mail address: [email protected]

JEPE Vol 2, 2016, p. 105-117

INFINITELY MANY PERIODIC SOLUTIONS FOR A FRACTIONAL

PROBLEM UNDER PERTURBATION

VINCENZO AMBROSIO

Abstract. We discuss the existence of infinitely many periodic weak solutions for a

subcritical nonlinear problem involving the fractional operator (−∆ + I)s on the torusTN . By using an abstract critical point result due to Clapp [14], we prove that, in spite

of the presence of a perturbation h ∈ L2(TN ) which breaks the symmetry of the problemunder consideration, it is possible to find an unbounded sequence of periodic (weak)

solutions.

1. Introduction

In the past years there has been a considerable amount of research related to the roleof symmetry in obtaining multiple critical points of symmetric functionals associated toordinary and partial differential equations. For instance, semilinear problems of the type

(1.1)

Lu = f(x, u) + h in Ωu = 0 on ∂Ω

,

where L is uniformly elliptic, Ω is a smooth bounded domain in RN , f(x, u) behaves like|u|q−2u with q ∈ (2, 2N

N−2 ), and h ∈ L2(Ω) is a perturbation, has been investigated by many

authors by using topological and variational methods; see for instance [8, 9, 19, 25].In this paper we focus our attention on the effect of a perturbation which destroys thesymmetry of the following nonlinear fractional problem

(1.2) (−∆ + I)su = f(x, u) + h(x) on TN ,where TN = RN/ZN is the N -dimensional torus, N ≥ 2, s ∈ (0, 1), and f : TN × R→ R isa function satisfying the following hypotheses:

(f1): f is a continuous function and f(x,−t) = −f(x, t) for all x ∈ TN and t ∈ R;(f2): there exist p ∈ (1, 2∗s − 1), where 2∗s = 2N

N−2s , and a1, a2 > 0 such that for any

x ∈ TN and t ∈ R|f(x, t)| ≤ a1 + a2|t|p;

(f3): there exist µ > 2 and r0 > 0 such that

0 < µF (x, t) ≤ tf(x, t)

for x ∈ TN , |t| ≥ r0, where F (x, t) =∫ t

0f(x, τ)dτ .

2010 Mathematics Subject Classification. Primary: 35A15, 35R11; Secondary: 47G20, 45G05.

Key words and phrases. Periodic solutions, Fractional Laplacian on torus, Critical Points.Received 10/09/2016, accepted 12/10/2016.

105

106 VINCENZO AMBROSIO

Here we assume that the perturbation

(1.3) h ∈ L2(TN )

and that p satisfies the following condition

(1.4)(N + 2s)− p(N − 2s)

N(p− 1)>

µ

µ− 1.

We notice that (f1) and (f3) imply the existence of constants a3, a4, a5 > 0 such that

(1.5)1

µ(tf(x, t) + a3) ≥ F (x, t) + a4 ≥ a5|t|µ

for all t ∈ R.The operator (−∆ + I)s on TN is defined for any u ∈ C∞(TN ) by setting

(−∆ + I)su(x) =∑k∈ZN

(|k|2 + 1)sckeıkx

where ck =∫TN u(x)e−ık·xdx are the Fourier coefficients of u. This operator can be extended

by density on the Hilbert space

Hs(TN ) =

u =

∑k∈ZN

ckeık·x ∈ L2(TN )

∣∣∣[u]2Hs(TN ) :=∑k∈ZN

|k|2s|ck|2 <∞

.

The study of fractional and non-local operators of elliptic type received immensely growingattention recently, because of their strong connection with real-world problems. These op-erators, arise in several contexts such as phase transition phenomena, population dynamics,game theory, mathematical finance, chemical reactions of liquids, geophysical fluid dynamics,quantum mechanics; see [16] and references therein for more details and applications.In spite of the fact that there are many papers dealing with superlinear problems involvingnon-local operators [5, 10, 11, 17, 18, 22, 23], there are few results concerning the multiplicityof solutions for a non-local boundary problem under the effect of a perturbation. The onlyresults which we know are due to Servadei [21], that proved the existence of infinitely manysolutions to the problem

(−∆)sRNu− λu = f(x, u) + h in Ωu = 0 on RN \ Ω

,

where s ∈ (0, 1), Ω ⊂ RN is a Lipschitz bounded open set, (−∆)sRN is the fractional Lapla-

cian, λ ∈ R, f is a subcritical nonlinearity and h ∈ L2(Ω) is a perturbation, and Coloradoet al. [15] which studied existence and multiplicity of solutions for the following fractionalcritical problem involving the spectral Laplacian (−∆)sΩ

(−∆)sΩu = |u|4s

N−2su+ h in Ωu = 0 on ∂Ω

under appropriate conditions on the size of h. We point out that the non-local operators(−∆)sRN and (−∆)sΩ appearing in the above problems are different; see [24].

The aim of this paper is to give a further result in this direction, considering a non-localproblem with periodic boundary conditions, under the effect of a not small perturbation.Our main result can be stated as follows:

INFINITELY MANY PERIODIC SOLUTIONS 107

Theorem 1. Let f satisfying (f1)-(f3) and let h ∈ L2(TN ). Assume that p satisfiesthe relation in (1.4). Then, (1.2) possesses an unbounded sequence of periodic solutions(uj)j∈N ⊂ Hs(TN ).

Let us observe that when h = 0, the existence of infinitely many solutions to (1.2) can beobtained by using standard critical point theory for even functionals [1].Our purpose is to investigate (1.2) in the case h 6= 0, that is when we have a lack of symmetry.In order to do this, we will use a variant of the Mountain Pass Theorem due to Clapp [14]:

Theorem 2. [14] Let V be a G-Hilbert space with V G = 0, and let V1 ⊂ V2 ⊂ · · · ⊂Vk ⊂ . . . be a sequence of finite dimensional G-invariant linear subspaces of V . Here V G :=x ∈ V : gx = x for all g ∈ G is the set of fixed points of V in G. Let J : V → R be aC1-functional which satisfies the following conditions

(i): J verifies the Palais-Smale condition (PS)a above a for some a > 0, that is anysequence (xn) in V such that J(xn) ⊂ [a, b] for some b ∈ R and such that J ′(xn)→ 0as n→∞ has a convergent subsequence;

(ii): There are constants γ > 0 and µ > 1 such that for all x ∈ V and g ∈ G

|J(x)− J(gx)| ≤ γ(|J(x)|1µ + 1);

(iii): There are constants β > 0, θ > µµ−1 , j0 ≥ 1 such that for all j ≥ j0

supρ≥0

infJ(x) : x ∈ V ⊥j−1, ‖x‖ = ρ ≥ βjθ;

(iv): For every j ≥ 1 there exists Rj > 0 such that Φ(x) ≤ 0 for all x ∈ Vj: ‖x‖ ≥ Rj;(v): There exists a fixed admissible representation W of G such that for all j ≥ j0,

Vj ∼= ⊕ji=1W .

Then J has an unbounded sequence of critical values.

This result can be read as follows: if J is not too far away from being G-invariant and ifthe mountain range is steep enough, then J can still have an unbounded sequence of criticalvalues.In order to prove Theorem 1, we will introduce the following functionals defined on Hs(TN )

I(u) =1

2‖u‖2Hs(TN ) −

∫TN

F (x, u)dx−∫TN

hudx

and

J(u) =1

2‖u‖2Hs(TN ) −

∫TN

F (x, u)dx−∫TN

ψ(u)hudx,

where ψ is a suitable functional such that ψ(u) = 1 if u is a critical point of I.By considering the antipodal action of G = Z2 on Hs(TN ), we will show that J satisfies theassumptions of Theorem 2 and that large critical values of the modified functional J arecritical values of I.We would like to note that in [2, 3, 4, 6, 7] the existence of periodic solutions to fractionalproblems of the type

(1.6) (−∆ + I)su = f(x, u) on TN ,

has been obtained by using variational methods after transforming (1.6) in a degenerateelliptic equation with nonlinear Neumann boundary conditions via a Caffarelli-Silvestre type


extension [12] in periodic setting. In this paper however, we prefer to analyze the problemdirectly in Hs(TN ) so that we can adapt the techniques developed in [19].

The paper is organized as follows: in Section 2 we present some preliminary facts concerningthe fractional Sobolev spaces on torus, and in Section 3 we give the proof of Theorem 1.

2. Preliminaries

2.1. Fractional Sobolev spaces on torus. In this section we collect some preliminaryresults concerning the fractional Sobolev spaces on torus.Let s ∈ (0, 1) and N ≥ 2. Let u ∈ C∞(TN ). As usual, we identify TN with [0, 2π]N , and thefunctions on TN with functions on RN that are periodic with period 2π in each coordinatex1, . . . , xN , that is u(x+ 2πei) = u(x) for all x ∈ RN and i = 1, . . . , N .Then we know that

u(x) =∑k∈ZN

ckeık·x,

where

ck =

∫TN

u(x)e−ık·xdx (k ∈ ZN )

are the Fourier coefficients of u. We define the fractional Sobolev space Hs(TN ) as theclosure of C∞(TN ) under the norm

‖u‖2Hs(TN ) :=∑k∈ZN

(|k|2 + 1)s |ck|2.

Let us observe that Hs(TN ) is a Hilbert space with respect to the inner product

〈u, v〉Hs(TN ) =∑k∈ZN

(|k|2 + 1)sckdk

for any u =∑k∈ZN cke

ık·x and v =∑k∈ZN dke

ık·x belonging to Hs(TN ). Finally we use thenotation

[u]2Hs(TN ) =∑k∈ZN

|k|2s |ck|2

to indicate the semi-norm of u.Now, we recall the following embeddings

Theorem 3. (Fractional Sobolev embeddings on torus) The inclusion of Hs(TN ) in Lq(TN )is continuous for any q ∈ [1, 2∗s] and compact for any q ∈ [1, 2∗s).

Proof. We give a simple proof of this result. Let u =∑k∈ZN cke

ık·x be a smooth function

on TN such that∫TN u dx = 0, and let v =

∑k∈ZN dke

ık·x ∈ L2NN+2s (TN ).

By applying the Cauchy-Schwartz inequality, we can see that

|(u, v)L2(TN )| =∣∣∣∑|k|≥1

ckdk

∣∣∣ =∣∣∣∑|k|≥1

|k|s|k|−sckdk∣∣∣

≤(∑|k|≥1

|k|2s|ck|2) 1

2(∑|k|≥1

|k|−2s|dk|2) 1

2

= [u]Hs(TN )‖(−∆)−s2 v‖L2(TN ).(2.1)


Now, by the Hardy-Littlewood-Sobolev inequality we know that

(2.2) ‖(−∆)−s2 v‖L2(TN ) ≤ C‖v‖

L2NN+2s (TN )

for some constant C > 0.Combining (2.1) and (2.2) we get

|(u, v)L2(TN )| ≤ C[u]Hs(TN )‖v‖L

2NN+2s (TN )

.(2.3)

Taking v = |u|N+2sN−2s−1u ∈ L

2NN+2s (TN ), we have

|(u, v)L2(TN )| = ‖u‖2NN−2s

L2NN−2s (TN )

and

‖v‖L

2NN+2s (TN )

= ‖u‖N+2sN−2s

L2NN−2s (TN )

,

so (2.3) becomes

(2.4) ‖u‖L

2NN−2s (TN )

≤ C[u]Hs(TN ).

This allows us to deduce that the embedding of Hs(TN ) into Lq(TN ) is continuous for anyq ∈ [1, 2∗s].Finally, we show that Hs(TN ) is compactly embedded in Lq(TN ) for any q ∈ [1, 2∗s). Byusing the interpolation inequality and (2.4), we know that for every q ∈ (2, 2∗s)

‖u‖Lq(TN ) ≤ ‖u‖θL2(TN )‖u‖1−θL2∗s (TN )

≤ C‖u‖θL2(TN )‖u‖1−θHs(TN )

for some θ ∈ (0, 1). Therefore, it suffices to verify that Hs(TN ) is compactly embedded inL2(TN ) to obtain the desired result.Let uj 0 in Hs(TN ) as j →∞. Then

(2.5) limj→∞

|cjk|2(|k|2 + 1)s = 0 ∀k ∈ ZN

and

(2.6)∑k∈ZN

|cjk|2(|k|2 + 1)s ≤ C ∀j ∈ N.

Fix ε > 0. Then there exists ν > 0 such that (|k|2 + 1)−s < ε for |k| > ν. By (2.6) we have∑k∈ZN

|cjk|2 =

∑|k|≤ν

|cjk|2 +

∑|k|>ν

|cjk|2

=∑|k|≤ν

|cjk|2 +

∑|k|>ν

|cjk|2(|k|2 + 1)s(|k|2 + 1)−s

≤∑|k|≤ν

|cjk|2 + Cε.

By (2.5) we deduce that∑|k|≤ν |c

jk|2 < ε for j large. So uj → 0 in L2(TN ) as j →∞.


It is well known (see [20, 26]) that the powers of a non-negative and self-adjoint operatorin a bounded domain are defined through the spectral decomposition using the powers ofthe eigenvalues of the original operator. Since (−∆ + I)−s is a positive compact self-adjointoperator in L2(TN ), it is easy to show that the following result holds:

Theorem 4 (Spectral Theorem).

(i): The operator (−∆ + I)s has a countable family of eigenvalues λhh∈N which canbe written as an increasing sequence of positive numbers

0 < λ1 < λ2 ≤ · · · ≤ λh ≤ λh+1 ≤ . . . .

Each eigenvalue is repeated a number of times equal to its multiplicity (which isfinite).

(ii): λh = µsh for all h ∈ N, where µhh∈N is the increasing sequence of eigenvaluesof −∆ + I.

(iii): λ1 = 1 is simple, λh = µsh → +∞ as h→ +∞,(iv): The sequence uhh∈N of eigenfunctions corresponding to λh is an orthonormal

basis of L2(TN ) and an orthogonal basis of Hs(TN ).Let us note that uh, µhh∈N are the eigenfunctions and eigenvalues of −∆ + I.

(v): For any h ∈ N, λh has finite multiplicity, and there holds

λh = minu∈V ⊥h \0

‖u‖2Hs(TN )

‖u‖2L2(TN )

(Rayleigh’s principle)

where

Vh = spanu1, · · · , uhand

V ⊥h = u ∈ Hs(TN ) : 〈u, uj〉Hs(TN ) = 0, for j = 1, . . . , h− 1.

(vi): For any h ∈ N, the h-eigenvalue can be characterized as follows:

λh = maxu∈Vh\0

‖u‖2Hs(TN )

‖u‖2L2(TN )

.


This last section is devoted to the proof of our main result.Let us introduce the following functional

I(u) =1

2‖u‖2Hs(TN ) −

∫TN

F (x, u) dx−∫TN

hu dx(3.1)

defined for u ∈ Hs(TN ). Clearly, I ∈ C1(Hs(TN ),R) in view of the assumptions on f .We begin proving the following

Lemma 1. Let u be a critical point of I. Then there is a constant a6 depending on ‖h‖L2(TN )

such that

(3.2)

∫TN

[F (x, u) + a4] dx ≤ 1

µ

∫TN

[uf(x, u) + a3] dx ≤ a6(I(u)2 + 1)1/2


Proof. By using (1.5) and the fact that u is a critical point of I, we can see that

I(u) = I(u)− 1

2I ′(u)u

≥(1

2− 1

µ

)∫TN

[uf(x, u) + a3] dx− 1

2‖h‖L2(TN )‖u‖L2(TN ) − a7.

(3.3)

Since µ > 2 and by applying the Holder and Young inequalities we deduce that for any ε > 0

I(u) ≥ a8

∫TN

(uf(x, u) + a3) dx− a9 − Cε‖h‖νL2(TN ) − ε‖u‖µLµ(TN )

,(3.4)

where 1µ + 1

ν = 1. Choosing ε such that 2ε = µa5a8, and by using (3.4), (1.5) and the

Schwartz inequality, we obtain the claim.

Now, we modify the functional I as follows. Let χ ∈ C∞(R,R) such that χ(t) = 1 for t ≤ 1,χ(t) = 0 for t > 2 and −2 < χ′ < 0 for t ∈ (1, 2).For u ∈ Hs(TN ), we set

Q(u) = 2a6(I(u)2 + 1)12

and we define the following functionals on Hs(TN )

ψ(u) = χ

(Q(u)−1

∫TN

[F (x, u) + a4] dx

)and

J(u) =1

2‖u‖2Hs(TN ) −

∫TN

F (x, u) dx−∫TN

ψ(u)hu dx.

We notice that (3.2) implies that ψ(u) = 1 if u is a critical point of I, and in particularJ(u) = I(u).Let us consider the antipodal action of G = Z2 on W = R, which is admissible by theBorsuk-Ulam Theorem.In order to show that J verifies the condition (ii) of Theorem 2, we give the followingpreliminary result

Lemma 2. If u ∈ suppψ, then

(3.5)∣∣∣∫

TNhu dx

∣∣∣ ≤ α1(|I(u)|1µ + 1)

where α1 depends on ‖h‖L2(TN ).

Proof. By using the Schwartz and Holder inequalities and (1.5), we obtain that for anyu ∈ Hs(TN ) ∣∣∣∫

TNhu dx

∣∣∣ ≤ ‖h‖L2(TN )‖u‖L2(TN ) ≤ α2‖u‖Lµ(TN )

≤ α3

(∫TN

(F (x, u) + a4) dx) 1µ

.(3.6)

Let us note that, if u ∈ suppψ, we get

(3.7)

∫TN

(F (x, u) + a4) dx ≤ 4a6(I(u)2 + 1)1/2 ≤ α4(|I(u)|+ 1).


Then, taking into account (3.6) and (3.7), we have the thesis.

At this point we can prove that J satisfies the following property:

Lemma 3. There is a constant β1, depending on ‖h‖L2(TN ), such that for all u ∈ Hs(TN ),

(3.8) |J(u)− J(−u)| ≤ β1(|J(u)|1µ + 1).

Proof. By using the definition of J and the assumption (f1), we can see that

(3.9) |J(u)− J(−u)| = (ψ(u) + ψ(−u))∣∣∣∫

TNhu dx

∣∣∣.Then, by Lemma 2, we deduce that

(3.10) ψ(−u)∣∣∣∫

TNhu dx

∣∣∣ ≤ α1ψ(−u)(|I(u)|1µ + 1).

Let us observe that by the definitions of I(u) and J(u) we know that

(3.11) |I(u)| ≤ |J(u)|+ 2∣∣∣∫

TNhu dx

∣∣∣,so, by using (3.10) we deduce

(3.12) ψ(−u)∣∣∣∫

TNhu dx

∣∣∣ ≤ α2ψ(−u)(|J(u)|

1µ +

∣∣∣∫TN

hu dx∣∣∣ 1µ + 1

).

Thus, by using the Young’s inequality, we can see that the term∫TN hu dx on the right-hand

side of (3.12) can be absorbed by the left-hand side. Similarly, we can deduce a correspondingestimate for the ψ(−u) term in (3.9), so we can infer that (3.8) holds.

Now, we show that large critical values of J are critical values of I. Firstly we prove thefollowing preliminary result:

Lemma 4. There are constants M0, α0 > 0, depending on ‖h‖L2(TN ), such that if M ≥M0,J(u) ≥M and u ∈ suppψ, then I(u) ≥ α0M .

Proof. Clearly, if u ∈ suppψ, then

(3.13) I(u) ≥ J(u)−∣∣∣∫

TNhu dx

∣∣∣.Hence (3.5) and (3.13) imply

I(u) + α1|I(u)|1/µ ≥ J(u)− α1 ≥M/2(3.14)

for M0 large enough. If I(u) ≤ 0,

αν1ν

+1

µ|I(u)| ≥ α1|I(u)|1/µ ≥M/2 + |I(u)|(3.15)

which gives a contradiction if M0 > 2αν1ν−1. As a consequence, I(u) > 0 and

I(u) > M/4 or I(u) ≥( M

4α1

)µwhich implies the Lemma since µ > 2.


Lemma 5. There is a constant M1 > 0 such that if J(u) ≥ M1 and J ′(u) = 0, thenJ(u) = I(u) and I ′(u) = 0.

Proof. We are going to prove that ψ(u) = 1 and ψ′(u) = 0. Taking into account thedefinition of ψ, this happens if

(3.16) Q(u)−1

∫TN

(F (x, u) + a4) dx ≤ 1.

Now, we show that (3.16) is satisfied.Let us note that

J ′(u)u = ‖u‖2Hs(TN ) −∫TN

uf(x, u)− (ψ(u) + ψ′(u)u)hu dx,(3.17)

where

ψ′(u)u = χ′(Q(u)−1

∫TN

(F (x, u) + a4) dx)

Q(u)−2[Q(u)

∫TN

uf(x, u) dx− (2a6)2(∫

TN(F (x, u) + a4) dx

)Q(u)−1I(u)I ′(u)u

].

Then, we can regroup the terms as

J ′(u)u = (1 + T1(u))‖u‖2Hs(TN ) − (1 + T2(u))

∫TN

uf(x, u) dx− (ψ(u) + T1(u))

∫TN

hu dx,

(3.18)

where

T1(u) = χ′(Q(u)−1

∫TN

(F (x, u) + a4) dx)

(2a6)2Q(u)−3I(u)

∫TN

(F (x, u) + a4) dx

∫TN

hu dx

and

T2(u) = χ′(Q(u)−1

∫TN

(F (x, u) + a4) dx)[Q(u)−1

∫TN

hu dx]

+ T1(u).

Now, we consider

(3.19) J(u)− 1

2(1 + T1(u))J ′(u)u.

If T1(u) = T2(u) = 0 and ψ(u) = 1, then (3.19) reduces to the left-hand side of (3.3), so(3.16) follows from (3.2). Since 0 ≤ ψ(u) ≤ 1, if T1(u) and T2(u) are both small enough,the calculation made in (3.3) when carried out for (3.19) leads to (3.2) with a6 replaced bya larger constant which is smaller than 2a6. But this gives (3.16). So, in order to concludethe proof of Lemma, it is enough to prove that T1(u), T2(u)→ 0 as M1 →∞.Firstly, we can note that

|T1(u)| ≤ |χ′(· · · )|4a6Q(u)−1∣∣∣∫

TNhu dx

∣∣∣.If u /∈ suppψ, T1(u) = 0 = T2(u). Otherwise, by using Lemma 2 and Lemma 4 we get

|T1(u)| ≤ α2Q(u)1µ−1 ≤ (M1 + 1)

1µ−1 → 0 as M1 →∞.

By the structure of T2, we also deduce T2(u)→ 0 as M1 →∞.


Taking into account the previous lemma, in order to verify (i) of Theorem 2, we need toprove the following result

Lemma 6. J ∈ C1(Hs(TN ),R) and there is a constant M2 > 0 such that J satisfies (PS)

on AM2≡ u ∈ Hs(TN ) : J(u) ≥M2.

Proof. By using (f1) and (f2), it is clear that I ∈ C1(Hs(TN ),R). Since χ ∈ C∞, and fverifies (f1) and (f2), we can see that ψ and therefore J ∈ C1(Hs(TN ),R).Now, let (um) ⊂ Hs(TN ) such that M2 ≤ J(um) ≤ K and J ′(um)→ 0.Then for all large m,

ρ‖um‖Hs(TN ) +K ≥ J(um)− ρJ ′(um)um

=(1

2− ρ(1 + T1(um))

)‖um‖2Hs(TN )

+ ρ(1 + T2(um))

∫TN

umf(x, um) dx−∫TN

F (x, um) dx

+ [ρ(ψ(um) + T1(um))− ψ(um)]

∫TN

hum dx(3.20)

where ρ is free for the moment.For M2 sufficiently large, and therefore T1, T2 small, by (f3) we can choose ρ ∈ ( 1

µ ,12 ) and

ε > 0 such that

(3.21)1

2(1 + T1(um))> ρ+ ε > ρ− ε > 1

µ(1 + T2(um))

uniformly in m.Putting together (3.20), (3.21) and (1.5), and by using the Holder and Young inequalitiesas in (3.4), we obtain

(3.22) ρ‖um‖Hs(TN ) +K ≥ ε‖um‖2Hs(TN ) + c1‖um‖µLµ(TN )− c2‖um‖Hs(TN ) − c3

which yields um is bounded in Hs(TN ).Now, it is easy to see that

(3.23) J ′(um) = (1 + T1(um))um − P(um)

where P is a compact operator. Taking M2 so large such that |T1(um)| ≤ 12 and by using the

facts (um) is bounded and J ′(um) → 0, we can infer that (1 + T1(um))−1P(um) convergesalong a subsequence. In virtue of (3.23), also (um) converges along a subsequence, and we

can conclude that J fulfills (PS) on AM2.

Therefore, in order to prove Theorem 1, it is enough to show that J has an unboundedsequence of critical values. For this reason, we are going to check (iii) and (iv) of Theorem2. Regarding the condition (iv), it is easy to see that for every u ∈ W, with W ⊂ Hs(TN )finite dimensional, there exist positive constants c1, c2, c3 and c4 (depending on W) suchthat

J(u) ≤ c1‖u‖2Hs(TN ) − c2‖u‖µHs(TN )

+ c3‖u‖Hs(TN ) + c4 → −∞ as ‖u‖Hs(TN ) →∞(3.24)


since µ > 2. We note that in (3.24), we used (1.5), |ψ(u)| < 1 and Holder inequality toestimate the term

∫TN hu dx.

Finally, we prove the following result:

Lemma 7. There are constants β2 > 0 and j0 ∈ N depending on ‖h‖L2(TN ) such that forall j ≥ j0,

(3.25) supρ≥0

infJ(u) : u ∈ V ⊥j−1, ‖u‖Hs(TN ) = ρ ≥ β2j(N+2s)−(N−2s)p

N(p−1) .

Proof. Let u ∈ ∂Bρ ∩ V ⊥j−1. Then by (f2), we can deduce that

(3.26) J(u) ≥ 1

2ρ2 − α2‖u‖p+1

Lp+1(TN )− α3 − ‖h‖L2(TN )‖u‖L2(TN ).

By using the interpolation inequality and Theorem 3, we get for all u ∈ Hs(TN )

(3.27) ‖u‖Lp+1(TN ) ≤ a7‖u‖aHs(TN )‖u‖1−aL2(TN )

,

where 2a = N(p−1)s(p+1) .

From Theorem 4, we also have

(3.28) ‖u‖L2(TN ) ≤ λ− 1

2j ‖u‖Hs(TN )

for all u ∈ V ⊥j−1.Putting together (3.26), (3.27) and (3.28), we can see that

(3.29) J(u) ≥ 1

2ρ2 − α4λ

− (1−a)(p+1)2

j ρp+1 − α3 − ‖h‖L2(TN )λ− 1

2j ρ.

Taking

ρ = ρj =1

(4α4)1p−1

λ(1−a)

2 ( p+1p−1 )

j ,

we deduce that

(3.30) J(u) ≥ 1

4ρ2j − ‖h‖L2(TN )λ

− 12

j ρj − α3.

Recalling [13] that for the compact manifold M = TN the following Weyl’s formula for theasymptotic distribution of the eigenvalues µj(M) of −∆ on M holds

µN2j (M) ∼ (2π)N

ωN

j

V ol(M)as j →∞,

and by using (ii) of Theorem 4, we can see that there exist j0 ∈ N and α5 independent of jsuch that

λj ≥ α5j2sN for j ≥ j0.

This together with (3.30) completes the proof of lemma.

Proof of Theorem 1. We consider the antipodal action of G = Z2 on Hs(TN ), and we takeW = R. Let us observe that for all j ∈ N, Vj is a G-invariant linear subspace of Hs(TN ),dimVj = j and that V G = 0. Putting together Lemma 1-Lemma 7, and (3.24), we cansee that the assumptions of Theorem 2 are satisfied. Then, there exist a sequence of critical


values (dj) ⊂ R and (uj) ⊂ Hs(TN ) such that I(uj) = dj →∞ and I ′(uj) = 0. In particular,being I ′(uj)uj = 0, we have

‖uj‖2Hs(TN ) =

∫TN

f(x, uj)ujdx+

∫TN

hujdx

= 2dj + 2

∫TN

F (x, uj)dx+

∫TN

hujdx.(3.31)

Then, by using (3.31), (f1), (f3) and h ∈ L2(TN ), it is easy to show that there exist α, β > 0independent of j ∈ N such that ‖uj‖2Hs(TN ) ≥ αdj − β →∞ as j →∞.

References

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120 (2015), 262–284.[3] V. Ambrosio, Periodic solutions for the non-local operator (−∆ + m2)s −m2s with m ≥ 0, to appear

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Sci. Math. 136 (2012), 521–573.

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Universita degli Studi di Napoli “Federico II”,, Dipartimento di Matematica e Applicazioni“Renato Caccioppoli”, Via Cintia 1, 80126 Napoli, Italy


JEPE Vol 2, 2016, p. 119-129

THE EFFECTIVENESS FACTOR OF REACTION-DIFFUSION

EQUATIONS: HOMOGENIZATION AND EXISTENCE OF OPTIMAL

PELLET SHAPES

JESUS ILDEFONSO DIAZ, DAVID GOMEZ-CASTRO, AND CLAUDIA TIMOFTE

Dedicated to an exceptional mathematician, David Kinderlehrer, with admiration.

Abstract. We study the asymptotic behaviour of the so-called effectiveness factor ηεof a nonlinear diffusion equation that occurs on the boundary of periodically distributed

inclusions (or particles) in an ε-periodic medium. Here, ε is a small parameter related tothe characteristic size of the inclusions, which, in the homogenization process, will tend

to 0. The inclusions are modeled as homothecy of a fixed pellet T , rescaled by a factor

r(ε). We study the cases in which r(ε) = O(εα), known as big holes, for α = 1, as well asnon-critical small holes, for 1 < α < n

n−2. We will prove the existence of some convex

shapes which maximize the effectiveness of the homogenized problem. In particular, we

deduce that for small holes the sphere is the domain of highest effectiveness.

1. Introduction

We study the asymptotic behaviour of the so-called effectiveness factor ηε of nonlineardiffusion equations for which a reaction occurs on the boundary of periodically distributedinclusions (or particles) in an ε-periodic medium.

To be more precise, let Ω ⊂ RN , with N ≥ 3, be a bounded connected open set suchthat |∂Ω| = 0 and let Y = (− 1

2 ,12 )N be the reference cell in RN . Let T be another open

bounded subset of RN , with the boundary ∂T of class C2. T will be called the elementaryparticle. We assume that 0 belongs to T and that T is star-shaped with respect to 0. SinceT is bounded, without loss of generality, we can assume that T ⊂ Y . We point out that,even though the usual term in homogenization theory for inclusions is holes (in order togive the idea that something has been removed from the domain), here we will avoid thisterminology. For us, these inclusions will be pellets, for example the ones that can be foundin fixed bed chemical reactors and towers. Therefore, we will refer to these holes as pellets,

2010 Mathematics Subject Classification. homogenization, effectiveness factor, semilinear elliptic

equations.

Key words and phrases. 35J61, 35B27, 25B40, 49K20.Received 28/09/2016, accepted 08/11/2016.The research of D. Gomez-Castro is supported by a FPU fellowship from the Spanish government. The

research of J.I. Dıaz and D. Gomez-Castro was partially supported by the project ref. MTM 2014-57113-Pof the DGISPI (Spain).

119

120 JESUS ILDEFONSO DIAZ, DAVID GOMEZ-CASTRO, AND CLAUDIA TIMOFTE

particles or even inclusions and obstacles.

Let ε be a real parameter taking values in a sequence of positive numbers converging tozero. Here ε represents a small parameter related to the characteristic size of the particles.For each ε and for any vector i ∈ ZN , we shall denote by T εi the translated image of r(ε)T bythe vector εi: T εi = εi+ r(ε)T. Also, let us denote by T ε the set of all the pellets containedin Ω, i.e.

T ε =⋃

T εi | T εi ⊂Ω, i ∈ ZN

and n(ε) = #i ∈ ZN : T εi ⊂Ω

be let the number of pellets. Set Ωε = Ω \ T ε. Therefore,

Ωε is a periodically perforated structure with pellets of the size r(ε). Let us notice thatthe inclusions do not intersect the fixed boundary ∂Ω. Let Sε = ∪∂T εi | T εi ⊂Ω, i ∈ ZN.Hence, we can write ∂Ωε = ∂Ω ∪ Sε.

We shall consider the homogenization of problems in which the reaction takes place onthe boundary of the particles. More precisely, we shall start from the family of problems

(1.1)

−∆uε = f in Ωε,∂uε∂ν + µ(ε)g(uε) = 0 on Sε,uε = 1 on ∂Ω,

where ν is the exterior unit normal to Sε, g is a non-decreasing function and the rest of thedata µ(ε) and f will be specified later.

We shall assume that r(ε) = εα. Particles that scale with α = 1 are called big particlesand the ones corresponding to the case α > 1 are called small particles. There is a specialcase, α = N

N−2 , which is known as the critical case, in which the limit behaviour of the

solution of problem (1.1) changes (see, e.g. [16]). We shall deal here only with the cases1 ≤ α < N

N−2 . The case α = NN−2 will be the subject of a forthcoming paper. The case

α > NN−2 is not interesting, since the reaction term disappears at the limit and, therefore,

the solution of problem (1.1) converges to u0 ≡ 1 in Ω.

We address here the relevant cases in which µ(ε)|Sε| = O(1). In fact, if r(ε) = εα, thenµ(ε) = ε−γ , with γ = α(N − 1)−N . In particular, for big holes, since α = 1, it follows thatγ = −1.

The problem (1.1) will be analyzed for a broad class of nonlinear monotone kinetics suchthat

(1.2) g is a maximal monotone graph (single-valued or even multivalued), with g(0) = 0,

and for

(1.3) f ∈ L2(Ω), f ≥ 0.

We recall that, in the Chemical Engineering context, it is typically assumed that g(1) = 1,and hence the above conditions guaranty that 0 ≤ uε ≤ 1, which is the natural setting forchemical concentrations.

HOMOGENIZATION-EFFECTIVENESS 121

A particular case we shall discuss is the Freundlich isotherm kinetics

(1.4) g(u) = |u|p−1u, p ∈ (0, 1].

Also, we can consider the limit case of zero order reactions:

(1.5) g(u) =

0 u < 0,[0, 1] u = 0,1 u > 0.

in the context of maximal monotone graphs of R2 (see, e.g., [3, 22]).

Problems of the form (1.1) have been extensively studied in the literature. The case of biginclusions, i.e. the case in which α = 1, with nonhomogeneous constant Dirichlet conditionson the boundary of the inclusions, was addressed in [8]. For nonlinear problems, varioustechniques, such as the method of oscillating test function (see [9]) and, more recently, theperiodic unfolding method (see [7]), have been used.

For small inclusions, i.e. for α > 1, the linear Neumann homogeneous problem was stud-ied in [24], the Neumann nonhomogeneous problem in [11] and the nonlinear problem wasanalyzed, by different techniques, in [20, 27]. The case where the inclusions are substitutedby a connected set can also be analyzed.

We shall work here with relatively smooth nonlinear kinetics g, for which g(0) = 0 andwhich satisfy suitable growth conditions (see (1.8) and (1.12)) but, as indicated before,it seems that our results could be extended even to the case in which g is a multivaluedmaximal monotone graph on R2 (see Remark 1 below). Inspired by the definition given inthe linear case (p = 1) by the chemical engineer R. Aris (see [1]), we define the notion ofeffectiveness of the pellet in this more general setting as follows:

(1.6) ηε(T ) =1

|Sε|

∫Sε

g(uε)dσ.

This is well defined since, for g smooth and satisfying (1.8) or (1.12), g(uε) ∈ W 1,q(Ωε),with q = 2N

q(N−2)+N . Definition (1.6) can be naturally extended to the homogenized case

(which will be studied below), by putting

(1.7) η(T ) =1

|Ω|

∫Ω

g(u)dx.

It is known that there exists an extension P εuε of the solution uε to the unperforateddomain Ω such that P εuε u0 in H1(Ω). The limit u0 is the solution of a different ellipticproblem, defined over the whole domain Ω, in which the reaction term appears in the interiorequation and an effective diffusion matrix which is not the identity arises for the case α = 1,but not for α > 1.

We recall some previous homogenization results. For the case of big holes we assume thatr(ε) = ε and consider either a smooth kinetic

(1.8) |g′(v)| ≤ C(1 + |v|q), 0 ≤ q < N

N − 2,


or a not necessarily smooth one, but with bounded growth

(1.9) |g(v)| ≤ C(1 + |v|q), 0 ≤ q < N

N − 2.

Following the theory in [9] and [10], the solution uε of problem (1.1), properly extended tothe whole of Ω, converges weakly in H1(Ω), as ε→ 0, to u ∈ H1(Ω), i.e. P εuε u, whereu is the unique solution of the following homogenized problem:

(1.10)

−div(a0(T )∇u) + |∂T |

|Y \T |g(u) = f in Ω,

u = 1 on ∂Ω.

The proof of the existence and uniqueness of a weak solution for this problem can be found,e.g., in [13]. Here, a0(T ) ∈MN (R) (the set of N ×N matrices) is the classical homogenizedmatrix (see, e.g., [9]). If we write a0(T ) = (qij), then

qij = δij +1

|Y \ T |

∫Y \T

∂χj∂yi

dy,

where χi are the solutions of the so-called cell problems:

(1.11)

−∆χi = 0 in Y \ T,∂(χi+yi)

∂ν = 0 on ∂T,χi Y -periodic.

For the case of small non-critical holes (1 < α < NN−2 ), under the assumption that for

the nonlinear function g there exist two constants k1, k2 such that

(1.12) 0 < k1 ≤ g ′(u) ≤ k2,

Zubova and Shaposhnikova showed in [27] that

(1.13) ‖∇uε −∇u0‖L2(Ωε)N + ε−γ‖uε − u0‖L2(Sε) → 0,

where, for r(ε) = Cεα, u0 is the solution of

(1.14)

−∆u0 + CN−1|∂T |g(u0) = f in Ω,u0 = 1, on ∂Ω.

This paper (which develops a previous short presentation [17]) is organized as follows:in Section 2, we present the main results of the paper, which we prove in the remainingsections. Section 3 is devoted to the proof Theorem 1, in which we compute the limit η ofthe effectiveness factor ηε as ε → 0. In Section 4, we prove Theorem 2, in which we showthe existence of optimal shapes, and Theorem 3, which contains their characterization forsmall holes.

2. Statement of the main results

In order to perform the homogenization process, we need to impose some regularityconditions for g.

Assumption 1. Let the following regularity hold:

• If α = 1, (1.8),


• If 1 < α < NN−2 , (1.12).

We state here the main results of this paper. The first one is a homogenization result forthe effectiveness.

Theorem 1. Let 1 ≤ α < NN−2 and Assumption 1 hold. Then,

(2.1) ηε(T )→ η(T ), as ε→ 0.

Remark 1. It is an open problem whether or not this convergence remains true under moregeneral nonlinearities g. Our proof of the convergence for α = 1 relies on [6], in which onerequires differentiability of g(uε). We could, however, define the effectiveness ηε by meansof g(trSε(u

ε)). Nonetheless, the proof, in essence, requires that we consider trSε(g(uε)). Webelieve that a proof of the general case might need an extension of the results in [6] or acompletely new approach. A different proof, applying the periodic unfolding method, of theconvergence in the case α = 1 and g = 0 (with a source boundary term) can be found in [5].

For the cases α > 1 it is known that the convergence results can be extended to broaderfamilies of nonlinearities by applying the same techniques. In particular, by using thetechniques of [16] we think possible to get a similar result to Theorem 1 for some casesα > 1 under assumption (1.4) for any p ∈ [0, 1] but this would be a very technical task thatwe shall not consider here.

For the homogenized problem, we have the following optimality result:

Theorem 2. Let 1 ≤ α < NN−2 , 0 < θ < |Y |, C,D be fixed proper subsets of Y and ε > 0.

Let us assume that

T satisfies the uniform ε-cone property.(2.2)

We define

Uadm = C ⊂ T ⊂ D : T satisfies (2.2) and |T | = θ,Cθ(D) = T ⊂ D : T is open, convex and |T | = θ.

Then, at fixed volume θ ∈ (0, |Y |), there exists a domain of maximal (and minimal) effec-tiveness for the homogenized problem in the class of T ∈ Uadm ∩ Cθ(D).

For small (non-critical) holes, we can characterize the optimizer’s shape.

Theorem 3. For the case 1 < α < NN−2 , the ball is the domain T of maximal effectiveness

for a set volume in the class of star-shaped C2 domains with fixed volume.

Remark 2. This is opposed to the homogenization with respect to the exterior domain Ω.In this context, when Ω is a ball has least effectivity, as can be shown by rearrangementtechniques (see [13]). In the context of product domains, Ω = B×Ω′′ is the least effective onthe class Ω = Ω′×Ω′′ for set volume, at least for convex or concave kinetics (see [14, 15, 22]).

Through standard procedures in weak solution theory, one easily gets the following result(see, e.g., [3]).

Proposition 1 (Well-possedness). Under the assumptions (1.2) and (1.3), there exists aunique solution uε ∈ H2(Ωε) of (1.1).

Proposition 2 (Strong maximum principle). Under the assumptions (1.2) and (1.3), uε > 0in Ωε.


Proof. By the weak maximum principle, we have that uε ≥ 0. Now, we can apply thecomparison principle with uε, the non-negative solution of

−∆uε = f in Ωε,uε = 0 on ∂Ωε,

to obtain uε ≥ uε in Ωε. For uε, we can apply the estimate in [18]

uε(x) ≥ c(∫

Ω

f(y) d(y, ∂Ωε) dy

)d(x, ∂Ωε), x ∈ Ωε,

which proves the result.

Remark 3. We can illustrate a couple of the steps of the homogenization process by meansof the following COMSOL simulation.

(a) ε = 0.33 (b) ε = 0.11

Figure 1. Fixed bed reactors with big pellets (rε = ε) and the level setof the solution of problem (1.1) for f = 0, and kinetic (1.4) where p = 1

2 ,µ(ε) = ε.

3. Effectiveness Homogenization

Since the cases α = 1 and α > 1 require different techniques and provide different results,we will divide the proof of Theorem 1 in two parts.

3.1. Big holes (α = 1).

Proof of Theorem 1 in the case α = 1. From [6] (see also [27, 28, 9]), it holds that

ε

∫Sε

g(uε(x))dσ → |∂T ||Y |

∫Ω

g(u(x))dx, as ε→ 0.

Since, by explicit computation, |Sε| = n(ε)|∂(εT )| = n(ε)εN−1|∂T |, when the cells tend tocover the total volume,

n(ε)|Y |εN = n(ε)|εY | → |Ω|, as ε→ 0,

and we have that |Sε|ε→ |Ω||∂T |, as ε→ 0. Hence, as ε→ 0,

ηε(T ) =1

|Sε|

∫Sε

g(uε(x))dσ → 1

|Ω|

∫Ω

g(u)dx = η(T ),

which proves the result.


Remark 4. The above convergence remains true for the kinetic (1.4) in the case of domainsin which there exists δ > 0 such that uε ≥ δ uniformly on ε, that is, no dead core exists.For the solution u, the region where u = 0 (which might have positive measure) is knownin the literature as a dead core. Conditions for the existence and location of a dead corein this and other kinds of equations can be found in [13], [2] and in the references therein.In the case when a dead core exists, even though the limit theorem does not apply, thestrong maximum principle (Proposition 2) suggests that the effectiveness is higher prior tothe homogenization process.

3.2. Small non critical holes (1 < α < NN−2). In [28], the authors show that for ϕ ∈

H1(Ωε, ∂Ω) = ϕ ∈ H1(Ωε) : ϕ = 0 on ∂Ω and for rε = Cεα, one has the estimate:

(3.1)

∣∣∣∣ ε−γ

CN−1|∂T |

∫Sε

ϕds−∫

Ωεϕdx

∣∣∣∣ ≤ KεN−α(N−2)2 .

Taking into account the explicit computation of |Sε|, we have that

(3.2)

∣∣∣∣ 1

|Sε|

∫Sε

ϕds− 1

|Ω|

∫Ωεϕdx

∣∣∣∣→ 0.

Since we are concerned with fixed volumes, we can set C = 1.We are now in the position to prove Theorem 1 for α > 1.

Proof of Theorem 1 in the case α > 1. Let us consider the change wε = 1− uε. So, wε = 0on ∂Ω. We can apply the homogenization results in [27] and [28] for wε and deduce directlythe corresponding ones for uε. Thus, we have that∣∣∣∣ 1

|Sε|

∫Sε

g(uε) ds− 1

|Ω|

∫Ω

g(u0) dx

∣∣∣∣ ≤ ∣∣∣∣ 1

|Sε|

∫Sε

g(uε) ds− 1

|Sε|

∫Sε

g(u0) ds

∣∣∣∣+

∣∣∣∣ 1

|Sε|

∫Sε

g(u0) ds− 1

|Ω|

∫Ω

g(u0) dx

∣∣∣∣ ≤ Cε−γ‖uε − u0‖L2(Sε)

(3.3) +

∣∣∣∣ 1

|Sε|

∫Sε

g(u0) ds− 1

|Ω|

∫Ω

g(u0) dx

∣∣∣∣→ 0,

which concludes the proof.

4. Existence of optimal pellet shapes

Once we know the effect that a general obstacle T causes, it seems reasonable to performdomain optimization. First, we show an abstract result of existence of optimal particleshape. We will focus on the homogenized models (1.10) and (1.14). We can prove Theorem 3directly by applying the isoperimetric inequality.

Proof of Theorem 3. Applying the comparison principle, we can see that u is a decreasingfunction of |∂T | and, since g is increasing, we have that η is an increasing function of u.Therefore, η is a decreasing function of |∂T |. For fixed volume |Y \ T |, the volume of T isfixed. The isoperimetric inequality (see, e.g. [13]) guaranties that a ball is the domain ofminimum |∂T |, hence the domain of optimal effectiveness.


Remark 5. Optimization of the effectiveness considering the homogenized domain Ω (thechemical reactor) has also been studied (see [13, 14, 15] and the references therein). In thissituation, the existence of a dead core affects the effectiveness negatively.

Remark 6. Dealing with the optimization of the domain Ω, there exist no optimalshapes considering a general framework (see [2], [15]). We conjecture that new results maybe also obtained by applying methods analogous to the ones that follow.

Remark 7. As in [9], the problem in which we consider reactions inside the pellets can alsobe addressed for α = 1. Let us consider the system of equations

−Df∆uε = f in Ωε,−Dp∆v

ε + a g(vε) = 0 in Ω \ Ωε,

−Df∂uε

∂ν = Dp∂vε

∂ν on Sε,uε = vε on Sε,uε = 1 on ∂Ω,

with Df , Dp > 0 and f ∈ L2(Ω). If we introduce the matrix A = DfχY \T +DpχT , where Iis the identity matrix in MN (R), then the homogenized problem for big pellets is (see [9])

−div(A0∇u) + a |T ||Y \T |g(u) = f in Ω,

u = 1 on ∂Ω,

where A0 = (a0ij) is the homogenized matrix, whose entries are defined as follows:

a0ij =

∫Y

(aij + aik

∂χj∂yk

)dy,

in terms of the functions χj , i = 1, . . . , N , Y -periodic solutions of the cell problems

−div(A∇(yj + χj)) = 0.

In this context, the results would be analogous and the proofs perhaps even simpler. Theproblem has been recently studied without continuity conditions (see [19]).

Coming back to the proof of Theorem 2, we see in (1.10) that, for α = 1, the effect of Tis present in three terms: |∂T |, |Y \T | and a0(T ). Therefore, any sensible choice of topologyfor the set of admissible domains T in a search for optimal shape obstacles must make thisexpressions continuous.

A logical choice of topology in the “space of shapes” is the one given by the Hausdorffdistance

dH(Ω1,Ω2) = sup

supx∈Ω1

d(x,Ω2), supx∈Ω2

d(x,Ω1)

.

For the optimization, we will restrict ourselves to a general enough family of domains, butin which we can define a topology which makes the family compact. It is well known (see,for example, [25]) that the following result holds true.

Theorem 4 ([25]). The class of closed subsets of a compact set D is compact for theHausdorff convergence.

A proof for the continuity of the effective diffusion under the Hausdorff distance in Uadmcan be found in [21].


Lemma 1 ([21]). If Uadm is compact with respect to the Hausdorff metric and if (Tn) ⊂Uadm, Tn → T as n→∞, T ∈ Uadm, then a0(Tn)→ a0(T ) in MN (R).

The behaviour of the measure |Y \ T | is slightly more delicate (we include a commentaryeven though, in our case, this will be constant). For this, a distance with a definition similarto Hausdorff metric, the Hausdorff complementary distance

dHc(Ω1,Ω2) = supx∈Rn

|d(x,Ωc1)− d(x,Ωc2)|,

has the following property: for open domains (Ωn)n,Ω, dHc(Ωn,Ω) → 0 as n → ∞ implieslim infn |Ωn| ≥ |Ω|. However, lower semicontinuity of the measure of the boundary (|∂T |)is, in general, false (see [21] for some counterexamples). Nevertheless, the set of convexdomains has a number of very interesting properties (see [26]).

Lemma 2 ([26]). The topological spaces (Cθ(D), dH) and (Cθ(D), dHc) are equivalent.

The continuity of the boundary measure is provided by the following result, proved in [4].

Lemma 3 ([4]). Let (Ωn),Ω ∈ Cθ(D). If Ω1 ⊂ Ω2, then |∂Ω1| ≤ |∂Ω2|. Moreover, if

ΩndH→ Ω, then |Ωn| → |Ω| and |∂Ωn| → |∂Ω|, as n→∞.

For the continuity of solutions with respect to T , we need the following theorem on thecontinuity of the associated Nemitskij operators (see, for example, [12] and [23]).

Lemma 4 ([23]). Let G : Ω → R → R be a Caratheodory function such that (1.9) holdstrue for q = r

t with r ≥ 1 and t <∞. Then, the map

Lr(Ω)→ Lt(Ω) v 7→ G(x, v(x))

is continuous in the strong topologies.

Lemma 5. Let A be the set of elliptic matrices and let g satisfy (1.9). Let u(A, λ) be theunique solution of

−div(A∇u) + λg(u) = f, in Ω,u = 1, on ∂Ω,

Then, the application

A× R+ → H1(Ω) (A, λ) 7→ u(A, λ),

is continuous in the weak topology.

Proof. Let G(u) =∫ u

0g(s)ds and

JA,λ(v) =1

2

∫Ω

(A∇v) · ∇v +

∫Ω

λG(v)−∫

Ω

fv.

We know that u(A, λ) is the unique minimizer of this functional. Let An → A and λn →λ be two converging sequences. It is easy to prove that un = u(An, λn) is bounded inH1(Ω) and, up to a subsequence, un u in H1 as n → ∞. Therefore,

∫Ω

(A∇u) · ∇u ≤lim infn

∫Ω

(An∇un) · ∇un. We can apply Theorem 4 to show that G(un) → G(u) in L1 asn→∞ (see details for a similar proof, for example, in [9]) and we have that u = u(A, λ).

Corollary 1. The map (I, λ) 7→ u, where I is the identity matrix, is continuous in the weaktopology of H1.


Corollary 2. In the hypotheses of Lemma 5, the maps (A, λ) 7→∫

Ωg(u(A, λ)) and (I, λ) 7→∫

Ωg(u(I, λ)) are continuous.

With these tools, we can prove now our main result.

Proof of Theorem 2. First, we have that Lemmas 1 and 3 imply that the application T 7→(a0(T ), λ(T )) is continuous. Then, Corollary 2 implies that T 7→ η(T ) is continuous (witheither a0(T ) or I). Therefore, since Cθ(D) is closed and Uadm is compact, by Lemma 1 wehave the compactness of Uadm ∩ Cθ(D) and then the existence of maximizers.

Remark 8. Some numerical experiences comparing the effectiveness for different shapeswhere presented in [17].

References

[1] R. Aris and W. Strieder, Variational Methods Applied to Problems of Diffusion and Reaction, vol. 24

of Springer Tracts in Natural Philosophy, Springer-Verlag, New York, 1973.[2] C. Bandle, A note on Optimal Domains in a Reaction-Diffusion Problem, Zeitschrift fur Analysis und

ihre Answendungen, 4 (3) (1985), pp. 207–213.

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Press, Inc., New York, 1971, pp. 101–156.

[4] G. Buttazzo and P. Guasoni, Shape Optimization problems over classes of convex domains, Journalof Convex Analysis, 4,2 (1997), pp. 343–352.

[5] I. Chourabi and P. Donato, Homogenization and correctors of a class of elliptic problems in perforated

domains, Asymptotic Analysis, 92 (2015), pp. 1–43.[6] D. Cioranescu and P. Donato, Homogeneisation du probleme de Neumann non homogene dans des

ouverts perfores, Asymptotic Analysis, 1 (1988), pp. 115–138.

[7] D. Cioranescu, P. Donato, and R. Zaki, Asymptotic behavior of elliptic problems in perforateddomains with nonlinear boundary conditions, Asymptotic Analysis, 53 (2007), pp. 209–235.

[8] D. Cioranescu and J. S. J. Paulin, Homogenization in open sets with holes, Journal of MathematicalAnalysis and Applications, 71 (1979), pp. 590–607.

[9] C. Conca, J. I. Dıaz, A. Linan, and C. Timofte, Homogenization in Chemical Reactive Flows,

Electronic Journal of Differential Equations, 40 (2004), pp. 1–22.[10] C. Conca, J. I. Dıaz, and C. Timofte, Effective Chemical Process in Porous Media, Mathematical

Models and Methods in Applied Sciences, 13 (2003), pp. 1437–1462.

[11] C. Conca and P. Donato, Non-homogeneous Neumann problems in domains with small holes,Modelisation Mathematique et Analyse Numerique, 22 (1988), pp. 561–607.

[12] G. Dal Maso, An Introduction to Γ-Convergence, Progress in Nonlinear Differential Equations,

Birkhauser Boston, Boston, MA, 1993.[13] J. I. Dıaz, Nonlinear Partial Differential Equations and Free Boundaries, Vol.I.: Elliptic equations,

Research Notes in Mathematics, Pitman, London, 1985.

[14] J. I. Dıaz and D. Gomez-Castro, Steiner symmetrization for concave semilinear elliptic and parabolicequations and the obstacle problem, in Dynamical Systems and Differential Equations, AIMS Proceed-

ings 2015 Proceedings of the 10th AIMS International Conference (Madrid, Spain), American Instituteof Mathematical Sciences, nov 2015, pp. 379–386.

[15] , On the Effectiveness of Wastewater Cylindrical Reactors: an Analysis Through Steiner Sym-

metrization, Pure and Applied Geophysics, 173 (2016), pp. 923–935.[16] J. I. Dıaz, D. Gomez-Castro, A. V. Podol’skii, and T. A. Shaposhnikova, Homogenization of the

p-Laplace operator with nonlinear boundary condition on critical size particles: identifying the strangeterms for some non smooth and multivalued operators, Doklady Mathematics, 94 (2016), pp. 387–392.

[17] J. I. Dıaz, D. Gomez-Castro, and C. Timofte, On the influence of pellet shape on the effective-

ness factor of homogenized chemical reactions, in Proceedings Of The XXIV Congress On DifferentialEquations And Applications XIV Congress On Applied Mathematics, 2015, pp. 571–576.


[18] J. I. Dıaz, J.-M. Morel, and L. Oswald, An elliptic equation with singular nonlinearity, Communi-

cations in Partial Differential Equations, 12 (1987), pp. 1333–1345.[19] M. Gahn, M. Neuss-Radu, and P. Knabner, Homogenization of Reaction–Diffusion Processes in a

Two-Component Porous Medium with Nonlinear Flux Conditions at the Interface, SIAM Journal on

Applied Mathematics, 76 (2016), pp. 1819–1843.[20] M. V. Goncharenko, Asymptotic behavior of the third boundary-value problem in domains with fine-

grained boundaries, Proceedings of the Conference “Homogenization and Applications to Material Sci-

ences” (Nice, 1995), GAKUTO (1997), pp. 203–213.[21] J. Haslinger and J. Dvorak, Optimum Composite Material Design, ESAIM Mathematical Modelling

and Numerical Analysis, 1 (1995), pp. 657–686.

[22] D. Kinderlehrer and G. Stampacchia, An introduction to variational inequalities and their applica-tions, vol. 31, Academic Press, New York, 1980.

[23] J. L. Lions, Quelques Methodes de Resolution pour les Problemes aux Limites non Lineaires, Dunod,

Paris, 1969.[24] O. A. Oleinik and T. A. Shaposhnikova, On the homogenization of the Poisson equation in partially

perforated domains with arbitrary density of cavities and mixed type conditions on their boundary, Attidella Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti

Lincei. Matematica e Applicazioni, 7 (1996), pp. 129–146.

[25] O. Pironneau, Optimal Shape Design for Elliptic Equations, Springer Series in Computational Physics,Springer-Verlag, Berlin, 1984.

[26] N. Van Goethem, Variational problems on classes of convex domains, Communications in Applied

Analysis, 8 (2004), pp. 353–371.[27] M. N. Zubova and T. A. Shaposhnikova, Homogenization of boundary value problems in perforated

domains with the third boundary condition and the resulting change in the character of the nonlinearity

in the problem, Differential Equations, 47 (2011), pp. 78–90.[28] , Averaging of boundary-value problems for the Laplace operator in perforated domains with a

nonlinear boundary condition of the third type on the boundary of cavities, Journal of MathematicalSciences, 190 (2013), pp. 181–193.

Dpto. Matematica Aplicada & Instituto de Matematica Interdisciplinar,

Universidad Complutense de Madrid


Dpto. Matematica Aplicada & Instituto de Matematica Interdisciplinar,

Universidad Complutense de MadridE-mail address: [email protected]

Faculty of Physics, University of Bucharest, RomaniaE-mail address: [email protected]

JEPE Vol 2, 2016, p. 131-144

MULTISCALE WEAK COMPACTNESS IN METRIC SPACES

GIUSEPPE DEVILLANOVA

Abstract. The aim of this paper is to provide a useful tool for a better understanding

of the approach proposed in [11] to extend to the setting of metric spaces profile decom-position theorems. To this aim we shall deal with a less general context which has the

advantage of making the analogies with the linear case more evident.

1. Introduction

The aim of this brief survey is to link some notions appeared in some recent papers dealingwith Profile Decomposition (see for instance [9], [11] to which we refer, from now on, forfurther details) with the final purpose of showing the main ideas which let these notions,which, at first glance, heavily depend on the linear structure of the space, to be formulatedeven in the much more general setting of metric spaces.

This result has been proved in [11], but in this note we prefer to deal with a less generalcase which has the advantage to make the analogies with the linear case more evident.

This note, which contains the exposition presented by the author at the “9th Europeanconference on Elliptic and Parabolic Problems ” held in May 2016 in Gaeta (Italy), isorganized as follows. In Section 2 we give a brief sketch of the main papers which led to theso called profile decomposition theorems and point out some differences between the mainstatements. In Section 3 we put into evidence the main difficulties met in [25] (see also [26])in the generalization process from Hilbert spaces or some concrete functional spaces to thefunctional analytic setting of Banach spaces. In sections 4 and 5 we describe the main toolsused to overcome such difficulties. In Section 6 we recall some definitions given in [9] inthe setting of Lp spaces to point out some differences between the main statements aboutprofile decomposition in literature. In Section 7 we propose suitable surrogates for thegroup G of Lp invariant scalings and for the algebraic zero, which allow to generalize tometric spaces the definition of all the notions which are basic for profile decomposition andprofile reconstruction, and to recover the main ingredients (see (NB) and (EB) below) whichare necessary to get the multiscale polar compactness property. In Section 8 we give a briefsketch of profile decomposition theorems by means of a (polar) profile reconstruction definedby means of a characterizing formula (similar to [9, Formula (4.18)]) which does not requireany linear structure.

2010 Mathematics Subject Classification. Primary: 46E35, 46E30; Secondary: 35B33.Key words and phrases. Concentration-Compactness, Profile Decompositions, Multiscale Analysis.Received 12/10/2016, accepted 24/10/2016.

The author is partially supported by the INdAM - GNAMPA Project 2016 “Fenomeni non-locali: teoria,metodi e applicazioni”.

131

132 GIUSEPPE DEVILLANOVA

2. Origins of profile decompositions theorems

Let us briefly mention the main papers which led to the so called profile decompositiontheorems which are a “quantitative” counterpart of the concentration-compactness princi-ple of P.L. Lions (see [17] and [18]). Indeed, while from one side concentration-compactnessprinciple was originally introduced for studying defects of compactness for minimizing se-quences of functionals arising in the field of Calculus of Variations, profile decompositiontheorems are structural statements about the behavior of bounded sequences at the lightof concentration-compactness phenomena, which have been applied to a great number ofevolution equations leading to striking global existence and scattering results in criticalcontexts.

The first profile decomposition result has been given in 1984 by M. Struwe [28, Propo-sition 2.1] for Palais-Smale (P.S. for short) sequences related to the functional E(u) =12

∫Ω|∇u|2dx− λ

2

∫Ωu2dx− 1

2∗

∫Ω|u|2∗dx, corresponding to the elliptic problem

(2.1)

−∆u = |u|2∗−2u+ λu in Ω ,

u = 0 in ∂Ω ,

settled in a smooth bounded domain Ω ⊂ RN , N > 2, where λ ∈ R and, for p < Np∗ = Np

N−p is, as usual, the critical exponent for the Sobolev embedding of H1,p0 (Ω) into

Lp(Ω). In such a case the sequence is approximated trough a finite sum of suitably scaledsolutions to the limit problem corresponding to (2.1) (i.e. when Ω and λ are replaced by RNand 0 respectively).

In 1995 S. Solimini proved a result [24, Theorem 2] in the spirit of Struwe Theorem, wherethe sum of “deflated profiles” is possibly infinite, but which holds true for any boundedsequence (in the Sobolev Spaces H1,p with respect to the Lebesgue norm of index p∗ corre-sponding to the critical embedding).

The statement of [24, Theorem 2] guarantees that the embedding of H1,p into Lp∗

isin some sense “multiscale compact” (nowadays the terminology is cocompact, i.e. compactwith respect to a “scalings group”) and quantifies the remarks given by P.L. Lions in thedescription of the concentration-compactness phenomena. Indeed, any bounded sequencein the homogeneous Sobolev space H1,p(RN ) for 1 < p < N admits a subsequence whoseelements are approximated by the sums

∑i∈N ρ

in(ϕi) (which are convergent unconditionally

with respect to i and uniformly with respect to n) of fixed “profiles” ϕi each one modulatedby a sequence of scalings (ρin)n∈N (which leave the Sobolev norm invariant) and of a reminderterm which converges to zero in the critical Lebesgue space Lp

∗(RN ).

One of the main purposes of [24, Theorem 2] was to show how the concentration - com-pactness phenomena are a consequence of the nonoptimality of Lp

∗for the embedding of

Sobolev spaces in the wider class of Lorentz spaces. Indeed these phenomena disappear inthe case of the optimal Lorentz space L(p∗,p) as well as Rellich Theorem fails in Lp

∗. In

addition, Struwe Theorem can be very easily deduced from [24, Theorem 2] by just noticingthat profiles of a P.S. sequence are solutions to a limit problem (and therefore their norm isbounded from below), so they are in finite number (see the energy estimate (EB) below).

Solimini’s result was then rediscovered some years later by P. Gerard (see [12, Theorem1.1]) and by S. Jaffard (see [13, Theorem 1]) in the framework of Sobolev spaces (of fractionalorder) with respect to the L2 and Lp norm respectively, under a slightly weaker form (since

MULTISCALE WEAK COMPACTNESS 133

the series of scaled profiles are replaced by larger and larger finite sums) and trough differentmethods.

Extensions of results of this type to other classes of functional spaces such as Besov,Triebel-Lizorkin and BMO spaces are presented in [2], [14] and [15] respectively.

It is worth to remark that, even if profile decomposition theorems allow to identify levelsat which P.S. condition fails, its use goes far beyond the P.S. levels analysis. Indeed asubstantially different use of profile decomposition theorems (see [8, Lemma 8]) allows toget the existence of infinitely many solutions to the equation −∆u+ a(x)u = up in R2 withp > 1, without asking on the continuous positive potential a(x) any symmetry assumption asin [32] or any small oscillation assumption as in [5, 6, 33] (the strongest subsequent result inthis direction is [19]). Even in [7] the use of a suitable profile decomposition (see [7, Theorem8]) (jointly with a local Pohozaev identity) allows to prove the existence of infinitely manypositive solutions to the Brezis-Nirenberg problem in a bounded domain of RN when N ≥ 7by proving compactness for sequences of solutions of approximating problems which aresomething very similar to a P.S. sequence but not exactly the same.

3. Toward a functional analytic version

K. Tintarev and K.-H. Fiesler gave the first general functional analytic formulation of [24,Theorem 2] in the context of Hilbert spaces in [30]. Of course this setting just covers the casep = 2, as in Struwe Theorem, since the other values of p give only Banach Spaces. So, toset satisfactory functional-analytic grounds (which include at least the concrete functionalspaces) profile decomposition has been formulated also in Banach spaces by S. Solimini andK. Tintarev in [25] and [26].

Without going into details of the above cited papers, one can state that the main diffi-culties in passing from Hilbert to Banach setting are connected with the need of recoveringthe two following bounds:

(NB) ‖un − u‖ ≤ ‖un‖+ o(1) ∀un u,

which allows the “reduction” of the norm when subtracting from a sequence its weak limit,and

(EB)

+∞∑i=1

‖ϕi‖21,2 ≤ lim infn→+∞

‖un‖22,

which gives an “energy” bound on the “profile bulk” of a sequence (and allows to quantify theabove reduction, indeed, by combining (EB) with (NB), one gets the so called Kadec-Kleeproperty, i.e. ‖un − u‖ → 0 whenever un u and ‖un‖ → ‖u‖).

Bound (EB) guarantees that the saturation process of a bounded sequence (un)n∈N by its“deflated profiles” (ρin(ϕi))n∈N (see Definition 2 below) must stop. Indeed, roughly speaking,when, scaling term by term the elements of (un)n∈N, one finds a nonzero weak limit (profile),by subtracting it from the scaled (un)n∈N one gets a sequence with a better bound on thenorm (see (NB)). Then, by using (EB), one reaches a sequence from which no other profilecan be extracted.

Analogous estimates (see (5.3) and (7.13) below) will be employed to extend profile decom-position results in more general spaces. In some sense we can say that profile decompositiontheorems or, more in general, what we shall call multiscale compactness theorems will bebased on estimates like (EB).


Relation (NB) holds true in Hilbert Spaces by orthogonality

(3.1) ‖un‖2 = ‖un − u‖2 + ‖u‖2 + o(1) ∀un u

and in some concrete cases of Banach spaces, such as Lp spaces, by Brezis-Lieb Lemma(see [3]) if one replaces weak convergence by (the stronger) a.e. convergence

(3.2) ‖un‖pp = ‖un − u‖pp + ‖u‖pp + o(1) ∀un → u a.e.

In addition estimates of this kind have many and significant applications to PDEs, inparticular, in a lot of problems with lack of compactness, they (jointly with the fact thatprofiles are solutions to the corresponding limit problem) allow to specify the “bad levels” forP.S. (i.e. levels at which the P.S. condition, for the corresponding functional, does not hold).In the last decades of the last century several variational methods have been developed inorder to avoid such bad levels and get existence and multiplicity of solutions.

We remark that, in [24], which is concerned with the Banach space H1,p, formula (3.1) isreplaced by (3.2) and a bound analogous to (EB) is obtained by replacing ‖ · ‖21,2 by ‖ · ‖p1,p(i.e. by associating to any profile ϕ the energy ‖ϕ‖p1,p instead of ‖ϕ‖21,2). In both formulasthe index p plays a fundamental role and a value of p is not expected in the generic case ofBanach spaces.

4. How to recover (NB): the use of polar convergence

Unfortunately in more general Banach spaces weak convergence does not work properly,indeed we have not a “splitting formula” like (3.1) or (3.2) from which to get (NB). Indeed,if un u, we can only deduce that

‖u‖ ≤ ‖un‖+ o(1) and ‖un − u‖ ≤ 2‖un‖+ o(1)

which is clearly not enough.So it is necessary to use a different mode of convergence of weak type. In the preliminary

version of [25] a suitable notion, called polar convergence, was introduced, and substituted inthe final version by the very similar notion of ∆-convergence introduced several years beforeby T.C. Lim in [16]. In a joint work with S. Solimini and K. Tintarev we have surveyed polarconvergence with respect to other modes of convergence in [10] (where we also addressedthe problem of the existence of a topology related to polar (or ∆) convergence) and belowwe rephrase [10, Definition 2.7].

Definition 1 (Polar limit). Let (un)n∈N ⊂ E be a sequence in a metric space (E, d). Onesays that u ∈ E is the polar limit of (un)n∈N and we shall write un u, if for every v 6= uwe have

(4.1) d(un, u) < d(un, v) for all large n.

Note that in the linear case, where d is the distance induced by the norm, (NB) istautological if one replaces weak convergence by polar convergence (and takes v = 0 in(4.1)).

The notion of polar limit of a bounded sequence (un)n∈N can be further clarified bymeans of the notions of asymptotic centers (denoted by cenas((un)n∈N)) and asymptoticradius (denoted by radas((un)n∈N)). They are respectively the minimum points and the


infimum value of the following functional (depending on (un)n∈N)) defined on E by settingfor all v ∈ E

(4.2) Ias(v) = lim supn

d(un, v).

So,

(4.3) radas((un)n∈N) := infv∈E

Ias(v) = infv∈E

lim supn

d(un, v).

We emphasize that, while the asymptotic radius always exists and is uniquely determined,asymptotic centers may not exist or may be not uniquely determined. Therefore, the symbolcenas((un)n∈N) must be understood in the same sense as the limit symbol in a topologicalspace which is not assumed to be Hausdorff. Note that if un u then, see [10, Remark 2.4],u is one of the asymptotic centers of (un)n∈N, therefore the following equality holds true

(4.4) radas((un)n∈N) = lim supn

d(un, u) ∀un u.

When, as it will happen in the following section, the Banach space is assumed to beuniformly smooth and uniformly convex (see [21, Definition 1.e.1] or (5.1) below) polarconvergence has many nice properties. Indeed, in such setting, polar convergence is of“dual” type. Indeed, suitably defining the dual element u′ of u ∈ E we have (see [10,Theorem 5.5]) that

(4.5) un 0 in E iff u′n 0 in E′

(this is the reason for which we have chosen to flip the symbol , used for weak convergence,getting to denote polar convergence).

Moreover, since in Hilbert spaces each element coincide with its dual, it follows that, insuch a setting, polar and weak convergence agree. It is worth to underline that when thetwo “weak” modes of convergence agree the space is not necessarily Hilbert. In such a casewe say that the space is equipped with an Opial norm (see [22]). We should like to point outthat the usual Lp norm is not an Opial norm for p 6= 2, (i.e. polar and weak convergence donot coincide in Lp spaces), but, on the other hand, as a particular case of a general resultdue to D. van Dulst (see [31]), there exists an equivalent Opial norm (of course, unlike weakconvergence, polar convergence is not invariant on passing to equivalent norms).

Note that (4.5) allows to characterize polar convergence in Lp spaces by using its dualnature, i.e.

un 0 in Lp iff |un|p−2un 0 in Lp′.

Since a.e. convergence always implies weak convergence and since, in the case p = 2, weakand polar convergence agree, one can think to recover (3.2) for a general p by replacing a.e.convergence with both weak and polar convergence. Indeed, S. Solimini and K. Tintarevin [25, Theorem 4.2] proved that, when p ≥ 3 (or p = 2),

‖un‖pp ≥ ‖u‖pp + ‖un − u‖pp + o(1) ∀un u and un u,

(from which (NB) follows) i.e. a partial extension to the space of p-summable functions(defined on any measurable space) of the well known Brezis-Lieb Lemma holds true.


5. How to recover (EB): the use of uniform convexity

As recalled above, in a general Banach space we have no index p to use to get an energyestimate analogous to (EB) (where p = 2). To this aim we shall require uniform convexityand make use of the modulus of convexity (composed with the norm).

We recall that a normed vector space E is called uniformly convex if the following functionδ : [0, 2]→ R, called modulus of convexity of the space, defined by setting

(5.1) δ(ε) = infu,v∈E, ‖u‖=‖v‖=1,

‖u−v‖=ε

(1−

∥∥∥∥u+ v

2

∥∥∥∥)

is strictly positive on (0, 2].Recall that, by [21, Proposition 1.e.2], uniform smoothness corresponds to uniform con-

vexity of the dual.In the case of uniformly smooth and uniformly convex Banach spaces we have, see [25,

Formula (1.4)], the following bound (for ‖un‖, ‖u‖ ≤ 1)

(5.2) ‖un − u‖ ≤ ‖un‖ − δ(‖u‖) + o(1) ∀un u 6= 0,

which quantifies the “reduction” of the norm caused by the subtraction of the polar limit.So, a suitable energy has been introduced in [25] which employes the modulus of convexity δby associating to any profile ϕ the real number δ(‖ϕ‖). In this way, see [25, Formula (1.2)],the bound (EB) has been transformed (when lim infn→+∞ ‖un‖ ≤ 1) in the following energybound

(5.3)

+∞∑i=1

δ(‖ϕi‖) ≤ lim infn→+∞

‖un‖ ≤ 1.

Note that for every 1 < p < +∞ Lp spaces are uniformly convex (see Clarkson inequalitiesin [4]), moreover the asymptotic behavior of the modulus of convexity shows that, when2 ≤ p < +∞, δ ‖ · ‖ ' p−12−p‖ · ‖pp and so (5.3) takes again the “shape” of the energybound (EB).

6. Some recent terminology

The existence of the above mentioned distinct and slightly weaker formulations of [24,Theorem 2] due to P. Gerard and S. Jaffard (where the sum in the approximation is replacedby a finite sum whose number of addenda increases as n→ +∞) has probably contributed toa misunderstanding concerning the two statements. Both of them are stated for “suitable”sequences of scalings but such sequences, as will be clarified later, are not the same in the twocases. The difference between the two statements has been underlined also by T. Tao in [29],in which the finite sum version (in another setting) has been defined “more convenient” whileit clearly is a weaker result which can be stated under weaker assumptions (compare thestatements of [9, Corollary 6.3] and [9, Corollary 6.4]). This circumstance does not facilitatethe use of profile decomposition theorems in applications and partially justifies the stillcurrent specific arguments in giving the proof of the compactness of P.S. sequences relatedto distinct functionals, producing minor variants of Struwe Theorem which can all be easilydeduced by profile decomposition, as we have mentioned for the original one.

We want to recall some terminology introduced in [9] and mainly used to underline thedifferences of the two statements specifying what the term “suitable” means in the two cases.


In [9] we introduced the notions of profile and profile system of a bounded sequence, and thenotion of s.t.s. system (scale transition sequences system) or blowup system (related to aprofile system (ϕi)i∈I) and, among them, some specialized s.t.s. systems ((ρin)n∈N)i∈I whichwe call routed s.t.s. which (means that the sums

∑i∈I ρ

in(ϕi) are convergent unconditionally

in i and uniformly with respect to n and which), roughly speaking, allows to treat the sumin (EB) as a finite sum.

In the remaining part of this section G will denote the group generated by the Lp invariantscalings, i.e. the group generated by the functions g : Lp → Lp such that, for all u ∈ Lp,g(u) = λ

Np u(λ(· − x0)) for some λ > 0 and x0 ∈ RN .

Definition 2 (Profiles and s.t.s. sequences or blowup sequences). Let (un)n∈N ⊂ Lp(RN )be a given bounded sequence. We say that ϕ ∈ Lp(RN ) \ 0 is a profile of the sequence(un)n∈N if there exists ρ = (ρn)n∈N ⊂ G such that

(6.1) ρ−1n (un) ϕ.

In such a case we shall say that ρ = (ρn)n∈N is a scale transitions sequence (s.t.s. for short)or a blowup sequence related to the profile ϕ.

Note that if ϕ is a profile of the sequence (un)n∈N and ρ = (ρn)n∈N is a s.t.s. related toϕ, then any σ = (σn)n∈N such that σ−1

n ρn → id (id denotes the identity map on E) is stilla s.t.s. related to ϕ, while for all g ∈ G, g(ϕ) is still a profile of the sequence (un)n∈N and(ρn g−1)n∈N is a s.t.s. related to the profile g(ϕ).

Therefore we shall say that two profiles ϕ and ψ of a sequence (un)n∈N are distinct ifψ 6= g(ϕ) for all g ∈ G while they are copies if exists g ∈ G such that ψ = g(ϕ). So anyprofile can be thought as a whole orbit of copies G(ϕ) := (g(ϕ))g∈G. Finally, by taking intoaccount [9, Remark 2.1], we deduce that if (ρn)n∈N and (σn)n∈N are s.t.s. related to distinctprofiles they must be mutually diverging or quasi orthogonal (i.e. (σ−1

n ρn)n∈N is diverging,which, in turn, means that σ−1

n ρn → 0).

Definition 3 (Profile system). Let (un)n∈N ⊂ Lp(RN ) be a bounded sequence. A family(ϕi)i∈I of profiles of the sequence (un)n∈N is said to be a profile system (in Lp(RN )) of thesequence (un)n∈N if, for any profile ϕ, all elements ϕi which are copies of ϕ are equal andtheir number is (finite and) less or equal to the multiplicity m(ϕ) of the profile. (m(ϕ) isthe supremum of the cardinality of the sets of mutually diverging sequences related to ϕ).

It is worth to remark that any profile system is also a profile system of every subsequence.

Definition 4 (s.t.s. system (or blowup system), concentration system). Let Φ = (ϕi)i∈I bea profile system of a (bounded) sequence (un)n∈N ⊂ Lp(RN ). A s.t.s. system or a blowupsystem related to the profile system Φ is any family P = (ρi)i∈I such that

i) for all i ∈ I, ρi = (ρin)n∈N ⊂ G is a s.t.s. sequence related to the profile ϕi;ii) for all i, j ∈ I, i 6= j, ρi and ρj are mutually diverging.

In such a case the pair (Φ,P ) will be called a concentration system of (un)n∈N.

Definition 5 (Complete profile system, profile convergent sequence). We shall say that a(possibly empty) profile system (ϕi)i∈I of a bounded sequence (un)n∈N is complete if nosubsequence (ukn)n∈N has a “richer profile system” (i.e. a profile system which has a newprofile or a profile with a bigger multiplicity). If a sequence admits a complete profile systemwe shall say that it is profile convergent. Therefore a given bounded sequence (un)n∈N is


profile convergent if it does not admit any subsequence with a bigger number of profiles, orwith profiles with higher multiplicity.

The existence of a complete profile system for a bounded sequence in Lp is guaranteedby [9, Theorem 3.1] which is a “multiscale” version of Banach Alaouglu Theorem and isreported below for the reader’s convenience.

Theorem 1 (Multiscale weak compactness). Any bounded sequence in Lp(RN ) admits aprofile convergent subsequence.

Note that the multiscale weak compactness theorem, combined with the “theorem ofalternative” [24, Theorem 1] allows to immediately deduce profile decomposition in Sobolevspaces, namely [24, Theorem 2] from the “cocompact embedding” of H1,p in Lp

∗. Such

cocompactness result, thanks to the Sobolev embedding in Lorentz spaces, admits a simpleproof which is carried out on the Marcinkiewicz space of index p∗. This is not just a technicaldevice because this result is false in the case of the optimal embedding in the wider categoryof Lorentz spaces, as analogously happens with Rellich Theorem in the category of Lebesguespaces.

7. Passing to metric spaces

Now, we have concrete examples of functional spaces and a theory sufficiently general toinclude them. It may make sense to think to the more general context of metric spaces.Note that in the metric settlement the main cited theorems have no meaning since they arebased on the notion of sum and therefore they heavily exploit the (now lacking) algebraicstructure of the space. We shall show that, thanks to the reinterpretation of all relevantnotions (such as profile, scale transitions sequence or blowup sequence, profile system, etc.),given in [9] and recalled in the previous section, we can approach the problem by takingas scalings the elements of a suitable group G of isometries which acts on the metric space(E, d) and which will surrogate the group of the invariant scalings in Lp.

The lack of a zero (in the algebraic sense) can be then compensated by the request thatall elements in G admit a unique common fixed point z which leads to the request of thefollowing axiom.

Axiom G1. There exists a unique (“zero”) z ∈ E such that g(z) = z for all g ∈ G.

This request, by following the concrete example given by the Lp invariant scalings, canbe reduced to the group H of translations which is a commutative invariant subgroup of Gas shown in the following result.

Theorem 2. Let G be a group of isometries on a metric space E. Let H ⊂ G be aninvariant commutative subgroup of G which satisfies

i) ∀f ∈ H, f 6= id, f has at most two fixed points,ii) ∃h ∈ H which admits a unique fixed point z ∈ E.

Then G admits a unique fixed point z.

Proof. We start by claiming that z in ii) is the unique fixed point of every f ∈ H, f 6= id.Indeed, given f ∈ H, we have, since z is fixed by h and H is commutative, that f(z) =f(h(z)) = h(f(z)) and deduce that also f(z) is fixed by h, then by assumption ii), we deducef(z) = z, i.e. z is fixed by f . Moreover, if f(x) = x for some x ∈ E, and f ∈ H, being H


commutative, we have f(h(x)) = h(f(x)) = h(x) i.e. h(x) is fixed by f . If x 6= z we haveh(x) 6= h(z) = z and, by ii), since x cannot be fixed by h, that h(x) 6= x. So x, h(x) and zare three distinct fixed points of f , in contradiction to i) and this proves the claim.

We prove now that z is fixed by every map g ∈ G. Indeed, given g ∈ G and f ∈ H \ id,we have, since H is invariant, that the map ϕ = gf g−1 ∈ H and ϕ(g(z)) = g(f(z)) = g(z),so g(z) is fixed by ϕ ∈ H and so g(z) = z. The uniqueness of z is a trivial consequenceof ii).

On the other hand, to simulate the behavior of Lp scalings we shall ask also the followingcompactness axiom on G of dichotomy (alternative) type which furthermore guarantees thatthe group G is closed with respect to the (strong pointwise) convergence (see [11]).

Axiom G2. For any sequence (gn)n∈N ⊂ G, if (gn)n∈N is not diverging (i.e. if for all u ∈ E(gn(u))n∈N does not polarly converge to the “zero” z ∈ E), then there exists a subsequence(gkn)n∈N of (gn)n∈N which is converging.

Since Lp spaces are, for 1 < p < +∞, uniformly smooth and uniformly convex Banachspaces (see [4]), we shall consider the corresponding metric counterpart known as StapleRotundity (see [10, Definition 3.1 and Remark 3.2]).

Definition 6. A metric space (E, d) is a (uniform) SR (“Staples rotund”) metric space (orsatisfies (uniformly) property SR) if there exists a continuous function δ : (R+)2 → R+

(called modulus of rotundity of the space) such that for any r, d > 0, for any u, v ∈ E withd(u, v) ≥ d:

(SR) rad(Br(u) ∩Br(v)) ≤ r − δ(r, d),

where, for any subset X ⊂ E, rad(X) := infu∈E supv∈X d(u, v) is the Chebyshev radius ofthe set X.

We take the opportunity to recall that the Chebyshev center of a set X ⊂ E, denoted bycen(X), is one of the points (if they exist) in E such that supv∈X d(cen(X), v) = rad(X) (i.e.are the centers (if they exist) of balls with minimal radius (the Chebyshev radius) containingthe set X).

Note that, given r > 0 and d > d′> 0, since if d(u, v) ≥ d then also d(u, v) > d

′

we get, by (SR), that rad(Br(u) ∩ Br(v)) not only is bounded by r − δ(r, d) but also by

r− δ(r, d ′). Therefore we deduce that δ(r, d′) cannot be bigger than δ(r, d) and so we shall

always assume that the modulus of rotundity in monotone increasing with respect to d.It is worth to recall that, in complete metric spaces, property SR guarantees, among other

things, existence and uniqueness of the asymptotic center of a bounded sequence (see [27,Theorem 2.5 and Theorem 3.3]). Moreover, see [10, Section 3-Statement a) and Remark2.4], if the sequence is polar convergent, then the polar limit coincides with the asymptoticcenter of the sequence and therefore (4.4) holds true. Finally the space is (sequentially)compact with respect to polar convergence as stated in the following result proved in [10].

Theorem 3. Let (E, d) be a complete SR metric space. Then every bounded sequence in Ehas a polar convergent subsequence.

At this point the notions of profile, profile system, s.t.s. systems, etc. recalled in Section 6can be extended in the metric setting (by replacing Lp and 0 by a complete SR metric space


and by z respectively) and by replacing weak convergence ( sign) by polar convergence( sign).

Next step is to recover the polar version of Theorem 1.Let us briefly recall what we have get about profile decomposition in Banach spaces:

• a notion of polar convergence which perfectly fits in the metric context;• a notion of uniform convexity which has in the Staple rotundity its metric counter-

part;• a energy function δ ‖ · ‖ which is based on the modulus of convexity (and on the

norm) and which can be adapted to metric spaces by means of the modulus of Staplerotundity (and of the distance d(·, z) from the point z given by Axiom G1);• a notion of profile system (and related s.t.s. system) and a weak multiscale com-

pactness (see Theorem 1) which do not require any algebraic structure.

So, it has been quite natural to ask for the existence of profile decomposition theorems inthe more general context of metric spaces and in [11], with S. Solimini and K. Tintarev,we have positively answered the question acting on suitable (and more general) isometriesgroup.

The main difficulty to overcome has been the inability to “subtract” a profile from aprofile system in order to get a (less “rich”) new profile system (with a lacking profile).Indeed, in the linear setting, by subtracting the weak limit u (which is a particular profile)of a sequence (un)n∈N one gets a sequence (un − u)n∈N which admits a profile system Φwhich, due to (5.3) has an energy which, by (NB), is less than the original one Φ∪ u (seeRemark 1 below). This property, of course, corresponds to some “additivity property” ofthe distance function which, in general, does not hold since there is no additivity structureon the space.

So we have to reinterpret both (NB) and (5.3). In particular since our norm will bereplaced by the distance from z, i.e. ‖ · ‖ ' d(·, z), in order to evaluate the right handside of (NB) we introduce the following “asymptotic norm” or “z-radius”, denoted byradz((un)n∈N), of a (bounded) sequence (un)n∈N by setting

(7.1) radz((un)n∈N) := lim supn

d(un, z).

Moreover, since by (6.1) profiles of a bounded sequence (un)n∈N are obtained as polar limitof sequences (ρ−1

n (un))n∈N with (ρn)n∈N ⊂ G, we introduce also the multiscale asymptoticradius, denoted by radG((un)n∈N), of a (bounded) sequence (un)n∈N by setting

(7.2) radG((un)n∈N) := inf(ρn)n∈N∈G

radas((ρn(un))n∈N).

Note that, for any sequence (un)n∈N ⊂ E, we have

(7.3) radG((un)n∈N) ≤ radas((un)n∈N) ≤ radz((un)n∈N),

where radas((un)n∈N) is defined by (4.3).It is worth to remark that, in the linear case (where z = 0), since it is possible to subtract

u from each element of the sequence, we have, by (4.4),

radas((un)n∈N) = lim supn

d(un − u, 0) = rad0((un − u)n∈N) = radz((un − u)n∈N).

So in the linear case, by subtracting the polar limit u of a bounded sequence (un)n∈N onegets a new sequence (un−u)n∈N which has a less rich profile system (it just misses the polar


limit we have subtracted) and whose z-radius radz is equal to the asymptotic radius radasof the given sequence. If, instead of polar limit we want to speak more generally aboutprofiles, we shall consider radG instead of radas so, every time we subtract a profile we geta new sequence for which the radz is equal to the radG of the previous sequence.

Such result can be postulated, without any additive structure, by Axiom G3 below whichdeals with the extension of the above notions to the set of all sequences which admit a givenprofile system (and a related s.t.s. system). More precisely, given I ⊂ N, Φ = (ϕi)i∈I afamily in E \ z, and P = (ρi)i∈I = ((ρin)n∈N)i∈I a family of sequences of G, we define

(7.4) U(Φ) := (un)n∈N | Φ is a profile system of (un)n∈N

and

(7.5) U(Φ,P ) := (un)n∈N | (Φ,P ) is a concentration system of (un)n∈N.

When U(Φ) 6= ∅, i.e. when there exists a sequence of which Φ is a profile system, we shallabuse Definition 3 by saying that Φ = (ϕi)i∈I ⊂ E \ z is a “profile system”. Analogously,we shall abuse Definition 4 by saying that P is a “s.t.s. system” or a “blowup system”related to Φ if U(Φ,P ) 6= ∅. Moreover, in such a case, we shall also say that P is compatiblewith Φ or that (Φ,P ) is a “concentration system”. So, when (Φ,P ) is a “concentrationsystem” the following properties hold true:

i) if ∃g ∈ G and ∃i, j ∈ I such that ϕj = g(ϕi) then g is the identity map id on E,ii) ∀i, j ∈ I, i 6= j, ρi and ρj are quasiorthogonal.

With a clear abuse of terminology that does not give rise to possible misunderstandings,we shall say that a system Φ′ (resp. a concentration system (Φ′,P ′)) is included in - or is asubsystem of - Φ (resp. (Φ,P )) and we shall write Φ′ ⊂ Φ (resp. (Φ′,P ′) ⊂ (Φ,P )) if thereexists J ⊂ I such that Φ′ = (ϕi)i∈J (and, in the respective case, P ′ = (ρi)i∈J). Finally, weshall say that Φ′ (resp. (Φ′,P ′)) is a maximal subsystem of Φ (resp. (Φ,P ))) if the set I \ Jreduces to a single point.

In the remaining part of the paper we shall reserve the notation U(Φ) (resp. U(Φ,P )) toprofile systems Φ (resp. concentration systems (Φ,P )).

Remark 1. When (un)n∈N ∈ U(Φ) and ϕ is a profile of (un)n∈N and ρ = (ρn)n∈N is arelated s.t.s. we shall write that (un)n∈N ∈ U(Φ∪ ϕ) and we shall say that Φ∪ ϕ is theprofile system of (un)n∈N obtained by “adding” ϕ to Φ if one of the following alternativeshold true:

a) ϕ 6= ρ(ϕi) for all i ∈ I, ρ ∈ G (i.e. when we are really adding a new profile);b) if there exists i ∈ I and ρ ∈ G such that ϕ = ρ(ϕi) then ρ is quasiorthogonal to

every ρj such that ϕj = ϕi (i.e. when we are increasing the multiplicity of ϕ sinceit already is one of the profiles in Φ).

Denoting by U one of the sets U(Φ) or U(Φ,P ), we define the following asymptotic radiusnotions of the set U .

(7.6) radas(U) := inf(un)n∈N∈U

radas((un)n∈N),

(7.7) radz(U) := inf(un)n∈N∈U

radz((un)n∈N),


and

(7.8) radG(U) := inf(un)n∈N∈U

radG((un)n∈N).

Note that

(7.9) radas(U(Φ)) = inf(un)n∈N∈U(Φ)

radG((un)n∈N) = radG(U(Φ)),

and that (at the light of the surrogate d(·, z) for the “norm”) radz(U(Φ)) will be the goodcandidate to replace the right hand side of (5.3) (and therefore of (EB)).

Finally, remark that the above defined radii are monotone increasing functions with re-spect to inclusion. More precisely, if rad denotes one of the radii defined by (7.6), (7.7) and(7.8), then

(7.10) rad(U(Φ′,P ′)) ≤ rad(U(Φ,P )) ∀(Φ′,P ′) ⊂ (Φ,P ).

When strict inequality holds in (7.10) we shall say that rad is strictly increasing with respectto inclusion.

Now we are able to state the last axiom required to the group G.

Axiom G3. The function radz is strictly increasing with respect to inclusion. Moreover,any finite profile system Φ admits a maximal profile subsystem Φ′ such that

(7.11) radz(U(Φ′)) ≤ radas(U(Φ)).

By making use of the modulus of rotundity δ of the space, for any R > 0 we can definethe function δR : ]0, 2R]→ R+, by setting for all 0 < d ≤ 2R

(7.12) δR(d) := mind2≤r≤R

δ(r, d).

Note that, fixed R > 0, since the modulus of rotundity δ is monotone increasing withrespect to the variable d and since the interval [2−1 d,R] reduces when d increases, we deducethat the function δR is monotone increasing (in the variable d). On the contrary, fixed d > 0,the value δR(d) decreases if R ≥ 2−1 d increases. Finally, since δR(d) ≤ δ(2−1 d, d) ≤ 2−1 d,we can extend δR to 0 by setting δR(0) = 0.

This monotone cost function, depending on R (and composed with the “norm” d(·, z))allows to associate an energy VR(ϕ) = δR(d(ϕ, z)) to any profile ϕ of a given (bounded)sequence (un)n∈N such that radz((un)n∈N) < R and to provide an energy estimate (of thesum of the energies of the profiles of the sequence), similar to formula (5.3) (and thereforeto (EB)) stated in the following lemma which is proved (in a more general approach in [11]).

Lemma 1 (Energy estimate). Let R > 0 be given. Then, for any profile system Φ = (ϕi)i∈Isuch that radz(U(Φ)) < R, we have

(7.13) VR(Φ) :=∑i∈I

VR(ϕi) =∑i∈IδR(d(ϕi, z)) ≤ radz(U(Φ)) < R.

By taking into account that, as remarked above, the function δR is decreasing with respectto R, we get that the energy bound (7.13) still holds true if one replaces R by any larger realnumber, indeed the left hand side decreases while the right hand side increases (the centralterm is independent on R). Therefore smaller is R more significant is (7.13).

Thanks to the above estimate it is possible to recover without any relevant effort themaximality argument used in [9] to prove the multiscale compactness theorem in Lp spaces.


Theorem 4 (Multiscale polar compactness). Let E be a complete SR metric space, withan admissible group G of scalings satisfying axioms G1, G2 and G3. Then any boundedsequence in E admits a (polar) profile convergent subsequence.

8. What about profile decomposition in metric spaces

At this point the metric structure does not allow to speak about profile reconstructionas the sum of “deflated” profiles used in Lp (see [9, Definition 4.3]) and which can of coursebe written, without big troubles (the sum becomes a series), in the linear setting. Sincein general, in a metric space, we have not any algebraic structure, we shall use insteada suitable counterpart of the characterizing formula, given for Lp spaces, in [9, Formula(4.18)]. Therefore, as a first attempt, one can give as a definition of profile reconstruction(of a given concentration system (Φ,P )) the sequence in U(Φ,P ) which is closer to z, i.e.which minimizes radz(U(Φ,P )).

By introducing some extra axioms on the space, it is possible to prove that given anyconcentration system (Φ,P ) the related profile reconstruction is unique (modulo infinites-imal terms and modulo subsequences). Moreover the profile reconstruction approximates,modulo an infinitesimal term, a given profile convergent sequence which admits (Φ,P ) as acomplete concentration system.

In this way we get a metric version of profile decomposition theorems. The reader canrefer to [11] for more details.

References

[1] T. Aubin, Nonlinear analysis on manifolds, Monge-Ampere equations. Grundlehren 252. Berlin Heidel-

berg New York: Springer 1982.

[2] H. Bahouri, A. Cohen and G. Koch, A general wavelet-based profile decomposition in the critical embed-ding of function spaces, Confluentes Math. 3 (2011) 387-411.

[3] H. Brezis and E. Lieb, A relation between pointwise convergence of functions and convergence of func-tionals. Proc. Amer. Math. Soc. 88 (1983), 486-490.

[4] J. A. Clarkson, Uniformly convex spaces, Trans. Amer. Math. Soc., vol. 40 (1936), 396-414.

[5] G. Cerami, D. Passaseo, S. Solimini, Infinitely many positive solutions to some scalar field equationswith non-symmetric coefficients, Comm. Pure Appl. Math. 66 (2013), no. 3, 372-413.

[6] G. Cerami, D. Passaseo, S. Solimini, Nonlinear scalar field equations: existence of a positive solutionwith infinitely many bumps, Ann. Inst. H. Poincare Anal. Non Lineaire 32 (2015), no. 1, 23-40.

[7] G. Devillanova, S. Solimini, Concentration estimates and multiple solutions to elliptic problems at criticalgrowth, Adv. Differential Equations 7 (2002), no. 10, 1257-1280.

[8] G. Devillanova, S. Solimini, Infinitely many positive solutions to some nonsymmetric scalar field equa-tions: the planar case. Calc. Var. Partial Differential Equations 52 (2015), no. 3-4, 857-898, DOI:10.1007/s00526-014-0736-7.

[9] G. Devillanova, S. Solimini, Some remarks on profile decomposition theorems, Advanced Nonlinear Stud-ies 2016, DOI: 10.1515/ans-2015-5049.

[10] G. Devillanova, S. Solimini, C. Tintarev, On weak convergence in metric spaces, in Nonlinear Analysisand Optimization, Contemporary Mathematics, vol. 659, Amer. Math. Soc., Providence, RI, 2016, 43-63.

[11] G. Devillanova, S. Solimini, K. Tintarev, Profile decomposition in metric spaces, preprint.[12] P. Gerard, Description du defaut de compacite de l’injection de Sobolev. ESAIM: Control, Optimisation

and Calculus of Variations 3 (1998), 213-233.

[13] S. Jaffard Analysis of the lack of compactness in the critical Sobolev embeddings J. Funct. Anal., 161

(1999), 384-396.[14] G. S. Koch, Profile decompositions for critical Lebesgue and Besov space embeddings, Indiana Univ.

Math. J. 59 (2010), no. 5, 1801-1830.[15] G. Kyriasis, Nonlinear approximation and interpolation spaces, J. Approx. Theory 113 (2001) 110-126.


[16] T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976) , 179-182.

[17] P-L. Lions, The concentration-compactness principle in the calculus of variations. The Limit Case, Part1, Revista Matematica Iberoamericana 1.1 (1985), 145-201.

[18] P-L. Lions, The concentration-compactness principle in the calculus of variations. The Limit Case, Part

2, Revista Matematica Iberoamericana 1.2 (1985), 45-121.[19] R. Molle, D. Passaseo, Multiplicity of solutions of nonlinear scalar field equations, Atti Accad. Naz.

Lincei Rend. Lincei Mat. Appl., 26 (2015), no. 1, 75-82.

[20] F. Maddalena, S. Solimini, Synchronic and asynchronic descriptions of irrigation problems. Adv. Non-linear Stud. 13 (2013) no. 3, 583-623.

[21] J. Lindenstrauss, L. Tzafriri, Classical Banach spaces II, Springer-Verlag, New York/Berlin, (1996)

reprint.[22] Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings,

Bull. Amer. Math. Soc. 73 (1967), 591-597.

[23] P. Sacks, K. Uhlenbeck On the existence of minimal immersions of 2-spheres. Ann. Math. 113 (1981),1-24.

[24] S. Solimini, A note on compactness-type properties with respect to Lorentz norms of bounded subsetsof a Sobolev space. Ann. Inst. H. Poincare Anal. Non Lineaire 12 (1995), 319-337.

[25] S. Solimini, K. Tintarev, Concentration analysis in Banach spaces. Communications in Contemporary

Mathematics (2015), DOI: 10.1142/S0219199715500388.[26] S. Solimini, K. Tintarev, On defect of compactness in Banach Spaces. Comptes Rendus Mathematique,

353 (2015), no 10, 899-903.

[27] J. Staples, Fixed point theorems in uniformly rotund metric spaces, Bull. Austral. Math. Soc. 14 (1976),no. 2, 181-192.

[28] M. Struwe, A global compactness result for elliptic boundary value problems involving limiting nonlin-

earities. Math. Z. 187 (1984), 511-517.[29] T. Tao, Concentration compactness via nonstandard analysis. Therence Tao Blog: What’s new,

http://terrytao.wordpress.com/2010/11/29/concentration-compactness-via-nonstandard-analysis/, No-

vember 10, 2010.[30] K. Tintarev, K-H. Fieseler, Concentration Compactness: Functional-Analitic Grounds and Applications

Imperial College Press, London (2007).[31] D. van Dulst, Equivalent norms and the fixed point property for nonexpansive mappings, J. London

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tential, Calc. Var. Partial Differential Equations 51 (2014), no 3, 761-798.

Mechanics, Mathematics and Management Department, Politecnico di Bari, via Amendola,126/B, IT-70126 Bari


JEPE Vol 2, 2016, p. 145-155

MONOTONICITY OF BISTABLE TRANSITION FRONTS IN RN

HONGJUN GUO AND FRANCOIS HAMEL

Abstract. This paper is concerned with the monotonicity of transition fronts for bistable reaction-diffusion equations. Transition fronts generalize the standard notions of traveling fronts. Known exam-ples of standard traveling fronts are the planar fronts and the fronts with conical-shaped or pyramidallevel sets which are invariant in a moving frame. Other more general non-standard transition frontswith more complex level sets were constructed recently. In this paper, we prove the time monotonicityof all bistable transition fronts with non-zero global mean speed, whatever shape their level sets mayhave.

Dedicated to Professor David Kinderlehrer

1. Introduction

This paper is concerned with the monotonicity of generalized fronts for the semilinear parabolicequation

(1.1) ut = ∆u + f (u), (t, x) ∈ R × RN ,

where ut = ∂u∂t and ∆ denotes the Laplace operator with respect to the space variables x ∈ RN . The

function f is assumed to be of the bistable type, namely the states u = 0 and u = 1 are assumed tobe both stable stationary states (more precise assumptions will be given later). A typical example isthe cubic nonlinearity fθ(s) = s(1 − s)(s − θ) with 0 < θ < 1.

It is well known that in one dimension, under some assumptions on f , (1.1) admits standardtraveling fronts, that is, solutions of the type

u(t, x) = φ(x − c f t)

where the front speed c f ∈ R and the front profile φ : R→ [0, 1] satisfyφ′′ + c fφ

′ + f (φ) = 0,φ(−∞) = 1, φ(+∞) = 0.

(1.2)

For precise conditions for the existence and non-existence, we refer to Fife and McLeod [6]. It hasalso been proved that if a front (c f , φ) solving (1.2) exists, it is uniquely determined up to shifts for

2010 Mathematics Subject Classification. Primary: 35B08, 35C07, 35K57; Secondary: 35B45.Key words and phrases. Reaction-diffusion equations, Transition fronts, Monotonicity.Received 26/10/2016, accepted 28/10/2016.This work has been carried out thanks to the support of the A*MIDEX project (no ANR-11-IDEX-0001-02) and

Archimede Labex (no ANR-11-LABX-0033) funded by the “Investissements d’Avenir” French Government program, man-aged by the French National Research Agency (ANR). The research leading to these results has also received funding fromthe ANR within the project NONLOCAL (no ANR-14-CE25-0013) and from the European Research Council under theEuropean Union’s Seventh Framework Programme (FP/2007-2013) / ERC Grant Agreement n.321186 - ReaDi - Reaction-Diffusion Equations, Propagation and Modelling.

145

146 HONGJUN GUO AND FRANCOIS HAMEL

φ and there holds φ′(ξ) < 0 for ξ ∈ R. In particular, for such a traveling front u(t, x) = φ(x − c f t),observe that the time derivative ut(t, x) = −c fφ

′(x − c f t) has a constant sign for all (t, x) ∈ R × R.For higher dimensions N ≥ 2, an immediate extension of one-dimensional traveling fronts con-

sists in planar traveling frontsu(t, x) = φ(x · e − c f t)

for any given unit vector e of RN , where c f and φ are as above. The level sets of such travelingfronts are parallel hyperplanes which are orthogonal to the direction of propagation e. These frontsare invariant in the moving frame with speed c f in the direction e. The existence and uniqueness ofthese fronts can be referred to the one-dimensional traveling fronts. Besides, in RN with N ≥ 2, moregeneral traveling fronts exist, which have non-planar level sets. For instance, conical-shaped axisym-metric non-planar fronts are known to exist for some f , see [9, 18]. Fronts with non-axisymmetricshapes, such as pyramidal fronts, are also known to exist, see [26, 28]. For qualitative properties ofthese traveling fronts, we refer to [8, 9, 10, 18, 19, 22, 27, 28].

Even if the types of traveling fronts are various, they share some common properties. For all ofthem, the solutions u converge to the stable states 0 or 1 far away from their moving or stationarylevel sets, uniformly in time. This fact led to the introduction of a more general notion of travelingfronts, that is, transition fronts, see [2, 3, 7] and see [23] in the one-dimensional setting. In orderto recall the notion of transition fronts, one needs to introduce a few notations. First, for any twosubsets A and B of RN and for x ∈ RN , we set

d(A, B) = inf|y − z|; (y, z) ∈ A × B

and d(x, A) = d(x, A), where | · | is the Euclidean norm in RN . Consider now two families (Ω−t )t∈R

and (Ω+t )t∈R of open nonempty subsets of RN such that

∀t ∈ R,

Ω−t ∩Ω+

t = ∅,

∂Ω−t = ∂Ω+t =: Γt,

Ω−t ∪ Γt ∪Ω+t = RN ,

supd(x,Γt); x ∈ Ω+t = supd(x,Γt); x ∈ Ω−t = +∞

(1.3)

and inf

supd(y,Γt); y ∈ Ω+

t , |y − x| ≤ r; t ∈ R, x ∈ Γt

→ +∞

inf

supd(y,Γt); y ∈ Ω−t , |y − x| ≤ r

; t ∈ R, x ∈ Γt

→ +∞

as r → +∞.(1.4)

Notice that the condition (1.3) implies in particular that the interface Γt is not empty for every t ∈ R.As far as (1.4) is concerned, it says that for any M > 0, there is rM > 0 such that for any t ∈ R andx ∈ Γt, there are y± = y±t,x ∈ R

N such that

y± ∈ Ω±t , |x − y±| ≤ rM and d(y±,Γt) ≥ M.(1.5)

that is, y± ∈ B(x, rM) and B(y±,M) ⊂ Ω±t , where B(y, r) denotes the open Euclidean ball of center yand radius r > 0. In other words, not too far from any point x ∈ Γt, the sets Ω±t contain large balls.Moreover, the sets Γt are assumed to be made of a finite number of graphs: there is an integer n ≥ 1such that, for each t ∈ R, there are n open subsets ωi,t ⊂ R

N−1(for 1 ≤ i ≤ n), n continuous mapsψi,t : ωi,t → R and n rotations Ri,t of RN , such that

(1.6) Γt ⊂⋃

1≤i≤n

Ri,t

(x ∈ RN ; x′ ∈ ωi,t, xN = ψi,t(x′)

).

MONOTONICITY OF BISTABLE TRANSITION FRONTS IN RN 147

Definition 1. [2, 3] For problem (1.1), a transition front connecting 0 and 1 is a classical solutionu : R × RN → (0, 1) for which there exist some sets (Ω±t )t∈R and (Γt)t∈R satisfying (1.3), (1.4)and (1.6), and, for every ε > 0, there exists Mε > 0 such that∀t ∈ R, ∀x ∈ Ω+

t , (d(x,Γt) ≥ Mε)⇒ (u(t, x) ≥ 1 − ε),

∀t ∈ R, ∀x ∈ Ω−t , (d(x,Γt) ≥ Mε)⇒ (u(t, x) ≤ ε).(1.7)

Furthermore, u is said to have a global mean speed γ (≥ 0) if

d(Γt,Γs)|t − s|

→ γ as |t − s| → +∞.

This definition has been shown in [2, 3, 7] to cover and unify all classical cases. Moreover, itwas proved in [7] that, under some assumptions on f , any almost-planar transition front (in thesense that, for every t ∈ R, Γt is a hyperplane) connecting 0 and 1 is truly planar, and that anytransition front connecting 0 and 1 has a global mean speed γ, which is equal to |c f |. Non-standardtransition fronts which are not invariant in any moving frame were also constructed in [7]. For otherproperties of bistable transition fronts, we refer to [2, 3, 7]. There is now a large literature devoted totransition fronts in various homogeneous or heterogeneous settings or for other reaction terms, seee.g. [11, 12, 13, 14, 15, 16, 17, 20, 21, 24, 25, 29, 30, 31].

Referring to many works devoted to traveling fronts, we can notice that the monotonicity is ac-tually an important factor for proving further properties of the traveling fronts, but the monotonicityalso has its own interest. The aforementioned standard fronts, such as the planar fronts, the conical-shaped fronts and the pyramidal fronts, possess some monotonicity properties, especially they areall monotone in time and in their direction of propagation. Although in dimensions N ≥ 2 the spatialmonotonicity of a given transition front does not make sense in general, since the front may not havea privileged direction of propagation, it still makes sense to ask whether transition fronts of (1.1) aremonotone in time. The main goal of this paper is actually to give a positive answer to this questionfor all transition fronts, whatever shape their level sets may have.

Let us now make more precise the assumptions on the function f . Throughout the paper, weassume the following conditions:

(F1) f ∈ C1([0, 1]) satisfies f (0) = f (1) = 0, f ′(0) < 0 and f ′(1) < 0.(F2) There exist c f , 0 and φ ∈ C2(R, [0, 1]) that satisfy (1.2).

Without loss of generality, we can then assume that

c f > 0

even if it means replacing u by 1−u, f (u) by − f (1−u) and c f by −c f . For mathematical purposes, thefunction f is extended in R as a C1(R) function such that f (s) = f ′(0)s > 0 for all s ∈ (−∞, 0) andf (s) = f ′(1)(s − 1) < 0 for all s ∈ (0,+∞). From (F1), there exists then a real number σ ∈ (0, 1/2)such that

(1.8) f is decreasing in (−∞, σ] and [1 − σ,+∞), f < 0 in (0, σ] and f > 0 in [1 − σ, 1).

Notice that, in addition to (F1), condition (F2) is fulfilled in particular if there is θ ∈ (0, 1) such thatf < 0 in (0, θ), f > 0 on (θ, 1) and

∫ 10 f (s)ds , 0, see [1, 6]. However, conditions (F1)-(F2) may

also cover other more general nonlinearities f with multiple zeroes in the interval (0, 1), see [6].We always assume throughout the paper that

u is any transition front connecting 0 and 1


in the sense of Definition 1, with sets (Ω±t )t∈R and (Γt)t∈R satisfying (1.3)-(1.6). We point out that u isany transition front, which may or may not be a standard traveling front with planar, conical-shapedor pyramidal level sets, or which may be of none of these types (examples of other fronts have beenconstructed in [7]).

The main result of the paper is to establish the time monotonicity of the transition front u, underthese conditions (F1)-(F2).

Theorem 1. Under assumptions (F1)-(F2) and c f > 0, any transition front u connecting 0 and 1 issuch that ut(t, x) > 0 for all (t, x) ∈ R × RN .

Remark 1. When c f < 0, it follows immediately from Theorem 1 that any transition front u con-necting 0 and 1 satisfies ut < 0 in R × RN . When c f = 0 in (F2) with f < 0 in (0, θ) and f > 0 in(θ, 1) for some θ ∈ (0, 1), then time-increasing fronts, time-decreasing fronts and stationary frontsare known to exist [1, 4, 6], as well as non-monotone fronts [5]. In other words, the condition (F2),i.e. c f , 0, is optimal in order to get the time-monotonicity of all transition fronts connecting 0and 1.

Outline of the paper. In the next section, we prove some auxiliary lemmas on estimates of someparticular radially symmetric functions. Section 3 is devoted to the proof of Theorem 1.

2. Some preliminary lemmas

We first introduce auxiliary notations for some radially symmetric functions and we show someof their dynamical properties. We recall that f is assumed to satisfy (F1)-(F2). For any R > 0 andβ ∈ R, let vR,β denote the solution of the Cauchy problem

(2.1)

(vR,β)t = ∆vR,β + f (vR,β), t > 0, x ∈ RN ,

vR,β(0, x) =

β if |x| < R,

0 if |x| ≥ R.

Lemma 1. For any T > 0, δ > 0 and β ∈ [1 − σ, 1), where σ > 0 is given in (1.8), there existsR = R(T, δ) > 0 such that

v2R,β(t, x) ≥ β − δ for all 0 ≤ t ≤ T and |x| ≤ R.

Proof. Let T , δ and β be fixed as in the statement. Let %β : R→ (0, 1) denote the solution of%′β(t) = f (%β(t))%β(0) = β.

Since β ∈ [1 −σ, 1) and f (s) > 0 for s ∈ [1 −σ, 1), %β(t) is increasing in t and %β(t) ≥ β for all t ≥ 0.From the maximum principle and (F1), one infers that, for any R > 0,

1 ≥ %β(t) ≥ v2R,β(t, x) ≥ 0

for all t ≥ 0 and x ∈ RN . Then, the following differential inequality holds

(%β − v2R,β)t − ∆(%β − v2R,β) = f (%β) − f (v2R,β) ≤ L(%β − v2R,β),

where L = max[0,1] | f ′|. It follows from the maximum principle that

0 ≤ %β(t) − v2R,β(t, x) ≤eLt

(4πt)N/2

∫|y|≥2R

e−|x−y|2

4t dy for all t > 0 and x ∈ RN .


For 0 < t ≤ T and |x| ≤ R, one has

0 ≤ %β(t) − v2R,β(t, x) ≤eLt

(4π)N/2

∫|z|≥ R

√t

e−|z|24 dz ≤

eLT

(4π)N/2

∫|z|≥ R

√T

e−|z|24 dz.

Thus, there exists R = R(T, δ) > 0 large enough such that

0 ≤ %β(t) − v2R,β(t, x) ≤ δ

for all 0 < t ≤ T and |x| ≤ R. Then, it follows that

v2R,β(t, x) ≥ %β(t) − δ ≥ β − δ for all 0 < t ≤ T and |x| ≤ R.

Notice also that the inequality v2R,β(0, x) ≥ β−δ for |x| ≤ R is satisfied immediately at time t = 0.

Remark 2. Notice from the proof that the radius R(T, δ) can be chosen independently of β ∈ [1 −σ, 1).

The proof of Lemma 1 only used the profile of the function f on the interval [1 − σ, 1]. Theconclusion was concerned with the behavior of the solution v2R,β locally in time. Let us now recall abrief version of [7, Lemma 4.1] (see also [1, Theorem 6.2]), which deals with the large-time behaviorof the solutions vR,β and for which we recall that c f > 0.

Lemma 2. [7] Fix any β ∈ [1 − σ, 1), where σ > 0 is given in (1.8). There are some real numbersR > 0 and T > 0 such that

vR,β(t, x) ≥ β for all t ≥ T and |x| ≤ R.

Let us roughly explain the above lemmas, since they are helpful to the understanding of the fol-lowing proofs. On the one hand, Lemma 1 says that in a bounded time interval [0,T ], the functionv2R,β can not decrease too much in a ball B(0,R) by setting R large enough. On the other hand,Lemma 2 says that vR,β stays larger than β in a ball B(0,R) at large time. For our transition front u,Lemma 1 says that the region where u is close to 1 can not reduce too much as time runs. Further-more, we recall from [7] that

(2.2)d(Γt,Γs)|t − s|

→ c f > 0 as |t − s| → +∞,

whence d(Γt,Γs) → +∞ as |t − s| → +∞. Finally, it can eventually only happen that the state 1invades in some sense the state 0. These properties will be some essential steps in the proof of themonotonicity of u with respect to t. We show the explicit proofs in the following section.

3. Monotonicity: proof of theorem 1

This section is devoted to the proof of Theorem 1 on the monotonicity in time of all transitionfronts. We recall that f satisfies (F1)-(F2) with c f > 0 and u is an arbitrary transition front connecting0 and 1 in the sense of Definition 1, with sets (Ω±t )t∈R and (Γt)t∈R satisfying (1.3)-(1.6). One can easilycheck that equation (1.1) and the function f satisfy all assumptions of [3, Theorem 1.11]. That meansthat, in order to get the time monotonicity and the conclusion of Theorem 1, it is sufficient to showthat the transition front u is an invasion (of the state 0 by the state 1), in the sense that

(3.1) Ω+s ⊂ Ω+

t for all s < t and d(Γt,Γs)→ +∞ as |t − s| → +∞.

Notice that, for the transition front u, the sets (Ω±t )t∈R and (Γt)t∈R satisfying (1.3)-(1.7) are notuniquely determined, since bounded shifts of them still satisfy the same properties. It is therefore


enough to show that, for our given transition front u, some families (Ω±t )t∈R and (Γt)t∈R satisfy con-ditions (1.3)-(1.7), together with the invasion property (3.1), even if it means redefining the sets(Ω±t )t∈R and (Γt)t∈R.

In the following lemmas, we prove some properties of the sets (Ω±t )t∈R of the transition front u.The first key property shows that the interfaces (Γt)t∈R cannot move infinitely fast.

Lemma 3. For any T > 0, there holds

(3.2) supd(x,Γt−τ); t ∈ R, 0 ≤ τ ≤ T, x ∈ Γt

< +∞.

Proof. If the conclusion is not true, then, owing to (1.3), two cases may occur, that is, either

supd(x,Γt−τ); t ∈ R, 0 ≤ τ ≤ T, x ∈ Γt ∩Ω+

t−τ

= +∞,

orsup

d(x,Γt−τ); t ∈ R, 0 ≤ τ ≤ T, x ∈ Γt ∩Ω−t−τ

= +∞.

We only consider the first case, the second one can be handled similarly. Fix ε ∈ (0, σ) (rememberthat σ > 0 is given in (1.8)), set β = 1 − ε ∈ [1 − σ, 1) and let R = R(T, σ − ε) > 0 be sufficientlylarge such that the conclusion of Lemma 1 holds with T > 0 and δ = σ − ε > 0. Then, as the firstcase above is here considered, there are t0 ∈ R, τ0 ∈ [0,T ] and a point

(3.3) x0 ∈ Γt0 ∩Ω+t0−τ0

such that

d(x0,Γt0−τ0 ) ≥ rMε+2R + Mε + 2R,(3.4)

where Mε > 0 is given in (1.7) and rMε+2R > 0 is given in the property (1.5) with M = Mε + 2R.From (1.5), there exists y0 ∈ R

N such that

y0 ∈ Ω−t0 , |y0 − x0| ≤ rMε+2R and d(y0,Γt0 ) ≥ Mε + 2R,

which implies thatB(y0, 2R) ⊂ Ω−t0 and d(B(y0, 2R),Γt0 ) ≥ Mε.

Thus,

u(t0, y) ≤ ε < σ < 1 − σ for all y ∈ B(y0, 2R).(3.5)

From (3.3), (3.4) and |y0 − x0| ≤ rMε+2R, one also has

B(y0, 2R) ⊂ Ω+t0−τ0

and d(B(y0, 2R),Γt0−τ0 ) ≥ Mε.

Thus,u(t0 − τ0, y) ≥ 1 − ε for all y ∈ B(y0, 2R).

Let v2R,1−ε be as defined in (2.1) with 2R and β = 1 − ε ∈ [1 − σ, 1). Since

u(t0 − τ0, y) ≥ v2R,1−ε(0, y − y0) for all y ∈ RN ,

it follows from the comparison principle that

u(t0, y) ≥ v2R,1−ε(τ0, y − y0) for all y ∈ RN .

Furthermore, from Lemma 1 and the choice of R, we have

u(t0, y) ≥ v2R,1−ε(τ0, y − y0) ≥ 1 − ε − (σ − ε) = 1 − σ for all |y − y0| ≤ R.

This contradicts (3.5). The proof of Lemma 3 is thereby complete.


Remark 3. Similarly to Lemma 3, one can show that, for any T > 0, there holds

supd(x,Γt+τ); t ∈ R, 0 ≤ τ ≤ T, x ∈ Γt

< +∞.

This property, which is a priori not equivalent to (3.2), will actually not be used in the sequel. Butit is still stated since, together with Lemma 3, it implies that, for any T > 0, the Hausdorff distancebetween Γt and Γs is bounded uniformly with respect to t ∈ R and s ∈ R such that |t − s| ≤ T .

From [3, Theorem 1.2] and Lemma 3, we can get the following lemma immediately.

Lemma 4. For any C ≥ 0, the transition front u satisfies

0 < infu(t, x); d(x,Γt) ≤ C, (t, x) ∈ R × RN

≤ supu(t, x); d(x,Γt) ≤ C, (t, x) ∈ R × RN

< 1.

The second key property of the sets (Ω±t )t∈R is their τ-monotonicity for large τ > 0.

Lemma 5. There exists τ0 > 0 such that, for any t ∈ R and τ ≥ τ0,

Ω+t ⊂ Ω+

t+τ.

Proof. First of all, property (1.7) and Lemma 4 yield the existence of ε > 0 such that

(3.6) u(t, x) < 1 − ε for all t ∈ R and x ∈ Ω−t ∪ Γt.

Without loss of generality, one can assume that ε ≤ σ, with σ ∈ (0, 1/2) given in (1.8). Let thenR > 0 and T > 0 be some real numbers such that Lemma 2 holds true with β = 1 − ε ∈ [1 − σ, 1).Since d(Γt,Γs) → +∞ as |t − s| → +∞ by (2.2), there exists τ0 > 0 large enough such that τ0 ≥ Tand

d(Γt+τ,Γt) ≥ rMε+R + Mε + R (> 0) for all t ∈ R and τ ≥ τ0,(3.7)

where Mε > 0 and rMε+R > 0 are given in (1.7) and (1.5) respectively.In this paragraph, we fix any real number τ such that τ ≥ τ0. We claim that Γt ⊂ Ω+

t+τ for allt ∈ R. Assume not. Then, remembering (1.3) and (3.7), there is (t0, x0) ∈ R × RN such that

x0 ∈ Γt0 and x0 ∈ Ω−t0+τ.(3.8)

Then there is y0 ∈ RN such that

y0 ∈ Ω+t0 , |y0 − x0| ≤ rMε+R and d(y0,Γt0 ) ≥ Mε + R,(3.9)

which implies that B(y0,R) ⊂ Ω+t0 and d(B(y0,R),Γt0 ) ≥ Mε. Therefore, u(t0, y) ≥ 1 − ε for any

y ∈ B(y0,R). Thus,u(t0, y) ≥ vR,1−ε(0, y − y0) for all y ∈ RN ,

where vR,1−ε is defined in (2.1) with β = 1 − ε. From the comparison principle, one gets that

u(t0 + t, y) ≥ vR,1−ε(t, y − y0) for all t > 0 and y ∈ RN .

From Lemma 2 and the choice of R > 0 and T > 0, it follows that

u(t0 + t, y0) ≥ vR,1−ε(t, 0) ≥ 1 − ε for all t ≥ T.(3.10)

Meanwhile, from (3.7), (3.8) and (3.9), one has

y0 ∈ Ω−t0+τ and d(y0,Γt0+τ) ≥ Mε + R ≥ Mε.

This implies that u(t0 + τ, y0) ≤ ε. Since ε ≤ σ < 1/2 < 1 − ε, this contradicts (3.10) witht = τ ≥ τ0 ≥ T . So, we conclude that

(3.11) Γt ⊂ Ω+t+τ for all t ∈ R and τ ≥ τ0.


Finally, assume by contradiction that the conclusion of Lemma 5 does not hold with τ0 > 0 givenabove. Then, there are t1 ∈ R, τ1 ≥ τ0 and x1 ∈ Ω+

t1 such that x1 < Ω+t1+τ1

. Since x1 ∈ Γt1+τ1 ∪ Ω−t1+τ1

and Γt1 ⊂ Ω+t1+τ1

by (3.11), one infers that

d(x1,Γt1 ) ≥ d(Γt1+τ1 ,Γt1 ).

Hence, by (3.7),d(x1,Γt1 ) ≥ rMε+R + Mε + R ≥ Mε + R.

Since x1 ∈ Ω+t1 , this also implies that

B(x1,R) ⊂ Ω+t1 and d(B(x1,R),Γt1 ) ≥ Mε.

Therefore, u(t1, x) ≥ 1 − ε for all x ∈ B(x1,R) and u(t1, x) ≥ vR,1−ε(0, x − x1) for all x ∈ RN . Fromthe comparison principle, one gets that

u(t1 + t, x) ≥ vR,1−ε(t, x − x1) for all t > 0 and x ∈ RN .

From Lemma 2 and the choice of R and T , it follows from τ1 ≥ τ0 ≥ T > 0 that

u(t1 + τ1, x1) ≥ vR,1−ε(τ1, 0) ≥ 1 − ε.

Since x1 ∈ Γt1+τ1 ∪ Ω−t1+τ1, the above inequality contradicts (3.6). The proof of Lemma 5 is thereby

complete.

Now we are going to redefine the sets (Ω±t )t∈R and (Γt)t∈R so that the transition front u is aninvasion in the sense of (3.1). To do so, let τ0 > 0 be as in Lemma 5 and set

(3.12)

Ω±kτ0+t := Ω±kτ0for any k ∈ Z and 0 ≤ t < τ0,

Γt := ∂Ω+t = ∂Ω−t for any t ∈ R.

Proposition 1. The solution u is a transition front with the families (Ω±t )t∈R and (Γt)t∈R, and then itis an invasion in the sense of (3.1) with (Ω±t )t∈R and (Γt)t∈R.

Proof. Observe first that, owing to the definitions of the sets Ω±t and Γt, one has d(Γt, Γs) → +∞ as|t − s| → +∞, since d(Γt,Γs) → +∞ as |t − s| → +∞. Hence, from Lemma 5, we immediately getthat u is an invasion with the families (Ω±t )t∈R and (Γt)t∈R, that is, these sets satisfy (3.1).

It is obvious that, owing to their definition, the sets (Ω±t )t∈R and (Γt)t∈R satisfy the properties (1.3),(1.4) and (1.6). Therefore, we only need to show that u satisfies (1.7) with (Ω±t )t∈R and (Γt)t∈R, atleast for all ε > 0 small enough.

First of all, we claim that there is ε0 ∈ (0, 1/2) such that

(3.13)

∀ ε ∈ (0, ε0), ∀ (t, x) ∈ R × RN , ∀ s ∈ [0, τ0], (u(t, x) ≥ 1 − ε

)=⇒

(u(t + s, x) > ε

),(

u(t, x) ≤ ε)

=⇒(u(t + s, x) < 1 − ε

).

We only show the first property (the second one can be proved similarly). If it does not hold, thereexists a sequence (tn, xn, sn)n∈N in R × RN × [0, τ0] such that u(tn, xn) → 1 and u(tn + sn, xn) → 0 asn→ +∞. From standard parabolic estimates, the functions

un(t, x) := u(t + tn, x + xn)

converge in C1,2t,x;loc(R×RN), up to extraction of a subsequence, to a solution 0 ≤ u∞(t, x) ≤ 1 of (1.1)

such that u∞(0, 0) = 1 and u∞(s∞, 0) = 0 for some s∞ ∈ [0, τ0]. The strong maximum principle


implies that u∞ = 1 in (−∞, 0] × RN and then in R × RN by uniqueness of the solutions of theassociated Cauchy problem. This is impossible, since u∞(s∞, 0) = 0. Thus, there is ε0 ∈ (0, 1/2)satisfying (3.13).

Now, set

(3.14) D := supd(x,Γkτ0 ); k ∈ Z, 0 ≤ t < τ0, x ∈ Γkτ0+t

,

which is a well-defined real number by Lemma 3. In the sequel, fix any real number ε such that0 < ε < ε0 (< 1/2) and define

(3.15) Mε := Mε + D > 0,

where Mε > 0 is given in (1.7) for the families (Ω±t )t∈R and (Γt)t∈R. Since any t ∈ R can be written ast = kτ0 + s for some k ∈ Z and 0 ≤ s < τ0, it follows from (1.7) that we only need to show that

(3.16) ∀ k ∈ Z, ∀ 0≤ t<τ0, ∀x ∈ Ω+kτ0+t, d(x, Γkτ0+t)≥ Mε =⇒

(x ∈ Ω+

kτ0+t and d(x,Γkτ0+t)≥Mε)

and

(3.17) ∀ k ∈ Z, ∀ 0≤ t<τ0, ∀x∈Ω−kτ0+t, d(x, Γkτ0+t)≥ Mε =⇒(x ∈ Ω−kτ0+t and d(x,Γkτ0+t)≥Mε

).

We only prove (3.16), the property (3.17) being proved similarly thanks to the second propertyin (3.13). To show (3.16), we first claim that, for any given k ∈ Z and 0 ≤ t < τ0, there holds

Γkτ0+t ⊂x ∈ RN ; d(x, Γkτ0+t) ≤ D

.(3.18)

Indeed, otherwise, there is a point x0 ∈ Γkτ0+t such that d(x0, Γkτ0+t) > D. Since Γkτ0+t = Γkτ0 bydefinition, this yields d(x0,Γkτ0 ) > D, which contradicts the definition of D in (3.14).

Then, we claim that, for any given k ∈ Z and 0 ≤ t < τ0, there holdsx ∈ Ω+

kτ0+t; d(x, Γkτ0+t) ≥ Mε⊂ Ω+

kτ0+t.(3.19)

Indeed, let x ∈ Ω+kτ0+t be such that d(x, Γkτ0+t) ≥ Mε. In other words, x ∈ Ω+

kτ0and

d(x,Γkτ0 ) ≥ Mε = Mε + D.

Hence, d(x,Γkτ0 ) ≥ Mε and

(3.20) u(kτ0, x) ≥ 1 − ε

by definition of Mε. Furthermore, d(x,Γkτ0 ) ≥ Mε = Mε + D and (3.14) imply that

d(x,Γkτ0+t) ≥ Mε.

Therefore, either x ∈ Ω−kτ0+t and u(kτ0 + t, x) ≤ ε, or x ∈ Ω+kτ0+t (and u(kτ0 + t, x) ≥ 1−ε). The former

case is impossible due to (3.13) and (3.20). Thus, x ∈ Ω+kτ0+t and (3.19) is proved.

Finally, from (3.12), (3.15), (3.18), (3.19), we easily get (3.16). As already emphasized, the proofof Proposition 1 is thereby complete.

Proof of Theorem 1. From [3, Theorem 1.11] and the fact that, by Proposition 1, the transition frontu is an invasion in the sense of (3.1), with the sets (Ω±t )t∈R and (Γt)t∈R, we immediately get the desiredmonotonicity property ut > 0 in R × RN .


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AixMarseille Univ, CNRS, CentraleMarseille, I2M, Marseille, FranceE-mail address: [email protected], [email protected]

JEPE Vol 2, 2016, p. 157-169

REMARKS ON LOCAL BOUNDEDNESS AND LOCAL HOLDER

CONTINUITY OF LOCAL WEAK SOLUTIONS TO ANISOTROPIC

p-LAPLACIAN TYPE EQUATIONS

EMMANUELE DIBENEDETTO, UGO GIANAZZA, AND VINCENZO VESPRI

Abstract. Locally bounded, local weak solutions to a special class of quasilinear, ani-sotropic, p-Laplacian type elliptic equations, are shown to be locally Holder continuous.

Homogeneous local upper bounds are established for local weak solutions to a general

class of quasilinear anisotropic equations.

1. Introduction

Consider quasi-linear, elliptic differential equations of the form

(1.1) divA(x, u,Du) = 0 weakly in sone open set E ⊂ RN

where the function A = (A1, . . . , AN ) : E × RN+1 → RN is only assumed to be measurableand subject to the structure conditions

(1.2)Ai(x, u,Du) · uxi ≥ Co,i|uxi |pi ,|Ai(x, u,Du)| ≤ C1,i|uxi |pi−1,

where pi > 1 and Co,i and C1,i are given positive constants. Such elliptic equations aretermed anisotropic, their prototype being

(1.3)N∑i=1

(|uxi |

pi−2uxi)xi

= 0 in E.

For a multi-index p = p1, . . . , pN, pi ≥ 1, let

W 1,p(E) = u ∈ L1(E) : uxi ∈ Lpi(E), i = 1, . . . , N,

and

W 1,po (E) = W 1,p(E) ∩W 1,1

o (E).

A function u ∈W 1,ploc (E) is a local, weak solution to (1.1) if for every compact set K ⊂ E

(1.4)

∫K

A(x, u,Du) ·Dϕdx = 0 for all ϕ ∈ C∞o (K) .

2010 Mathematics Subject Classification. Primary: 35J70, 35J92, 35B65; Secondary 35B45.

Key words and phrases. Anisotropic p-Laplacian, Elliptic, Holder continuity.Received 27/10/2016, accepted 22/11/2016.E. DiBenedetto’s research supported under NSF grant DMS-1265548.

157

158 EMMANUELE DIBENEDETTO, UGO GIANAZZA, AND VINCENZO VESPRI

The parameters N, pi, Co,i, C1,i are the data, and we say that a generic constant γ =γ(N, pi, Co,i, C1,i) depends upon the data, if it can be quantitatively determined a priorionly in terms of the indicated parameters.

Define,

(1.5)1

p=

1

N

N∑i=1

1

pi,

pmin = minp1, . . . , pN,pmax = maxp1, . . . , pN.

For a compact set K ⊂ E introduce the intrinsic, elliptic p-distance from K to ∂E by

p− dist(K; ∂E)def= inf

x∈Ky∈∂E

(N∑i=1

‖u‖pi−pmin

pi

∞,E |xi − yi|1pi

).

Theorem 1. Let u be a bounded, local, weak solution to (1.1)–(1.2), and assume p < N .There exists a positive quantity q > 1, depending only on the data, such that if

(1.6) pmax − pmin ≤1

q,

then u is locally Holder continuous in E, i.e. there exist constants γ > 1 and α ∈ (0, 1)depending only on the data, such that for every compact set K ⊂ E,

|u(x1)− u(x2)| ≤ γ‖u‖∞,E

( N∑i=1

‖u‖pi−pmin

pi

∞,E |x1,i − x2,i|1pi

p− dist(K; ∂E)

)αfor every pair of points x1, x2 ∈ K.

Remark 1. For a general distribution of the pj , unbounded weak solutions might exist([7, 11]). In [6, 8, 1] it is shown that local weak solutions are locally bounded provided

(1.7) p < N, pmax ≤Np

N − p.

In Section 6 we revisit and improve these boundedness estimates.

Remark 2. The constants γ and α deteriorate as either pi → ∞ or pi → 1, in the sensethat γ(p)→∞ and α→ 0 as either pi →∞, or pi → 1.

Acknowledgements - We thank Prof. P. Marcellini, E. Mascolo and G. Cupini for enlight-ening conversations on embeddings for anisotropic Sobolev spaces.

2. Novelty and Significance

If the coefficients in (1.1)–(1.2) are differentiable, and satisfy some further, suitable struc-ture conditions, Lipschitz estimates have been derived by Marcellini [12, 13]. If the coeffi-cients are merely bounded and measurable, Holder continuity has been established in [10]in the special case of p1 = 2 < p2 = p3 = · · · = pN , i.e., the pj are all the same except thesmallest one. The main idea is to regard the equation as “parabolic” with respect to thevariable x1, corresponding to p1, and to apply the techniques of [3, 4]. An extension to thecase 1 < p1 < p2 = p3 = · · · = pN , by the same techniques, is in [5].

Theorem 1 is a further step in understanding the regularity of solutions of anisotropicelliptic equations, with full quasi-linear structure. Our approach is “elliptic” in nature, it is

ANISOTROPIC p-LAPLACIAN EQUATION 159

modelled after [2], no variable is regarded as “parabolic”, and no restriction is placed on thedistribution of the pj other that pmax−pmin 1. In particular, the pj could all be different.

While partial, Theorem 1 disproves the claim in [7] by which, Holder continuity of weaksolutions to (1.1)–(1.2), holds if and only if pmin = pmax, i.e., if no anisotropy is present.

Finally, Theorem 1 can be seen as a stability result of the Holder continuity of solutions,when pi → pmin, and correspondingly, the anisotropic p-laplacian tends to the pmin-laplacian.

3. Preliminaries and Intrinsic Geometry

Lemma 1 (Sobolev-Troisi Inequality, [15]). Let E ⊂ RN be a bounded, open set and consideru ∈W 1,p

o (E), pi > 1 for all i = 1, . . . , N . Assume p < N and let

(3.1) p∗ =Np

N − p.

Then there exists a constant c depending only on N,p, such that

‖u‖NLp∗ (E) ≤ cN∏i=1

‖uxi‖Lpi (E) .

For ρ > 0 consider the cube Kρ = (−ρ, ρ)N , with center at the origin of RN and edge 2ρ,and set

(3.2) µ+ = ess supK2ρ

u; µ− = ess infK2ρ

u; ω = µ+ − µ− = ess oscK2ρ

u.

These numbers being determined, construct the cylinder

(3.3) Qρ =N∏j=1

(−ρj , ρj),

with 0 < ρj ≤ ρ to be determined. This implies that Q2ρ ⊂ K2ρ and hence ess oscQρ u ≤ ω.

3.1. Basic Equation and Energy Inequalities. For σ ∈ (0, 1) let ζj be a non-negative,piecewise smooth cutoff function in the interval (−ρj , ρj) which equals 1 on (−σρj , σρj),vanishes at ±ρj , and such that |ζ ′j | ≤ [(1− σ)ρj ]

−1.

Set ζ =∏Nj=1ζ

pjj , and in the weak formulation of (1.1)–(1.2) take the testing function

±(u− k)±ζ. This gives, after standard calculations,

(3.4)N∑j=1

∫Qρ

∣∣∣ ∂∂xj

[(u− k)±ζ

1pj ]∣∣∣pjdx ≤ γ N∑

j=1

1

(1− σ)pjρpjj

∫Qρ

(u− k)pj± dx.

The constant γ depends only upon the data, and is independent of ρ.

4. DeGiorgi Type Lemmas

Taking k = µ+ − ω2s , for s ≥ 1, and (u− k)+ in (3.4) yields

(4.1)

N∑j=1

∫Qρ

∣∣∣[(u− (µ+ − ω

2s

))+ζ

1pj

]xj

∣∣∣pjdx≤ γ

(1− σ)pmax

N∑j=1

1

ρpjj

( ω2s

)pj ∣∣Qρ ∩ [u > µ+ − ω

2s]∣∣.


Likewise, taking k = µ− + ω2s , for s ≥ 1, and −(u− k)− in (3.4) yields

(4.2)

N∑j=1

∫Qρ

∣∣∣[(u− (µ− +ω

2s

))−ζ

1pj

]xj

∣∣∣pjdx≤ γ

(1− σ)pmax

N∑j=1

1

ρpjj

( ω2s

)pj ∣∣Qρ ∩ [u < µ− +ω

2s]∣∣.

Choose

(4.3) ρj =( ω

2q

)ραpj for some q > 0 and α ≥ pmax to be chosen

and let Qρ the cylinder in (3.3) for such a choice of ρj . Without loss of generality, mayassume ω ≤ 1, so that 0 < ρj ≤ ρ as required.

Lemma 2. There exists a number ν ∈ (0, 1) depending only upon the data, such that if

(4.4)∣∣∣[u > µ+ − ω

2q

]∩Qρ

∣∣∣ < ν∣∣Qρ∣∣,

for some q ∈ N, then

(4.5) u ≤ µ+ − ω

2q+1a.e. in Q 1

2ρ.

Likewise

Lemma 3. There exists a number ν ∈ (0, 1) depending only upon the data, such that if

(4.6)∣∣∣[u < µ− +

ω

2q

]∩Qρ

∣∣∣ < ν∣∣Qρ∣∣,

for some q ∈ N, then

(4.7) u ≥ µ− +ω

2q+1a.e. in Q 1

2ρ.

We prove only Lemma 2, the proof of Lemma 3 being analogous.

Proof. For each j ∈ 1, . . . , N consider the sequence of radii

(4.11) ρj,n =1

2ρj

(1 +

1

2n

), for n = 0, 1, . . . .

This is a decreasing sequence with ρj,o = ρj and ρj,∞ = 12ρj . The corresponding cylinders

(4.12) Qndef= Qρn =

∏Nj=1

(− ρj,n, ρj,n

)for n = 0, 1, . . .

are nested, i.e., Qn+1 ⊂ Qn, with Qo = Qρ and Q∞ = Q 12ρ

, since α ≥ 1. For each

j ∈ 1, . . . , N let ζj,n be a standard non-negative cutoff function in (−ρj,n, ρj,n) which

equals 1 on (−ρj,n+1, ρj,n+1), vanishes at ±ρj,n and such that |ζ ′j,n| ≤ 2n+2ρ−1j,n. Then set

ζn =∏Nj=1ζ

pjj,n to be a cutoff function in Qn that equals 1 on Qn+1. Consider also the

increasing sequence of levels

(4.13) kn = µ+ − ω

2q+1

(1 +

1

2n

), for n = 0, 1, . . .


and in the weak formulation of (1.1)–(1.2), take the test function (u−kn)+ζn. This leads toanalogues of (4.1) over the cylinders Qn, with 1−σ > 2−(n+2), and q ≤ s < q+1. Rewriting(4.1) with these specifications gives

(4.14)N∑j=1

∫Qn

∣∣[(u− kn)+ζ1pjn

]xj

∣∣pjdx ≤ γ 2npmax

ρα∣∣Qn ∩ [u > kn]

∣∣.Since (u− kn)+ζn vanishes on ∂Qn, by the anisotropic embedding of Lemma 1

(4.15)(∫

Qn

[(u− kn)+ζn

]p∗dx) 1p∗ ≤ c

N∏j=1

(∫Qn

∣∣[(u− kn)+ζn]xj

∣∣pjdx) 1Npj

.

where p∗ is as in (3.1), pj > 1 for j = 1, . . . , N , p < N . Since 0 ≤ ζn ≤ 1 and pj > 1estimate ∫

Qn


∣∣pjdx ≤ γ ∫Qn


]xj

∣∣pjdx.Therefore, combining this with (4.15) and (4.14) gives

(kn+1 − kn)∣∣Qn+1 ∩ [u > kn+1]

∣∣ ≤ ∫Qn+1∩[u>kn+1]

(u− kn)+dx

≤∫Qn

(u− kn)+ζndx ≤(∫

Qn

[(u− kn)+ζn

]p∗dx) 1p∗ ∣∣Qn ∩ [u > kn]

∣∣1− 1p∗

≤ γN∏j=1

(∫Qn


]xj

∣∣pjdx) 1Npj

.∣∣Qn ∩ [u > kn]

∣∣1− 1p∗

≤ γ(

2pmaxp

)n 1

ραp

∣∣Qn ∩ [u > kn]∣∣1+ 1

p−1p∗

= γbn

( ω2q

)∣∣Qρ∣∣ 1

N

∣∣Qn ∩ [u > kn]∣∣1+ 1

N

where we have set b = 2pmax/p. By the definition of kn in (4.13), the first term in roundbrackets on the left hand side is

kn − kn+1 =( ω

2q

) 1

22+n.

Combining these remarks and inequalities, and setting

Yn =

∣∣Qn ∩ [u > kn]∣∣

|Qρ|yields the recursive inequalities

Yn+1 ≤ C(2b)nY1+ 1

Nn

for constants C and b depending only upon the data. It follows from these and Lemma 5.1of [4, Chapter 2], that there exists a number ν ∈ (0, 1) depending only on C, b,N, andhence only upon the data, such that Yn → 0 as n→∞ provided

Yo =

∣∣Qρ ∩ [u > µ+ − ω2q

]∣∣|Qρ|

≤ ν.


By these lemmata the Holder continuity of u will follow by standard arguments, if onecan determine q, and hence the intrinsic cylinders Qρ, for which either (4.4) or (4.6) holds.


Assume that

(5.1)∣∣[u < µ− + 1

2ω]∩Qρ

∣∣ ≥ 12

∣∣Qρ∣∣.For each s ∈ N with s ≤ q, introduce the two complementary sets

(5.2) As =[u > µ+ − ω

2s

]∩Qρ; Qρ −As =

[u ≤ µ+ − ω

2s

]∩Qρ

and consider the doubly truncated function

(5.3) vs =

0 for u < µ+ − ω

2s,

u−(µ+ − ω

2s

)for µ+ − ω

2s≤ u < µ+ − ω

2s+1,

ω

2s+1for µ+ − ω

2s+1≤ u.

By construction vs vanishes on Qρ −As. Pick any two points

x = (x1, . . . , xN ) ∈ As and y = (y1, . . . , yN ) ∈ Qρ −As

and construct a polygonal joining x and y and sides parallel to the coordinate axes, say forexample PN = x and

PN−1 = (x1, . . . , xN−1, yN ); PN−2 = (x1, x2, . . . , yN−1, yN ); · · ·P1 = (x1, y2, . . . , yN ); Po = (y1, . . . , yN ).

By elementary calculus

vs(x) = [vs(PN )− vs(PN−1)] + · · ·+ [vs(P1)− vs(Po)]

=

∫ xN

yN

∂

∂xNvs(x1, . . . , xN−1, t)dt+

∫ xN−1

yN−1

∂

∂xN−1vs(x1, . . . , xN−2, t, yN )dt

+ · · ·+∫ x1

y1

∂

∂x1vs(t, y2, . . . , yN )dt

≤N∑j=1

∫ ρj

−ρj

∣∣vs,xj ∣∣(x1, . . . , t︸︷︷︸j−th variable

, . . . , yN )dt

where the quantities ρj are defined in (4.3). Integrate in dx over As and in dy over Qρ−As,and take into account (5.1) to get

1

2

∣∣Qρ∣∣ ∫Qρ

vsdx ≤ 2∣∣Qρ∣∣ N∑

j=1

ρj

∫Qρ

∣∣vs,xj ∣∣dx.


From this, by the definitions (5.2) and (5.3) of As and vs,

(5.4)

ω

2s+1

∣∣As+1

∣∣ ≤ 4N∑j=1

ρj

∫As−As+1

∣∣uxj ∣∣dx≤ 4

N∑j=1

ρj

(∫As−As+1

∣∣uxj ∣∣pmindx

) 1pmin ∣∣As −As+1

∣∣1− 1pmin

≤ 4N∑j=1

ρj

(∫As−As+1

∣∣uxj ∣∣pjdx) 1pj ∣∣Qρ∣∣ 1

pmin− 1pj

∣∣As −As+1

∣∣1− 1pmin .

For each j fixed, the integrals involving uxj are estimated by means of (4.1) applied overthe pair of cubes Qρ and Q2ρ, as follows:(∫

As−As+1

∣∣uxj ∣∣pjdx) 1pj

≤

(∫Qρ

∣∣∣∣ ∂∂xj(u−

(µ+ − ω

2s

))+

∣∣∣∣pj dx) 1pj

≤ γ(

1

ρα

N∑=1

( ω2s

)p` ( ω2q

)−p` ∣∣Qρ∣∣) 1pj

≤ γ(

1

ρα

N∑=1

(2q

2s

)p` ∣∣Qρ∣∣) 1pj

= γ

(1

ρpjj

( ω2q

)pj N∑=1

(2q

2s

)p` ∣∣Qρ∣∣)1pj

.

If p` ≤ pj , since s ≤ q estimate

(5.5)

(2q

2s

)p`≤(

2q

2s

)pj=( ω

2s

)pj ( ω2q

)−pj, ( case of p` ≤ pj).

If p` > pj since s ≤ q compute and estimate

(5.6)

(2q

2s

)p`=

(2q

2s

)pj (2q

2s

)p`−pj≤( ω

2s

)pj ( ω2q

)−pj2q(pmax−pmin)

(case of p` > pj).

Assume momentarily that the number q has been chosen. Then stipulate that q(pmax −pmin) ≤ 1. For such a choice we have in all cases(∫

As−As+1

∣∣uxj ∣∣pjdx) 1pj

≤ γ 1

ρj

( ω2s

) ∣∣Qρ∣∣ 1pj .

Combining these estimates in (5.4) yields∣∣As+1

∣∣ ≤ γ∣∣Qρ∣∣ 1pmin

(∣∣As∣∣− ∣∣As+1

∣∣)1− 1pmin .

Take the(

pmin

pmin−1

)power and add for s = 1, . . . (q − 1) to get

(q − 1)∣∣Aq∣∣ pmin

pmin−1 ≤ γpminpmin−1

∣∣Qρ∣∣ 1pmin−1

∣∣Ao∣∣.


From this ∣∣Aq∣∣ ≤ γ

(q − 1)pmin−1

pmin

∣∣Qρ∣∣.In the DeGiorgi-type Lemma, the number ν is independent of q. Now choose q so that

(5.7)∣∣Aq∣∣ ≤ ν∣∣Qρ∣∣, for ν =

γ

(q − 1)pmin−1

pmin

.

Notice that q is determined in terms of pmin and not in terms of the difference (pmax−pmin).Thus, one determines first q from (5.7) in terms only of the data. Then (1.6), for such achoice of q, serves as a condition of Holder continuity for u.

6. Boundedness

Continue to denote by u ∈ W 1,ploc (E) a local weak solution to (1.1)–(1.2), in the sense of

(1.4). The estimations below use that u ∈ Lp∗loc(E). When pmax < p∗ this is insured bythe embeddings in [9, Theorem 1] or [14]. If pmax = p∗ in what follows the membershipu ∈ Lp∗loc(E), is assumed.

6.1. Some General Recursive Inequalities. Let ρj as in (3.3) to be defined, and foreach j ∈ 1, . . . , N consider the sequence of radii,

(6.1) ρj,n =1

2ρj

(1 +

1

2n

), for n = 0, 1, . . . .

This is a decreasing sequence, with ρj,n = ρj and ρj,∞ = 12ρj . The corresponding cylinders

(6.2) Qndef= Qρn =

∏Nj=1

(− ρj,n, ρj,n

)for n = 0, 1, . . .

are nested, i.e., Qn+1 ⊂ Qn, with Qo = Qρ and Q∞ = Q 12ρ

, since α ≥ pj . For each

j ∈ 1, . . . , N let ζj,n be a standard non-negative cutoff function in (−ρj,n, ρj,n) which

equals 1 on (−ρj,n+1, ρj,n+1), vanishes at ±ρj,n and such that |ζ ′j,n| ≤ 2n+2ρ−1j,n. Then set

ζn =∏Nj=1ζ

pjj,n to be a cutoff function in Qn that equals 1 on Qn+1.

Consider also the increasing sequence of levels

(6.3) kn =(

1− 1

2n

)k, and kn =

kn+1 + kn2

for n = 0, 1, . . . , with k > 0 to be chosen. By the definition ko = 0 and k∞ = k. Write (3.4)for (u − kn)+ζn, over the cylinders Qn, with 1 − σ > 2−(n+2). Since (u − kn)+ζn vanisheson ∂Qn, by the anisotropic embedding of Lemma 1

(6.4)(∫

Qn

[(u− kn)+ζn

]p∗dx) 1p∗ ≤ γ

N∏j=1

(∫Qn


∣∣pjdx) 1Npj

.

where p∗ has been defined in (3.1).Since 0 ≤ ζn ≤ 1 and pj ≥ 1 estimate∫

Qn


∣∣pjdx ≤ γ ∫Qn


]xj

∣∣pjdx.


Therefore, combining this with (3.4) and (6.4) gives(∫Qn+1

(u− kn)p∗+ dx

) 1p∗

≤(∫

Qn

[(u− kn)+ζn

]p∗dx

) 1p∗

≤ γN∏j=1

(∫Qn


]xj

∣∣pjdx) 1Npj

≤ γN∏j=1

(N∑=1

2p`n

ρp``

∫Qn

(u− kn)p`+ dx

) 1N

1pj

= γ

(N∑=1

2p`n

ρp``

∫Qn

(u− kn)p`+ dx

) 1p

.

From this homogenizing with respect to the measure of Qn and with respect to the integrand,

(6.5)

(1

kp∗−∫Qn+1

(u− kn)p∗+ dx

) 1p∗

≤ γ(|Qρ|

pN

N∑=1

2p`nkp`−p

ρp``

1

kp`−∫Qn

(u− kn)p`+ dx

) 1p

.

For each ` ∈ 1, . . . , N, estimate

1

kp`−∫Qn

(u− kn)p`+ dx ≤(

1

kp∗−∫Qn

(u− kn)p∗+ dx

) p`p∗(∣∣[u > kn] ∩Qn

∣∣|Qn|

)1− p`p∗.

Also1

kp∗−∫Qn

(u− kn)p∗+ dx ≥ 1

kp∗−∫Qn∩[u>kn]

(kn − kn

)p∗dx

≥ 1

2p∗(n+2)

∣∣[u > kn] ∩Qn∣∣

|Qn|.

Therefore,1

kp`−∫Qn

(u− kn)p`+ dx ≤ 2(p∗−p`)(n+2) 1

kp∗−∫Qn

(u− kn)p∗+ dx.

Combine these calculations in (6.5), to get(1

kp∗−∫Qn+1

(u− kn+1)p∗+ dx

) 1p∗

≤ γ2p∗p n

[|Qρ|

pN

N∑=1

kp`−p

ρp``

] 1p

[(1

kp∗−∫Qn

(u− kn)p∗+ dx

) 1p∗] p∗p

.

Set

(6.6) Yn =

(1

kp∗−∫Qn

(u− kn)p∗+ dx

) 1p∗

,


and rewrite the previous inequalities in the form

(6.7) Yn+1 ≤ γ2p∗p n

[|Qρ|

pN

N∑=1

kp`−p

ρp``

] 1p

Y1+ p∗−p

pn .

Recall that the radii ρj are still to be chosen.

6.2. A Quantitative, Homogeneous Estimate for pmax < p∗. Choose

(6.8) ρj = ραpj ,

where α is an arbitrary positive parameter. Stipulate to take k ≥ 1 and estimate[|Qρ|

pN

N∑=1

kp`−p

ρp``

] 1p

≤ 2Nkpmax−p

p .

For such choices (6.7) yield

(6.9) Yn+1 ≤ γ2p∗p nk

pmax−pp Y

1+ p∗−pp

n

for a new constant γ depending only on N, p1, . . . , pN. It follows from these that Yn → 0as n→∞, provided

Yo =1

k

(−∫Qρ

up∗+ dx

) 1p∗

≤ γ−p

p∗−p 2−p∗p ( p

p∗−p )2

k−pmax−pp∗−p .

Thus, choosing

k = γp

p∗−pmax 2p∗

p∗−pmax

pp∗−p

(−∫Qρ

up∗+ dx

) 1p∗

p∗−pp∗−pmax

yields

(6.10) ess supQ 1

2ρ

u+ ≤ 1 ∧ C

(−∫Qρ

up∗+ dx

) 1p∗

p∗−pp∗−pmax

,

whereC = γ

pp∗−pmax 2

p∗p∗−pmax

pp∗−p .

Write now (6.10) over the pair of cubes Qσρ ⊂ Qρ, where σ ∈ ( 12 , 1) is an interpolation

parameter. Then

(6.11) ess supQ 1

2ρ

u+ ≤ 1 ∧ C ′(−∫Qρ

upmax

+ dx

) 1p

p∗−pp∗−pmax

.

Remark 3. The estimates in (6.10) and (6.11) are homogeneous with respect to the cubeQρ, i.e., they are invariant for dilations of the variables (x1, . . . , xN ) that keep invariantthe relative intrinsic geometry of (3.3) and (6.8). In this sense they are an improvementwith respect to the estimates of Kolodıı [8, Theorem 2]. If pj = p for all j = 1, . . . , N thisreproduces the classical estimate for isotropic elliptic equations.

Remark 4. The constants C and C ′ in (6.10) and (6.11), can be quantitatively determinedonly in terms of N and the pj for j = 1, . . . , N . However, they tend to infinity as pmax p∗.


6.3. A Quantitative, Homogeneous Estimate for pmax = p∗. Redefine the levels in(6.3) as

(6.12) kn =(

1− 1

2n+1

)k, and kn =

kn+1 + kn2

, for n = 0, 1, . . . .

This implies that ko = 12k and k∞ = k. All estimations remain unchanged and yield (6.7),

with a slight modification of the constant γ and with the same definition (6.6) of the Yn.

Continue to choose ρj = ραpj , and stipulate to take k ≥ 1. This yields the analogues of (6.9)

with pmax = p∗, i.e.,

(6.9)pmax=p∗ Yn+1 ≤ γ2p∗p nk

p∗−pp Y

1+ p∗−pp

n .

Taking into account the definition (6.6) of the Yn, in this last inequality, the parameter kscales out. Thus, setting

Xn =

(−∫Qn

(u− kn)p∗+ dx

) 1p∗

,

the recursive inequalities (6.9)pmax=p∗ are

(6.13) Xn+1 ≤ γ2p∗p nX

1+ p∗−pp

n .

It follows from these that Xn → 0 as n→∞, provided

(6.14) Xo =

(−∫Qρ

(u− 1

2k)p∗

+dx

) 1p∗

≤ γ−p

p∗−p 2−p∗p ( p

p∗−p )2

.

Since u ∈ Lp∗loc(E), such a k can be quantitatively, although not explicitely, determined, interms of ‖u‖Lp∗ (Qρ), and then,

(6.15) ess supQ 1

2ρ

u+ ≤ 1 ∧ k.

Remark 5. The estimate in (6.15) is homogeneous with respect to the cube Qρ, i.e., it isinvariant for dilations of the variables (x1, . . . , xN ) that keep invariant the relative intrinsicgeometry of (3.3) and (6.8). In this sense, it is an improvement with respect to the estimatesof Fusco-Sbordone [6, Theorem 1].

Remark 6. In (6.13) the number κ = p∗−pp by which the power of Xn exceeds one, is

precisely determined by the estimations, and not arbitrary as it seems to be permitted in[6]. In view of this, the alternative, in the argument of [6], by which

2p∗p nX

1+ p∗−pp

n > 1 for all n ≥ no for some no ∈ N sufficiently large

is not needed. Since no in [6] is determined only qualitatively, the resulting boundednessestimates seem to be qualitative.

Remark 7. If one had the additional information that u ∈ Lqloc(E), for some q > p∗, thenk in (6.15) could be precisely quantified. Indeed, given a non negative function f ∈ Lq(E)and ε > 0, consider finding k > 0 such that∫

E

(f − k)p+dx < ε where 0 < p < q.


By Chebyshev’s inequality |[f > t]| ≤ t−q‖f‖qq,E , for all t > 0. Then for p < q,

∫E

(f − k)p+dx = p

∫ ∞0

sp−1∣∣[(f − k)+ > s]

∣∣ds= p

∫ ∞k

(t− k)p−1∣∣[f > t]

∣∣dt ≤ p ∫ ∞k

tp−1∣∣[f > t]

∣∣dt≤ p ‖f‖qq,E

∫ ∞k

t−(q−p)−1dt =p

q − p1

kq−p‖f‖qq,E .

Then choose

kq−p =p

ε

1

q − p‖f‖qq,E .

References

[1] G. Cupini, P. Marcellini, E. Mascolo, Regularity of minimizers under limit growth conditions, Nonlinear

Anal., in press.[2] E. De Giorgi, Sulla differenziabilita e l’analiticita delle estremali degli integrali multipli regolari, Mem.

Accad. Sci. Torino. Cl. Sci. Fis. Mat. Nat. 3 (3), (1957), 25–43.[3] E. DiBenedetto, Degenerate Parabolic Equations, Springer Verlag, Series Universitext, New York, 1993.

[4] E. DiBenedetto, U. Gianazza and V. Vespri, Harnack’s inequality for degenerate and singular parabolic

equations, Springer Monographs in Mathematics, Springer, 2012.[5] F.G. Duzgun, P. Marcellini and V. Vespri, Space expansion for a solution of an anisotropic p-Laplacian

equation by using a parabolic approach, Riv. Math. Univ. Parma, 5, (2014), 93–111.

[6] N. Fusco and C. Sbordone, Local boundedness of minimizers in a limit case, Manuscripta Math., 69,(1990) 19–25.

[7] M. Giaquinta, Growth conditions and regularity, a counter example, Manuscripta Math., 59, (1987),245–248.

[8] I.M. Kolodıı, The boundedness of generalized solutions of elliptic differential equations, Vestnik Moskov.

Univ. Ser. I Mat. Meh., 25, (1970), 44–52 (Russian). English transl.: Moscow Univ. Math. Bull. 25(1970), 31–37.

[9] S.N. Kruzhkov and I.M. Kolodıı, On the theory of embedding of anisotropic Sobolev spaces, Uspekhi

Mat. Nauk, 38, (1983), 207–208 (Russian). English transl.: Russian Math. Surveys 38 (1983), 188–189[10] V. Liskevich and I.I. Skrypnik, Holder continuity of solutions to an anisotropic elliptic equation, Non-

linear Anal., 71, (2009), 1699–1708.

[11] P. Marcellini, Un exemple de solution discontinue d’un probleme variationel dans le cas scalaire,Preprint 11, Ist. Mat. “U. Dini”, Firenze, 1987–88.

[12] P. Marcellini, Regularity of minimizers of integrals of the calculus of variations with nonstandard growth

conditions, Arch. Ration. Mech. Anal., 105, (1989), 267–284.[13] P. Marcellini, Regularity and existence of solutions of elliptic equations with p, q-growth conditions, J.

Differential Equations, 90, (1991), 1–30.

[14] S.M. Nikol’skiı, Imbedding theorems for functions with partial derivatives considered in different met-rics, Izd. Akad. Nauk. SSSR, 22, (1958), 321–336 (Russian). English transl.: Amer. Math. Soc. Transl.,

90, (1970), 27–44.

[15] M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Ricerche Mat., 18, (1969), 3–24.


Department of Mathematics, Vanderbilt University, 1326 Stevenson Center, Nashville TN

37240, USAE-mail address: [email protected]

URL: http://www.math.vanderbilt.edu/~dibe/

Dipartimento di Matematica “F. Casorati”, Universita di Pavia, via Ferrata 1, I-27100 PaviaE-mail address: [email protected]

URL: http://arturo.imati.cnr.it/~gianazza

Dipartimento di Matematica e Informatica “U. Dini”, Universita di Firenze, viale Morgagni

67/A, I-50134 Firenze


URL: http://web.math.unifi.it/users/vespri/

JEPE Vol 2, 2016, p. 171-187

EXISTENCE AND REGULARITY RESULTS FOR FULLY NONLINEAR

OPERATORS ON THE MODEL OF THE PSEUDO PUCCI’S

OPERATORS

ISABEAU BIRINDELLI, FRANCOISE DEMENGEL

Abstract. This paper is devoted to the existence and Lipschitz regularity of viscositysolutions for a class of very degenerate fully nonlinear operators, on the model of the

pseudo p-Laplacian. We also prove a strong maximum principle.

1. Introduction

Recall that the pseudo-p-Laplacian, for p > 1 is defined by:

∆pu :=

N∑1

∂i(|∂iu|p−2∂iu).

When p > 2, it is degenerate elliptic at any point where even only one derivative ∂iu is zero.Using classical methods in the calculus of variations, equation

(1.1) ∆pu = f

has solutions in W 1,ploc , when for example f ∈ Lp

′

loc. Even if the existence results do notdiffer from the one for the usual p-Laplacian i.e. ∆pu = div(|∇u|p−2∇u), the regularityraises high difficulties. For the usual p-Laplacian , the reader can look at [16], [10], in anon exhaustive manner . However, coming back to the pseudo p-Laplacian, when p < 2,Lipschitz regularity is a consequence of [11].

When p > 2 things are more delicate. Note that in [7], for some fixed non negativenumbers δi, the following widely degenerate equation was considered

(1.2)∑i

∂i((|∂iu| − δi)p−1+

∂iu

|∂iu|) = f.

The authors proved that the solutions of (1.2) are in W 1,qloc for any q < ∞, when f ∈ L∞loc.

As a consequence, by the Sobolev Morrey’s imbedding, the solutions are Holder continuousfor any exponent γ < 1.

The Lipschitz interior regularity for (1.1) has been recently proved by the second authorin [9]. The regularity obtained concerns Lipschitz continuity for viscosity solutions. Since

2010 Mathematics Subject Classification. Primary: 06B10; Secondary: 06D05.

Key words and phrases. Degenerate elliptic, Lipschitz regularity.Received 17/08/2016, accepted 29/11/2016.This work was completed with the support of Laboratoire AGM8088 and GNAMPA-INDAM.

171

172 ISABEAU BIRINDELLI, FRANCOISE DEMENGEL

weak solutions are viscosity solutions, (see also [2]), she obtains Lipschitz continuity for weaksolutions when the forcing term is in L∞loc.

At the same time, in [6], the local Lipschitz regularity of the solutions of (1.2) has been

proved when either N = 2, p ≥ 2 and f ∈ W 1,p′

loc or N ≥ 3, p ≥ 4, and f ∈ W 1,∞loc . Remark

that (1.2) can also be written formally as∑i

(|∂iu| − δi)p−2+ ∂iiu =

f

(p− 1).

This expression has an obvious meaning in the framework of viscosity solutions and withthe methods used in [9], one can prove, in particular, that the solutions are locally Holder’scontinuous for any exponent γ < 1, when f ∈ L∞loc. Unfortunately the Lipschitz continuityfor viscosity solutions of (1.2) cannot be obtained in the same way.

We now state the precise assumptions on the fully nonlinear operators that will be con-sidered in this paper and we state our main result. Fix α > 0 and, for any q ∈ RN , letΘα(q) be the diagonal matrix with entries |qi|

α2 on the diagonal, and let S be in the space

of symmetric matrices in RN .In the following the norm |X| denotes for a symmetric matrix X, |X| =

∑i |λi(X)|,

sometimes for convenience of the computations we shall also use ||X|| = (∑|λi|2)

12 ≡

tr(tXX))12 .

Let F be defined on RN × RN × S, continuous in all its arguments, which satisfiesF (x, 0,M) = F (x, p, 0) = 0 and

(H1) For any M1 ∈ S and M2 ∈ S, M2 ≥ 0, for any x ∈ RN

(1.3) λtr(Θα(q)M2Θα(q)) ≤ F (x, q,M1 +M2)− F (x, q,M1) ≤ Λtr(Θα(q)M2Θα(q)).

(H2) There exist γF ∈]0, 1] and cγF > 0 such that for any (q,X) ∈ RN × S, for any(x, y) ∈ (RN )2

(1.4) |F (x, q,X)− F (y, q,X)| ≤ cγF |x− y|γF |q|α|X|.

(H3) There exists ωF a continuous function on R+ such that ωF (0) = 0, and as soon as(X,Y ) satisfy for some m > 0

−m(

I 00 I

)≤(X 00 Y

)≤ m

(I −I−I I

)then

F (x,m(x− y), X)− F (y,m(x− y), Y ) ≤ ωF (m|x− y|α+2α+1 ) + o(m|x− y|

α+2α+1 ).

(H4) There exists cF such that for any p, q ∈ RN , for all x ∈ RN , X ∈ S

|F (x, p,X)− F (x, q,X)| ≤ cF (

i=N∑i=1

||pi|α − |qi|α|)|X|

Example of operators that satisfy (H1) to (H4) are

F (x, p,X) := tr(L(x)Θα(p)XΘα(p)L(x)),

when L(x) is a Lipschitz and bounded matrix such that√λI ≤ L ≤

√ΛI.

EXISTENCE AND REGULARITY 173

Other examples are the pseudo-Pucci’s operators, for 0 < λ < Λ

M+α (q,X) = Λtr((Θα(q)XΘα(q))+)− λtr((Θα(q)XΘα(q))−)

= supλI≤A≤ΛI

tr(AΘα(q)XΘα(q)).

and

M−α (q,X) = −M+α (q,−X).

satisfy all the assumptions above. The case α = 0 reduces to the standard extremizinguniformly elliptic Pucci operators. In the appendix we shall check that M+

α (q,X) satisfies(H4).

We can also consider

F (x, p,X) := a(x)M±α (p,X),

where a is a Lipschitz function such that a(x) ≥ ao > 0.We shall also consider equations with lower order terms. Precisely, let h be defined on

RN ×RN , continuous with respect to its arguments, which satisfies on any bounded domainΩ

(1.5) |h(x, q)| ≤ ch,Ω(|q|1+α + 1)

Our main result is the following.

Theorem 1.1. Let Ω be a bounded domain and f be continuous and bounded in Ω andsuppose that (H1), (H2), (H4) and (1.5) hold. Let u be any viscosity solution of

(1.6) F (x,∇u,D2u) + h(x,∇u) = f in Ω,

Then, for any Ω′ ⊂⊂ Ω , there exists CΩ′ , such that for any x and y in Ω′

|u(x)− u(y)| ≤ CΩ′ |x− y|.

This will be a consequence of the more general result given in Theorem 2.1, Section 2.We shall construct in Section 3 a super-solution of (1.6) which is zero on the boundary.

Theorem 1.1, and the validity of the comparison principle, allows to prove, using Ishii’sversion of Perron’s method, the following existence result :

Theorem 1.2. Suppose that Ω is a bounded C2 domain and let F and h satisfy (H1), (H2),(H3), (H4) and (1.5). Then, for any f ∈ C(Ω), there exists u a viscosity solution of

F (x,∇u,D2u) + h(x,∇u) = f(x) in Ωu = 0 on ∂Ω.

Furthermore u is Lipschitz continuous in Ω .

Finally in the last section we prove that the strong maximum principle holds for solutionsof equation (1.6) under the hypothesis of Theorem 1.2.

We end this introduction by recalling that many questions concerning these very degen-erate operators are still open. For example it is not clear whether a sort of Alexandrov,Bakelman, Pucci ’s inequality hold true, similarly to the cases treated by Imbert in [12].Finally the next open question concerning the regularity of solutions would be to prove thatthe solutions are in fact C1. Even in the cases f = 0 and/or N = 2 it does not seem easy todo.


2. Proof of Lipschitz regularity.

Let F and Ω be as in Theorem 1.1. We shall now state and prove our main result:

Theorem 2.1. Let f and k be continuous and bounded in a bounded open set Ω. Let F andh satisfy (H1), (H2), (H4) and (1.5). Suppose that u is a bounded USC sub-solution of

F (x,∇u,D2u) + h(x,∇u) ≥ f in Ω

and that v is a bounded LSC super-solution of

F (x,∇v,D2v) + h(x,∇v) ≤ k in Ω.

Then, for any Ω′ ⊂⊂ Ω, there exists CΩ′ , such that for any (x, y) ∈ (Ω′)2

u(x) ≤ v(y) + supΩ

(u− v) + CΩ′ |x− y|.

We start by recalling some general facts.If ψ : RN × RN → R, let D1ψ denotes the gradient in the first N variables and D2ψ the

gradient in the last N variables.In the proof of Theorem 2.1 we shall need the following technical lemma.

Lemma 2.2. Suppose that u and v are respectively USC and LSC functions such that, forsome constant M > 1 and for some C2 function Φ

ψ(x, y) := u(x)− v(y)−M |x− xo|2 −M |y − xo|2 −MΦ(x, y)

has a local maximum in (x, y).Then for any ι > 0, there exist Xι, Yι such that

(MD1Φ(x, y) + 2M(x− xo), Xι) ∈ J2,+u(x),

(−MD2Φ(x, y)− 2M(y − xo),−Yι) ∈ J2,−v(y)

with

−(1

ι+ |A|+ 1)

(I 00 I

)≤(Xι − 2M I 0

0 Yι − 2M I

)≤ (A+ ιA2) +

(I 00 I

)and A = MD2Φ(x, y).

This is a direct consequence of a famous Lemma by Ishii [14]. For the convenience of thereader the proof of Lemma 2.2 is given in the appendix. In the sequel, we will use Lemma2.2 with Φ(x, y) := g(x−y), and g is some radial function C2 except at 0, that will be chosenlater. Then

MD2Φ(x, y) = M

(D2g(x− y) −D2g(x− y)−D2g(x− y) D2g(x− y)

).

Choosing ι = 11+4M |D2g(x)| , and defining H(x) := D2g(x) + 2ιD2g2(x), one has

M( D2Φ + ι(D2Φ)2) = M

(H(x− y) −H(x− y)−H(x− y) H(x− y)

).

Remark that M |D2Φ(x, y)| = 2M |D2g(x− y)|. We give some precisions on the choice of g.We will assume that there exists ω ∈ C(R+) ∪ C2(R+?), such that g(x) = ω(|x|) and :

(2.1) ω(0) = 0, ω(s) > 0, ω′(s) > 0 and ω′′(s) < 0 on ]0, so[, for some given so ≤ 1.


For x 6= 0, it is well known that Dg(x) = ω′(|x|) x|x| and

D2g(x) =

(ω′′(|x|)− ω′(|x|)

|x|

)x⊗ x|x|2

+ω′(|x|)|x|

I.

For ι ≤ 14|D2g(x)| , defining γH(r) = 1 + 2ι

(ω′(r)r

)∈ [ 1

2 ,32 ], and βH(r) = 1 + 2ιω′′(r) ∈ [ 1

2 ,32 ]

then

(2.2) D2g + 2ι(D2g)2(x) =

(βH(|x|)ω′′(|x|)− γH(|x|)ω

′(|x|)|x|

)x⊗ x|x|2

+ γH(|x|)ω′(|x|)|x|

I.

For |x| < 1 and ε > 0, we shall use the following set:

I(x, ε) := i ∈ [1, N ], |xi| ≥ |x|1+ε

and the diagonal matrix Θ(x) := Θα(q) for q = M ω′(|x|)xi|x| i.e. with entries Θii(x) =

Mα2

∣∣∣ω′(|x|)xi|x|

∣∣∣α2 . From now on, if X is a symmetric matrix, µi(X) for i = 1, . . . , N indicate

the ordered eigenvalues of X.A consequence of (2.2) is the following Proposition proved in [9].

Proposition 2.3 ([9]). Using the notations above,

(1) If α ≤ 2, for all x 6= 0, |x| < so,

(2.3) µ1

(Θ(x)H(x)Θ(x)

)≤ M1+α

2N−

α2 ω′′(|x|)(ω′(|x|))α < 0.

(2) If α > 2, for all x 6= 0, |x| < so, for any ε > 0 such that I(x, ε) 6= ∅, and such that

(2.4) βH(|x|)ω′′(|x|)(1−N |x|2ε) + γH(|x|)N |x|2εω′(|x|)|x|

≤ ω′′(|x|)4

< 0,

then

µ1

(Θ(x)H(x)Θ(x)

)≤M1+α 1−N |x|2ε

#I(x, ε)(ω′(|x|))αω

′′(|x|)4|x|(α−2)ε.(2.5)

[Proof of Theorem 2.1] It is clear that it is sufficient to prove the result when Ω = B1 isthe ball of center 0 and radius 1 and Ω′ = Br for some r < 1.

Borrowing ideas from [1], [5], [15], [13], for some xo ∈ Br we define the function

ψ(x, y) = u(x)− v(y)− sup(u− v)−Mω(|x− y|)−M |x− xo|2 −M |y − xo|2;

M is a large constant and ω is a function satisfying (2.1), both to be defined more preciselylater .

If there exists M , independent of xo ∈ Br, such that ψ(x, y) ≤ 0 in B21 , by taking x = xo

and, using |xo − y| ≤ 2, one gets

u(xo)− v(y) ≤ sup(u− v) + 3Mω(|xo − y|).So making xo vary we obtain that, for any (x, y) ∈ B2

r ,

u(x)− v(y) ≤ sup(u− v) + 3Mω(|x− y|).This proves the theorem when ω(s) behaves like s near zero. Note that this will be

obtained once the case where ω(s) = sγ is treated for γ ∈]0, 1[, i.e the Holder’s analogousresult.


In order to prove that ψ(x, y) ≤ 0 in B2r , suppose by contradiction that the supremum of

ψ is positive and achieved on (x, y) ∈ B12

. For some δ > 0, with δ < so, we choose M suchthat

(2.6) M(1− r)2 > 8(|u|∞ + |v|∞), and M > 1 +2|u|∞ + 2|v|∞

ω(δ).

This implies that |x−xo|, |y−xo| < 1−r2 . Hence, by (2.6), x and y are in B 1+r

2in particular

they are in B1. Furthermore, always using (2.6), the positivity of the supremum of ψ, thevalue chosen for M and the increasing behaviour of ω before so, lead to |x− y| < δ.

As it will be shown later the contradiction will be found by choosing δ small enoughdepending on (r, α, λ,Λ, N).

We proceed using Lemma 2.2 and so, for all ι > 0 there exist Xι and Yι such that

(qx, Xι) ∈ J2,+u(x) and (qy,−Yι) ∈ J

2,−v(y)

with qx = q + 2M(x − xo), qy = q − 2M(x − xo), q = Mω′(|x − y|) x−y|x−y| . Furthermore,

still using the above notations i.e. g(x) = ω(|x|), and recalling that we have chosen ι ≤1

1+4M |D2g(x−y)| , for H = (D2g(x− y)) + 2ιD2g(x− y))2), we have that

(2.7)

−( 1ι + 2M |H|)

(I 00 I

)≤

(Xι − (2M + 1)I 0

0 Yι − (2M + 1)I

)

≤M(

H −H−H H

).

From now on we will drop the ι for X and Y . Recall that Θ(q) := Θα(q) is the diagonal

matrix such that (Θ)ii(q) = (|qi|)α2 .

In order to end the proof we will prove the following claims.Claims. There exists c > 0 depending only on α,N, λ,Λ, r and there exists τ > 0, such

that, if δ is small enough and |x− y| < δ, the matrix Θ(X + Y )Θ satisfies

(2.8) µ1(Θ(X + Y )Θ) ≤ −cMα+1|x− y|−τ

There exist τi < τ and ci for i = 1, . . . , 4 depending on α,N, λ,Λ, r such that the fourfollowing assertions hold :

(2.9) for all j ≥ 2, µj(Θ(X + Y )Θ) ≤ c1Mα+1|x− y|−τ1 ,

(2.10) |F (x, qx, X)− F (x, q,X)| ≤ c2Mα+1|x− y|−τ2

(similarly |F (y, qy,−Y )− F (y, q,−Y )| ≤ c2Mα+1|x− y|−τ2)

(2.11) |F (x, q,X)− F (y, q,X)| ≤ c3Mα+1|x− y|−τ3 ,(similarly |F (x, q,−Y )− F (y, q,−Y )| ≤ c3Mα+1|x− y|−τ3)

(2.12) |h(x, qx)|+ |h(y, qy)| ≤ c4Mα+1|x− y|−τ4 .From all these claims, by taking δ small enough such that for c > 0 defined in (2.8),c2δ−τ2+τ + c3δ

τ−τ3 + c4δτ−τ4 + Λc1δ

τ−τ1 < λc2 , one gets


F (x, qx, X)− F (y, qy,−Y ) + h(x, qx)− h(y, qy) ≤ −λc2Mα+1|x− y|−τ .

Observe that δ depends only on λ,Λ, α,N, r. Finally, one can conclude as follows

f(x) ≤ F (x, qx, X) + h(x, qx)

≤ F (y, qy,−Y ) + h(y, qy)− λc

2Mα+1|x− y|−τ

≤ −λc2Mα+1|x− y|−τ + k(y).

This contradicts the fact that f and k are bounded, as soon as δ is small or M is largeenough.

In conclusion, in order to end the proof it is sufficient to prove (2.8), (2.9), (2.10), (2.11),(2.12). But we will need to distinguish the cases ω(s) = sγ and ω(s) ' s both when α ≤ 2and when α ≥ 2.

To prove the claims, we will use inequality (2.7) which has three important consequencesfor Θ(X + Y − 2(2M + 1)I)Θ:

(1) As it is well known the second inequality in (2.7) gives(X + Y − 2(2M + 1)I) ≤ 0, then also Θ(X + Y − 2(2M + 1)I)Θ ≤ 0. In particular,for any j = 1, . . . , N

(2.13) µj(Θ(X + Y )Θ) ≤ 6M |Θ|2.

(2) By Proposition 2.3, Θ(H)Θ has a large negative eigenvalue, given respectively by(2.3) in the case α ≤ 2 and by (2.5) when α ≥ 2. Let e be a corresponding

eigenvector. Multiplying by Θ

(e−e

)on the right and by its transpose on the left

of (2.7), one gets, that

teΘ(X + Y − 2(2M + 1)Id)Θe ≤ 4te(Θ(H)Θ)e.

In particular, using (2.3), one obtains that when α ≤ 2,

(2.14) µ1(Θ(X + Y − 2(2M + 1)I)Θ) ≤ 2N−1M1+αω′′(|x− y|)(ω′(|x− y|))α;

which in turn implies that

(2.15) µ1(Θ(X + Y )Θ) ≤ 2N−1M1+αω′′(|x− y|)(ω′(|x− y|))α + 6M |Θ|2.

When α > 2, if (2.4) holds, using (2.5), one obtains

(2.16) µ1(Θ(X + Y )Θ) ≤1−N |x− y|2ε

#I(x− y, ε)M1+αω′′(|x− y|)(ω′(|x− y|))α|x− y|(α−2)ε + 6M |Θ|2.

(3) Finally, using (2.7), we obtain an upper bound for |X|, |Y | i.e.

(2.17) |X|, |Y | ≤ 6M(|D2g(x− y)|+ 1),

remarking that |H| ≤ 32 |D

2g(x− y)|.


Proofs of the claims when ω(s) = sγ and α ≤ 2.In this case, ω′(s) = γsγ−1 and ω′′(s) = −γ(1− γ)sγ−2,

q = Mγ|x− y|γ−1 x− y|x− y|

, qx = q + 2M(x− xo), qy = q − 2M(y − xo).

By (2.15), since γ ∈ (0, 1),

µ1(Θ(X + Y )Θ) ≤ −2γ(1− γ)N−1Mα+1|x− y|γ−2+(γ−1)α + 6M |Θ|2,

while 6|Θ|2 ≤ 6Mαγα|x− y|(γ−1)α. Consequently, as soon as δ is small enough,

µ1(Θ(X + Y )Θ) ≤ −2γ(1− γ)

NMα+1|x− y|γ−2+(γ−1)α + 6M1+αγα|x− y|(γ−1)α

≤ −γ 1− γN

Mα+1|x− y|γ−2+(γ−1)α.

This proves (2.8) with τ = 2− γ + (1− γ)α, and c = γ 1−γN .

Now using (2.13) and the above estimate on M |Θ|2, (2.9) holds with τ1 = (1− γ)α < τ .Recall that by (2.17),

(2.18) |X|, |Y | ≤ 6M(γ(N − γ) + 1)|x− y|γ−2.

Consequently (2.11) holds with τ3 = (2−γ)+(1−γ)α−γF and c3 = 6cγF γα(γ(N−γ)+1)

using hypothesis (1.4).To prove (2.10) we will use the following universal inequality : For any z and t in R

||z|α − |t|α| ≤ sup(1, α)|z − t|inf(1,α)(|z|+ |t|)(α−1)+

in the form ( for any i ∈ [1, N ]),

(2.19) ||qxi |α − |qi|α| ≤ sup(1, α)Mα|xi − yi|(γ−1)(α−1)+ .

Hence using (H4) and (2.18), (2.10) holds with τ2 = (2 − γ) + (1 − γ)(α − 1)+, andc2 = 6cFN sup(1, α)γα(γ(N − γ) + 1). Finally, (2.12) holds with τ4 = (1 − γ)(1 + α) andc4 = 2ch,Ω((γ + 3)1+α + 1) .

Proofs of the claims when ω(s) = sγ and α ≥ 2.The function ω is the same than in the previous case. In order to use the result in Proposition2.3 we need (2.4) to be satisfied. For that aim we take ε > 0 such that ε < inf

(γF2 ,

1−γ2

).

Let

(2.20) δN :=

[(1− γ)

2(4− γ)N

] 12ε

and assume δ < δN . In particular, for α ≥ 2, using the definition of δN in (2.20), for|x− y| < δ ≤ δN the set I(x− y, ε) 6= ∅, indeed observe that there exists i ∈ [1, N ] such that

|xi − yi|2 ≥|x− y|2

N≥ |x− y|2+2ε.


Furthermore,

1

2ω′′(|x− y|)(1−N |x− y|2ε) +

3N

2|x− y|2εω

′(|x− y|)|x− y|

≤ 1

2ω′′(|x− y|)

+N

2|x− y|2ε(γ(1− γ) + 3γ)|x− y|γ−2

≤ 1

4γ(γ − 1)|x− y|γ−2

=ω′′(|x− y|)

4,

and then (2.4) is satisfied. We are in a position to apply (2.16), and Θ(X + Y )Θ satisfies

µ1(Θ(X + Y )Θ) ≤ −(

1−N |x− y|2ε

#I(|x− y, ε)

)γ(1− γ)Mα+1|x− y|γ−2+(γ−1)α+ε + 6M |Θ|2,

hence remarking that 1−N ||x−y|2ε#I(|x−y,ε) ≥

12N

µ1(Θ(X + Y )Θ) ≤ −(γ(1− γ)

2N)Mα+1|x− y|γ−2+(γ−1)α+ε

+6M1+αγα|x− y|(γ−1)α

≤ −(γ(1− γ)

4N)Mα+1|x− y|γ−2+(γ−1)α+ε

for |x− y| ≤ δ small enough. Hence (2.8) holds with τ = 2− γ + (1− γ)α− ε.Note that (2.9), (2.11) (2.10) and (2.12) have already been proved in the previous case,

since the sign of α − 2 does not play a role. Recall then that τ1 = (−γ + 1)α, whileτ3 = (2− γ) + (1− γ)α− γF < τ by the choice of ε.

Finally τ2 = (2− γ) + (α− 1)(γ − 1) and (2.12) still holds with τ4 = (1− γ)(1 + α).

Let us observe that in the hypothesis of Theorem 2.1 we have proved that u and v satisfy,for any γ ∈ (0, 1),

(2.21) u(x) ≤ v(y) + supΩ

(u− v) + cγ,r|x− y|γ .

This will be used in the next cases.Proofs of the claims when ω(s) ' s and α ≤ 2.We choose τ ∈ (0, inf(γF ,

12 ,

α2 )) and γ ∈] τ

inf( 12 ,α2 ), 1[. We define, for s ≤ so, ω(s) = s−ωos1+τ

and, for s > so, ω(s) = soτ1+τ , ωo is chosen so that ω is extended continuously.

We suppose that δ < 1 and δτωo(1 + τ) < 12 , which ensures that

(2.22) for s < δ,1

2≤ ω′(s) < 1, ω(s) ≥ s

2.

We suppose that

(2.23) M = sup

((1 + τ)2(|u|∞ + |v|∞)

δτ, 1 +

4(|u|∞ + |v|∞)

(1− r)2

)


which implies in particular (2.6). So we derive that |x− y| ≤ δ and x, y ∈ B 1+r2

.

Here

|D2g(x− y)| = N − 1

|x− y|+ ωoτ(1 + τ)|x− y|−1+τ ≤ (N − 1 + ωoτ(1 + τ))|x− y|−1,

and |H| ≤ 3

2|D2g(x− y)|.

Then (2.17) is nothing else but

(2.24) |X|,|Y | ≤ 6M(|D2g(x− y)|+ 1) ≤ 6M(N + ωoτ(1 + τ))|x− y|−1.

Furthermore q = Mω′(|x− y|) x−y|x−y| , q

x = q + 2M(x− xo), qy = q − 2M(y − xo).Using (2.21) in B 1+r

2, for all γ < 1,

M |x− xo|2 +M |y − xo|2 + sup(u− v) ≤ u(x)− v(y) ≤ sup(u− v) + cγ,r|x− y|γ

and then

(2.25) |y − xo|+ |x− xo| ≤ 2

(cγ,r|x− y|γ

M

) 12

.

Then taking δ small enough, more precisely if (cγ,rδγ)

12 < 1

64 by (2.22),

(2.26)M

2≤ |q| ≤M,

M

4≤ |qx|, |qy| ≤ 5M

4.

Then we derive from (2.15) that

µ1(Θ(X + Y )Θ) ≤ −ωoτ(1 + τ)

NMα+1|x− y|τ−1 + 6M |Θ|2.

Since M |Θ|2 ≤ M1+α, (2.8) holds (as soon as δ is small enough) with τ = 1 − τ , and

c = ωoτ(1+τ)2N , (2.9) holds with

τ1 = 0 < 1− τ, and c1 = 6,

while (2.11) is satisfied with

τ3 = −γF + 1 < 1− τ, and c3 = 12cγF (N + ωoτ(1 + τ)).

To check (2.10), we use (2.19), (2.24) , (2.25) and (2.26)

| |qxi |α − |qi|α| |X| ≤ 6(N + ωoτ(1 + τ))M1+inf(α,1)

2 cinf(1,α)

2γ,r |x− y|

inf(1,α)γ2 |x− y|−1.

Hence, for inf(1, α)γ > 2τ , (2.10) holds with

τ2 = 1− inf(1, α)

2γ and c2 = 6cF sup(1, α)N(N − 1 + ωoτ(1 + τ))(cγ,r)

α2

if α ≤ 1 and

c2 = αcF 6N(N − 1 + ωoτ(1 + τ))(cγ,r)α2 3α−1), if α ≥ 1.

Finally τ4 = 0 and c4 = ch,Ω(21+α + 1) are convenient for (2.12).

Proofs of the claims when ω(s) ' s and α > 2.In order to use the result in Proposition 2.3 we need (2.4) to be satisfied. For that aim

we take τ , ε > 0 and γ such that


(2.27) 0 < τ <γFα, 1 > γ > τα, and

τ

2< ε < inf

( γ2 − τα− 2

,γF − τα− 2

).

Let us define ω, so, as in the case α ≤ 2. We suppose δ < δN where

δN :=

(ωo(1 + τ)τ

2N(3 + ωoτ(1 + τ))

) 12ε−τ

.(2.28)

In particular, since there exists i such that |xi − yi|2 ≥ 1N |x − y|

2 ≥ |x − y|2+2ε, by (2.28),

I(x− y, ε) 6= ∅. Furthermore, recall that by (2.28), 1 ≥ ω′(|x− y|) ≥ 12 and

1

2ω′′(|x− y|) +

N

2ωoτ(1 + τ)|x− y|τ−1+2ε +

3

2N |x− y|2ε−1ω′(|x− y|)

≤ 1

2ω′′(|x− y|) +

N

2(ωoτ(1 + τ) + 3)|x− y|2ε−1

≤ −1

4ωo(1 + τ)τ |x− y|−1+τ =

ω′′(|x− y|)4

,

and then (2.4) holds. We still assume that (2.23) holds.As in the case α ≤ 2, using (2.21), for δ small enough, (2.26) is still true.The hypothesis (2.28) ensures, using also (2.5) that

µ1(Θ(X + Y )I)Θ) ≤ −ωoτ(1 + τ)

2NM1+α|x− y|−1+τ+(α−2)ε + 6M |Θ|2

and then, by (2.13) and 6M |Θ|2 ≤ 6M1+α, by (2.27) and for δ small enough, (2.8) holds

with τ = (2−α)ε+1−τ and c = ωoτ(1+τ)4N . Furthermore (2.9) holds with τ1 = 0, and c1 = 6.

As in the previous case, (2.24) is true, and then, (2.11) holds with

τ3 = 1− γF < 1− τ + (2− α)ε and c3 = 6cγF (N − 1 + ωoτ(1 + τ)).

Now using (2.19), (2.24), (2.26), (2.25), one has

||qxi |α− |qi|α||X| ≤ 6α(N +ωoτ(1 + τ))M |x−xo|(5M

4)α−1M |x− y|−1 ≤ c3|x− y|

γ2−1M1+α

and then (2.10) holds with

τ2 = 1− γ

2< 1− τ + (2− α)ε and c2 = αNc

12γ,r(2)α−16(N − 1 + ωoτ(1 + τ)).

Note finally that

|h(x, qx)|+ |h(y, qy)| ≤ 2ch

(5M

4

)1+α

and then (2.12) holds with τ4 = 0 and c4 = 22+αch .


3. Existence of solutions.

Using Perron’s method, see e.g. [8], the existence’s Theorem 1.2 will be proved once thefollowing Propositions are known:

Proposition 3.1. Suppose that Ω is a bounded domain in RN and that F satisfies (H1),(H2), (H3), (H4). Suppose that h is continuous and it satisfies (1.5). Let u be a USCsub-solution of

F (x,∇u,D2u) + h(x,∇u)− β(u) ≥ f in Ω

and v be a LSC super-solution of

F (x,∇v,D2v) + h(x,∇v)− β(v) ≤ k in Ω

where β, f and k are continuous. Suppose that either β is increasing and f ≥ k in Ω, or βis nondecreasing and f > k in Ω.

If u ≤ v on ∂Ω, then u ≤ v in Ω.

Proposition 3.2. Suppose that the assumptions in Proposition 3.1 hold, and that f iscontinuous and bounded and β is increasing. Suppose that u is a USC sub-solution, and uis a LSC super-solution of the equation

F (x,∇u,D2u) + h(x,∇u)− β(u) = f, in Ω,

such that u = u = ϕ on ∂Ω. Then there exists u a viscosity solution of the equation withu ≤ u ≤ u in Ω, and u = ϕ on ∂Ω.

The proofs of these two Propositions can be done by using the classical tools, see [8].

Remark 1. One can get the same existence’s result when β = 0, by using a standardapproximation procedure and the stability of viscosity solutions.

Nevertheless the proof of Theorem 1.2 requires the existence of a super-solution which iszero on the boundary when β = 0 which is the object of the next proposition:

Proposition 3.3. Suppose that Ω is a bounded C2 domain, and that F and h satisfy thehypothesis in Proposition 3.1. Then for any f continuous and bounded, there exist a super-solution and a sub-solution of

F (x,∇u,D2u) + h(x,∇u) = f in Ω

which are zero on the boundary.

Proof of Proposition 3.3 : Let diam(Ω) denote the diameter of Ω and we recall that thedistance to the boundary d satisfies everywhere that d is semi concave or equivalently thereexists C1 such that

D2d ≤ C1I.

In the following we will make the computations as if d be C2, it is not difficult to see thatthe required inequalities hold also in the viscosity sense.

Recall that∑Ni=1(∂id)2 = 1, hence

N∑i=1

|∂id|α ≤ N, while

N∑1

|∂id|α+2 ≥ N−αα+2 .


For some M large that will be chosen later, we define

ψ(x) = M(1− 1

(1 + d(x))k).

Clearly

∇ψ = Mk∇d

(1 + d)k+1, D2ψ =

Mk

(1 + d)k+2((1 + d)D2d− (k + 1)∇d⊗∇d)

and then, one has

F (x,∇ψ,D2ψ) ≤ (Mk)α+1

(1 + d)k+2+(k+1)α[(1 + d)M+

α (∇d,D2d)

−(k + 1)M−α (∇d,∇d⊗∇d)]

≤ (Mk)α+1

(1 + d)k+2+(k+1)α[N(1 + d)ΛC1

∑|∂id|α − (k + 1)λ

∑|∂id|α+2]

≤ (Mk)α+1

(1 + d)k+2+(k+1)α[N2(1 + diam(Ω))ΛC1 − λ(k + 1)N−

αα+2 ]

and

h(x,∇ψ) ≤ Ch(Mk)α+1

(1 + d)(k+1)(1+α).

In particular we can choose k such that

1

2λ(k + 1)N−

αα+2 = (1 + diam(Ω))(ΛC1N

2 + Ch).

Hence

F (x,∇ψ,D2ψ) + h(x,∇ψ) ≤ − (k + 1)λN−αα+2 (Mk)α+1

4(1 + d)k+2+(k+1)α.

For k as above we can choose M large enough in order that

F (x,∇ψ,D2ψ) + h(x,∇ψ) ≤ −‖f‖∞.A similar computation leads to:

F (x,∇(−ψ), D2(−ψ)) + h(x,∇(−ψ)) ≥ ‖f‖∞.

4. The strong Maximum Principle

Theorem 4.1. Under the hypothesis of Theorem 1.1, suppose that u is a supersolution ofthe equation F (x,∇u,D2u) ≤ 0 in a domain Ω and that u ≥ 0. Then either u > 0 in Ω oru ≡ 0.

Proof. Without loss of generality we suppose that u > 0 on B(x1, R), with R = |x1 − xo|and u(xo) = 0, and we can assume that the annulus R

2 ≤ |x − x1| ≤ 3R2 is included in Ω.

Let w be defined asw(x) = m(e−c|x−x1| − e−cR)

for some c and m to be chosen.For simplicity of the calculation we will suppose that x1 = 0 and we denote by r :=

|x − x1| = |x|. We choose m so that on r = R2 , w ≤ u in the same spirit of simplicity we

replace m by 1.


One has

∇w =−cxre−cr, D2w = e−cr

((c2

r2+

c

r3)(x⊗ x)− c

rI

)and then, using the usual notation Θ(∇w), H := Θ(∇w)D2wΘ(∇w), i.e.

Hec(α+1)r =( cr

)α((c2

r2+

c

r3)~i⊗~i− c

r~j ⊗~j

)where ~i =

∑|xi|

α2 xiei and ~j =

∑|xi|

α2 ei.

Since, by hypothesis (H1), F (x,∇w,D2w) ≥ e−c(α+1)rM−(H), whereM−(X) := infλI≤A≤ΛI(trAX) is the extremal Pucci operator, we need to evaluate theeigenvalues of H and in particular prove that

M−(H) > 0.

For that aim let us note that (~i,~j)⊥ is in the kernel of H. We introduce a = c2

r2 + cr3 and

b = − cr . Then the non zero eigenvalues of Hc−αecr(1+α) are given by

µ± =a|~i|2 + b|~j|2

2±

√√√√(a|~i|2 + b|~j|22

)2

− ab(|~i|2|~j|2 − (~i ·~j)2).

Note that there exist constants ci(N,α) for i = 1, · · · 4, such that

c1(N,α)

(R

2

)α+2

≤ c1(N,α)rα+2 ≤ |~i|2 ≤ c2(N,α)rα+2 ≤ c2(N,α)

(3R

2

)α+2

and

c3(N,α)

(R

2

)α≤ c3(N,α)rα ≤ |~j|2 ≤ c4(N,α)rα ≤ c4(N,α)

(3R

2

)α.

Note that one can choose c large enough in order that for some constant c5(N,α)

a|~i|2 + b|~j|2 ≥ c1(N,α)

(R

2

)α+2c2

r2− c4(N,α)

(3R

2

)αc

r

≥ c5(N,α)c2.

On the other hand one can assume c large enough in order that

4|ab|(|~i|2|~j|2 − (~i ·~j)2) ≤ 4c3

r2c2(N,α)c4(N,α)

(3R

2

)2α+2

≤ c6(N,α)c3

<

[(λ+ Λ

Λ− λ

)2

− 1

](c5(N,α)c2))2

≤

[(λ+ Λ

Λ− λ

)2

− 1

](a|~i|2 + b|~j|2

)2

.


In particular this implies

λµ+ + Λµ− = (a|~i|2 + b|~j|2

2) ×(

(λ+ Λ) + (λ− Λ)

√1 + 4

|ab|(|~i|2|~j|2 − (~i ·~j)2)

(a|~i|2 + b|~j|2)2

)> 0

i.e. M−(H) > 0. Using the comparison principle in the annulus R2 ≤ |x − x1| ≤ 3R2 one

obtains that u ≥ w.Observe that w touches u by below on xo, and then, since w is C2 around xo, by the

definition of viscosity solution

F (xo,∇w(xo), D2w(xo)) ≤ 0.

This contradicts the above computation.

Remark 2. As it is well known, the above proof can be used to see that on a point of theboundary where the interior sphere condition is satisfied, the Hopf principle holds.

Appendix A. Proof of Lemma 2.2

The proof of Lemma 2.2 is based on the following Lemma by Ishii

Lemma A.1. [14] Let A be a symmetric matrix on R2N . Suppose that U ∈ USC(RN ) andV ∈ USC(RN ) satisfy U(0) = V (0) and, for all (x, y) ∈ (RN )2,

U(x) + V (y) ≤ 1

2(tx,t y)A

(xy

).

Then, for all ι > 0, there exist XUι ∈ S, XV

ι ∈ S such that

(0, XUι ) ∈ J2,+U(0), (0, XV

ι ) ∈ J2,+V (0)

and

−(1

ι+ |A|)

(I 00 I

)≤(XUι 0

0 XVι

)≤ (A+ ιA2).

We can now start the proof of Lemma 2.2. The second order Taylor’s expansion for Φaround (x, y) , gives that for all ε > 0 there exists r > 0 such that, for |x− x|2 + |y−y|2 ≤ r2,

u(x)− u(x) − 〈(MD1Φ(x, y) + 2M)(x− xo), x− x〉+

+v(y)− v(y) − 〈(MD2Φ(x, y) + 2M)(y − xo), y − y〉

≤ 1

2

(t(x− x),t (y − y)

)(MD2Φ(x, y) + εI)

(x− xy − y

)+M(|x− x|2 + |y − y|2).

We now introduce the functions U and V defined, in the closed ball |x− x|2 + |y− y|2 ≤ r2,by

U(x) = u(x+ x)− 〈MD1Φ(x, y) + 2M(x− xo), x〉 − u(x)−M |x|2

andV (y) = −v(y + y)− 〈MD2Φ(x, y) + 2M(y − xo), y〉+ v(y)−M |y|2

which we extend by some convenient negative constants in the complementary of that ball(see [14] for details). Observe first that


(0, XU ) ∈ J2,+U(0), (0, XV ) ∈ J2,−

V (0)

is equivalent to

(MD1Φ(x, y) + 2M(x− xo), XU + 2M I) ∈ J2,+u(x)

and

(−MD2Φ(x, y)− 2M(y − xo),−XV − 2M I) ∈ J2,−v(y).

We can apply Lemma A.1, which gives that, for any ι > 0, there exists (Xι, Yι) such that

(MD1Φ(x, y) + 2M(x− xo), Xι) ∈ J2,+u(x)

and

(−MD2Φ(x, y)− 2M(y − xo),−Yι) ∈ J2,−v(y)

Choosing ε such that 2ει|MD2Φ(x, y)|+ ε+ ι(ε)2 < 1, one gets

−(1

ι+ |MD2Φ|+ 1)

(I 00 I

)≤

(Xι − 2M I 0

0 Yι − 2M I

)≤ (MD2Φ + ι(MD2Φ)2) +

(I 00 I

).

This ends the proof of Lemma 2.2.

Finally, as promised in the introduction, we here check that M+α (q,X) satisfies (H4).

First, recalling the properties of the Pucci’s operators we get

M+α (p,X) ≤ M+

α (p,X) +M+(Θα(p)XΘα(p)−Θα(q)XΘα(q))

≤ M+α (p,X) + (Λ + λ)|(Θα(p)XΘα(p)−Θα(q)XΘα(q))|

= M+α (p,X) +

Λ + λ

2(|(Θα(p)−Θα(q))X(Θα(p) + Θα(q))

+ ((Θα(p) + Θα(q))XΘα(p)−Θα(q)|)

Then one has using for X symmetric ||X|| ≤ |X| ≤√N ||X||, and observing that for any

matrices A B, ||AB|| = ||BA||

|(Θα(p)−Θα(q))X(Θα(p) + Θα(q)) + ((Θα(p) + Θα(q))XΘα(p)−Θα(q)|≤√N ||(Θα(p)−Θα(q))X(Θα(p) + Θα(q))

+ ((Θα(p) + Θα(q))XΘα(p)−Θα(q)||≤ 2

√N ||X(Θα(p)−Θα(q))(Θα(p) + Θα(q))||

≤ 2√N ||X|| ||(Θα(p)−Θα(q))(Θα(p) + Θα(q))||

≤ 2√N |X| ||(Θα(p))2 − (Θα(q))2||

≤ 2√N |X|

∑i

||pi|α − |qi|α|

Acknowledgment The authors wish to thank the anonymous referee for his judicious remarkswhich improved this paper.


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[9] F. Demengel, Lipschitz interior regularity for the viscosity and weak solutions of the Pseudo p-LaplacianEquation. Advances in Differential Equations, (2016) 21, (3-4), .

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Dipartimento di Matematica ”G. Castelnuovo”, Sapienza Universita di Roma, Piazzale Aldo

Moro 5, 00185 Roma, and, Universite de Cergy-Pontoise, 2 av. Adolphe Chauvin, 95302, France


JEPE Vol 2, 2016, p. 189-206

CONTINUOUS TIME RANDOM WALK BASED THEORY FOR A

ONE-DIMENSIONAL COARSENING MODEL

DIEGO TORREJON, MARIA EMELIANENKO AND DMITRY GOLOVATY

Abstract. In this work we propose a master equation describing evolution of the veloc-ity statistics in a one-dimensional coarsening model motivated by the studies of polycrys-

talline materials. The model postulates the dynamics of a large number of intervals—

referred to as domains—on the real line. The length of the intervals changes duringevolution and the intervals are removed from the system once their length reaches zero.

The coarsening process observed in this model exhibits a number of interesting features,

such as nonhomogeneous inter-arrival times between reconfiguration events and develop-ment of spatiotemporally self-similar distributions.

We generalize the standard continuous time random walk (CTRW) theory to include

time-dependent jumps and subject it to time-dependent temporal rescaling to obtainan accurate non-homogeneous Poisson description of the coarsening process in the one-

dimensional model. The theory leads to the evolution equation having self-similar solu-

tions observed in simulations.The new framework allows to accurately estimate coarsening rates and characterize

resulting steady-state distribution for the domain energies described by a power law of a

uniformly distributed quantity. Although derived here in the context of a one-dimensionalsystems, this work naturally extends to higher dimensional CTRW coarsening models.

1. Introduction

Coarsening models are used to describe dynamics of physical systems consisting of multi-ple domains, where some regions grow at the expense of others. The average size of domainsgrows over time, hence the network experiences “coarsening”. While this process is verycomplex in three dimensions, lower-dimensional models often offer advantages in elucidatingcertain features of dynamics and are extremely instrumental at the early stages of theorydevelopment. Smoluchowski’s coagulation [1], Mullins model [2, 3] and curvature-drivenmodel by Lazar et al [4, 5] are only some of the one-dimensional examples that can be foundin literature. Here we focus our attention on a one-dimensional model originally introducedin [6], which is different from those in [1]-[5] in that it is aimed at capturing topologicalchanges in the evolving grain boundary network compared to the curvature-driven effects.This choice is motivated by the widely accepted fact that topological reconfigurations playa critical role during texture development in polycrystals.

2010 Mathematics Subject Classification. Primary: 82B41; Secondary: 82D35.Key words and phrases. CTRW, coarsening, polycrystals, self-similarity, master equation.Received 29/10/2016, accepted 01/11/2016.

DT was supported by the National Science Foundation Graduate Research Fellowship under Grant No.DGE-1356109. ME was partially supported by National Science Foundation grant DMS-1056821.

189

190 DIEGO TORREJON, MARIA EMELIANENKO AND DMITRY GOLOVATY

Scaling theories characterize scaling laws of self-similar behavior of statistical distributionsharvested from simulations or experiments. The procedure normally amounts to postulatingsuitable scaling hypothesis (ansatz) and deriving corresponding scaling exponents describingthe self-similar form for the quantities of interest. Our goal is to carry out this task inthe context of a one-dimensional model, which we do by means of interrogating the large-scale coarsening simulation for a variety of initial conditions and parameter choices. Havingevidence of self-similar behavior, we invoke the probabilistic description of the velocity jumpprocess first introduced in [6], which allows us to recover the scaling exponents. Whatemerges as a result of this work is a coherent theory in a form of a generalized masterequation capturing self-similar features of the system evolution.

An attempt to develop such theory in the context of materials applications dates back toMullins [7, 8] and has been a focus of many recent investigations [9]-[15]. Relatively speak-ing, starting from a discrete model, the continuous time random walk (CTRW) approachpresented herein attempts to avoid any physical assumptions and is grounded in purelystatistical and probabilistic analysis of numerical observations.

CTRW theory has found applications in many areas of science and technology. In partic-ular, it is widely used in financial applications such as in insurance risk theory, in Gibrat’smodel for growth and inequality, and in pricing financial markets [16]-[19]. In biology, itis applied to derive aggregation models [20, 21]. In physics, CTRWs are useful in model-ing transport in fusion plasmas [22], in reaction-diffusion models [23]-[25], and in processesinvolving anomalous diffusion [26]-[31].

In the standard CTRW formalism, a random walker inter-arrival times and jump sizesare drawn from a certain transition probability density [18, 20, 27]. In [23, 25], Angstmanet al. derive a generalized CTRW master equation on a lattice with non-stationary jumpsizes and space dependent inter-arrival times for a single particle and for an ensemble ofparticles undergoing reactions whilst being subject to an external force field. In [32], aCTRW master equation on a lattice is derived for the delayed forcing and instanteneousforcing under a biased nearest-neighbor jumps assumption. In [22], Milligen et al derive ageneralized CTRW master equation with space- and time dependence on the jump sizes andspace dependence on the inter-arrival times.

Our derivation of the generalized master equation happens to fall under the typical sce-nario in which the transition probability is separable [20, 23, 27, 33], but we are faced withthe challenge of having non-homogeneous inter-arrival times and time-dependent jump sizes.We use appropriate time-dependent scalings to deal with both of these complications. Whilethe exact form of these laws differs from one system to another, the overall framework re-mains the same as long as one can accurately compute the jump sizes and inter-arrival timesstatistics from a given simulation.

The paper is organized as follows. In Section 2 we generalize the standard CTRW theoryto the case of time-dependent jump sizes and derive the corresponding master equation.In Section 3 we show that our coarsening model falls precisely under the class of CTRWprocesses treated in Section 2. This allows us to derive the exact expressions for the self-similar solutions in Section 4. We implement the resulting steady state equation numericallyand show agreement with simulation.

1.1. Simulation description. In this section we define and analyze the properties of asimple coarsening network - a system of domains represented by intervals on the circle, as

ONE-DIMENSIONAL COARSENING MODEL 191

introduced in [6] and analyzed in a series of works [34, 35]. Following the same set-up, weassume that each domain is described by its length and a scalar “energy density” parameter.Only nearest neighbor interactions between the domains are considered, with the strengthof the interactions dependent on the energy densities of the neighboring domains.

To fix ideas, for a given L > 0 consider a circle of circumference L or, equivalently, aninterval [0, L] ⊂ R with periodic boundary conditions. Randomly select n points xini=1 ⊂[0, L] with xi ≤ xi+1 for i = 1, . . . , n − 1 and the point xn+1 identified with the point x1.Given the periodicity assumption, the interval [0, L] is thus subdivided into n sub-intervals[xi, xi+1], i = 1, . . . , n of lengths li = xi+1 − xi, i = 1, . . . , n − 1 and ln = L + x1 − xn,respectively. Note that the locations and the number of points will vary during evolution,however the total length L of all intervals remains fixed. Further, if a partition point leaves[0, L] through x = 0, it immediately re-enters [0, L] at x = L and vice versa.

xi−2 xi−1 xi xi+1 xi+2li−2 li−1 li li+1

φi−2 φi−1 φi φi+1

Figure 1. Set-up of the one dimensional coarsening model where xini=1,lini=1, φini=1 denote domain boundaries, domain lengths, and energydensities, respectively.

For each interval [xi, xi+1], i = 1, . . . , n, we select a random number φi ∈ R as seen inFigure 1. Here the interval [xi, xi+1] for i = 1, . . . , n can be thought of as a “domain” andthe points xi, xi+1 as the domain boundaries (DBs).

We define the total energy of the one-dimensional system by

(1.1) E(t) =

n∑i=1

φi [xi+1(t)− xi(t)]

and consider the gradient flow dynamics given by the system of ordinary differential equa-tions

(1.2) xi = φi − φi−1, i = 2, . . . , n, and x1 = φ1 − φn.

In what follows, we consider a non-negative energy density of the form

(1.3) φi = φ(αi) = |αi|γ , γ > 0.

Here the parameter αi, initially chosen for each domain according to the uniform randomdistribution in the interval (−π/4, π/4), does not change during the lifetime of the corre-sponding domain.

The rate of change of the domain length—referred to as a velocity of a domain i =1, . . . , n—can be computed from the relation

(1.4) vi = xi+1 − xi = φ(αi+1) + φ(αi−1)− 2φ(αi).

The set of domain velocities changes only at the times of collisions between adjacent DBseliminating the domains between these boundaries; we call each event of this type a disap-pearance event. The velocity corresponding to the collapsed domain is then removed from thelist of domain velocities while the velocities of its neighbors are adjusted appropriately. Note


that the lengths of the individual domains vary linearly in time between the disappearanceevents and depend entirely on the respective domain velocities.

An important feature of the thermodynamics of coarsening in materials is that it isenergy dissipative. The reduced gradient flow model (1.2) is specifically designed to enforcedissipation, as verified in [6].

1.2. Stabilization of statistics. In this section we describe the results of numerical sim-ulations of a system containing a large number of domains. The simulations reveal the setof coarse-grained characteristics of the system that develop over time which we will use insubsequent sections to establish an appropriate statistical model.

First, we find that there is a transient relaxation stage that occurs early in the simula-tion, when energetically unfavorable domains are quickly eliminated from the system. Afterthe relaxation stage, we observe stabilization of the respective statistics of relative domainlengths, DB velocities, domain velocities, and energy densities. The duration of the relax-ation stage varies depending on the value of γ. As γ decreases, as seen in Figure 2, weobserve that the steady-state distribution of the relative energy density is farther from theinitial distribution. Hence, it takes longer for the distribution to stabilize for lower valuesof γ. Note that the initial distribution of the energy density, in Figure 2, has a jump due tothe fact that α is chosen in the interval (−π/4, π/4).

Following the analysis done in [6], we reproduce the inter-arrival times and jump sizesstatistics, as seen in Figure 3. While the inter-arrival times plots are clearly of exponentialtype with growing means (cf. [6]), the distributions of jump sizes behave in a less intuitivemanner, prompting a further investigation.

The feature of primary interest in the present study is the development of the self-similarregime in this coarsening model, as alluded to in the introduction. Indeed, with a suitablechoice of spatiotemporal scaling laws resulting distributions of velocities and energies exhibitself-similarity. We give numerical evidence of this fact in Section 4, in Figures 9 and 10.Based on the stochastic properties of the appropriate random variables that we deducefrom numerical experiments, we develop a modification of the continuous time random walk(CTRW) theory. This theory leads to an integral-differential equation that accurately modelsthe evolution of the one-dimensional system of domains in the self-similar regime. Above all,it gives a formal characterization for the stochastic processes driving coarsening dynamics,as we are about to describe in the sections that follow.

2. Generalization of CTRW theory

2.1. Homogeneous master equation. The concept of CTRW theory became popularhalf a century ago as a rather general microscopic model for diffusion processes. Unlikediscrete time random walks, in the CTRW the number of jumps made by a walker during atime interval is a stochastic—often a homogeneous Poisson—process. The continuous timerandom walk was first introduced by Montroll and Weiss [36], Montroll and Scher [37], andlater on by Klafter and Silbey [38].

In the standard CTRW setting, both the inter-arrival times and the jump sizes are as-sumed to be independent and identically distributed. Another typical assumption is that thejump size are statistically independent of the inter-arrival times; the corresponding processis referred to as a decoupled CTRW.


Figure 2. Relative energy density distribution at various stages of evolu-tion as indicated by the percentage of domains eliminated from the system.Top left: γ = 0.5. Top right: γ = 1. Bottom left: γ = 1.5. Bottom right:γ = 2.

Figure 3. Inter-arrival times and jump sizes for γ = 2 at various stagesof evolution as indicated by the percentage of domains eliminated from thesystem. Left: Semi-log plot of inter-arrival times. Right: Evolution of jumpsizes.


We begin this section with a generalization of the CTRW framework obtained by droppingthe assumption of identically distributed jump sizes. The CTRW model is based on the ideathat the jump size and inter-arrival time are drawn from a joint p.d.f. θ which will be referredto as the transition probability density function. Hence the transition probability of taking ajump from velocity v0 at time τ0 to velocity v at time τ is represented by θ(v−v0, τ − τ0, τ).We assume that the transition p.d.f. is separable

θ(v − v0, τ − τ0, τ) = µ(v − v0, τ)w(τ − τ0) = µ(∆v, τ)w(s),(2.1)

with s = τ − τ0 and ∆v = v − v0. Here∫Rµ(∆v, τ) d(∆v) = 1 for all τ > 0 and

∫ ∞0

w(s) ds = 1.

We also introduce the function

ψ(τ) = 1−∫ τ

0

w(s) ds

that evaluates the probability that a walker remains at a given position at least for the timeperiod τ .

Let p(v, τ) be the probability density function such that p(v, τ) dv gives the probabilitythat the position of a walker lies inside the interval (v, v + dv) at time τ . We have thefollowing lemma.

Lemma 1. The probability distribution function p(·, τ) satisfies the master equation

p(v, τ) = ψ(τ)F (v) +

∫ τ

0

∫Rw(τ − τ ′)µ(∆v, τ ′)p(v −∆v, τ ′)d(∆v)dτ ′(2.2)

for all v ∈ R and τ > 0, where F (·) is the initial configuration of walkers.

Proof. We first state an evolution equation for the occupancy density function P (v, τ |0)defined so that P (v, τ |0)dv is the probability that the position of a walker (who was at theorigin at time τ = 0) lies in the interval (v, v + dv) at time τ . The occupancy densityP (v, τ |0) satisfies the following renewal equation

P (v, τ |0) = ψ(τ)δ(v) +

∫ τ

0

∫Rw(τ − τ ′)µ(∆v, τ ′)P (v −∆v, τ ′|0)d(∆v)dτ ′.(2.3)

Intuitively, the first term accounts for the walker who fails to move from the starting positionat v = 0 at least until time τ . The integral expresses the fact that a walker found at theposition v at time τ arrived to v after making his last jump from v −∆v where he was attime τ ′. If instead the walker starts at v0 6= 0, i.e., P (v, 0|v0) = δ(v−v0), then (2.3) changesto

P (v, τ |v0) = ψ(τ)δ(v − v0) +

∫ τ

0

∫Rw(τ − τ ′)µ(∆v, τ ′)P (v −∆v, τ ′|v0)d(∆v)dτ ′.(2.4)

If the initial state of the walker is given by an initial probability density distribution p(v, 0) =F (v), then

p(v, τ) =

∫RP (v, τ |v0)F (v0)dv0.


Hence p(v, τ) satisfies the master equation

p(v, τ) = ψ(τ)F (v) +

∫ τ

0

∫Rw(τ − τ ′)µ(∆v, τ ′)p(v −∆v, τ ′)d(∆v)dτ ′,

as follows from (2.4).

Lemma 2. Let the waiting times in Lemma 1 be exponentially distributed according to

w(τ) = λe−λτ . Then

∂

∂τp(v, τ) = λ

∫Rµ(∆v, τ)[p(v −∆v, τ)− p(v, τ)]d(∆v).(2.5)

Proof. It is convenient to work with equation (2.2) in Laplace space with respect to time.

We denote the Laplace variable as u and the Laplace transform of any function f(τ) by

f(u). Taking the Laplace transform of (2.2),

Lτ [p(v, τ)](u) = Lτ [ψ(τ)F (v)](u) + Lτ[∫

R

∫ τ

0

w(τ − τ ′)µ(∆v, τ ′)p(v −∆v, τ ′)dτ ′d(∆v)

](u)

= F (v)1− w(u)

u+ w(u)Lτ

[∫Rµ(∆v, τ)p(v −∆v, τ)d(∆v)

](u).(2.6)

After some algebra manipulation of (2.6),

Φ(u)(up(v, u)− F (v)) + p(v, u) = Lτ[∫

Rµ(∆v, τ)p(v −∆v, τ)d(∆v)

](u)(2.7)

where

Φ(u) =1− w(u)

uw(u).

Taking the inverse Laplace transform of (2.7),∫ ∞0

Φ(τ − τ ′) ∂∂τp(v, τ ′)dτ ′ + p(v, τ) =

∫Rµ(∆v, τ)p(v −∆v, τ)d(∆v).(2.8)

Φ(τ) denotes the memory function of the continuous time random walk [17]. Equation(2.8) reduces to a differential equation if the process is Markovian, i.e., the waiting time is

exponentially distributed with rate function λ. Then

w(τ) = λe−λτ ⇒ w(u) =λ

u+ λ⇒ Φ(u) =

1

λ⇒ Φ(τ) =

1

λδ(τ).

Under these assumptions, (2.8) derives

∂

∂τp(v, τ) = λ

∫Rµ(∆v, τ)[p(v −∆v, τ)− p(v, τ)]d(∆v).

Remark 1. Denote nt as the counting process of jumps by a walker. Assumming expo-nentially and identically distributed inter-arrival times implies that nt is a homogeneousPoisson process with the arrival rate λ.

Remark 2. Equation (2.5) can also be obtained by differentiating (2.2) and simplifying theresulting expression.


2.2. Nonhomogeneous master equation. The following theorem based on [39] relatesthe homogeneous Poisson process with the nonhomogeneous Poisson process,

Theorem 1. If nτ , τ ≥ 0 is a homogeneous Poisson process with rate λ = 1, then

nt = nm(t), t ≥ 0 with m(t) =∫ t0λ(s)ds is a nonhomogeneous Poisson process with rate

λ(t).

As a simple consequence of Theorem 1 and Lemma 2, we get the following analog of (2.5).

Corollary 1. The probability distribution function p(·, t) = p(·,m(t)) satisfies the masterequation

∂

∂tp(v, t) = λ(t)

∫R

[p(v −∆v, t)− p(v, t)]µ(∆v, t)d(∆v),(2.9)

for all v ∈ R and t > 0 with µ(∆v, t) = µ(∆v,m(t)), and λ(t) being the arrival rate of thenonhomogeneous Poisson process nt.

Proof. Denote by nτ the homogeneous Poisson process with arrival rate λ. Without loss ofgenerality assume λ = 1; otherwise include λ as a factor in λ(t). We apply Theorem 1 with

τ = m(t) where m(t) =∫ t0λ(s)ds. Then nt = nτ(t) is a nonhomogeneous Poisson process

with rate λ(t). We have that (2.5) holds with the rate λ = 1, that is

∂

∂τp(v, τ) =

∫R

[p(v −∆v, τ)− p(v, τ)]µ(∆v, τ)d(∆v).

In terms of the time variable t this equation takes the form (2.9).

3. Coarsening model as a CTRW

3.1. Ensemble statistics vs. individual statistics. In this section, we apply the theorydeveloped in Section 2 to our one-dimensional coarsening model. Considering each DBvelocity as a walker, we can denote by nt the stochastic process counting the number ofjumps experienced by this walker up to time t. The rate associated with this process will bedenoted by λ(t). LetMt be the stochastic process describing the evolution of velocity jumpsizes with probability density function µ(∆v, t) and Vt be the stochastic process describingthe evolution of velocities with probability density function p(v, t). Here v and ∆v denoteDB velocity and DB velocity jump size, respectively. In what follows, we demonstrate thestochastic processes Mt, Vt, and nt satisfy the assumptions that lead to (2.9). Hence, weexpect p, µ, and λ to be connected via (2.9).

Experimentally, we observe a collection of domains at any given time, so any statisticsthat we collect is the statistics for the entire collection. In order to recover the quantitiesfor a single walker, we therefore need to relate the statistical properties of a collection to thestatistical properties of an individual walker. Numerical experiments given in [6] indicatethat the coarsening model dynamics is ergodic and the DB velocities are i.i.d. at any giventime. We conclude that the DB velocity distribution in the system of many boundaries isdescribed by the same equation (2.9) as that for a single boundary.

In order to recover the arrival rate for a walker corresponding to velocity jumps of a singleDB, consider the stochastic process Nt counting the number of domains at time t. Then,


as it is shown in Theorem 3 in the Appendix, the arrival rate λN (t) is related to the arrivalrate of a single walker by

λN (t) = 2N(t)λ(t),

where N(t) is the number of domains at time t and the factor of 2 comes from the fact thateach disappearance event corresponds to two DB velocity jumps.

3.2. Separability. We validate the assumption that the joint distribution of non-homoge-neous inter-arrival times and velocity jump sizes is separable, i.e., θ(∆v, s, t) = w(s, t)µ(∆v, t)for any t > 0. We start by checking the correlation of jump sizes and the inter-arrival timesat different stages of the coarsening simulation. In Figure 4, we plot the p-values and thecorrelation coefficients corresponding to the test of no correlation between jump sizes andinterarrival times. Since the majority of p-values are greater than 0.05 and the correlationcoefficients are small, there is no evidence of correlation in the data.

Figure 4. Left: p-values for the null hypothesis of no correlation betweenjump sizes and inter-arrival times at different stages of the simulation forγ = 2. p-values below black line denote significant values. Right: Cor-relation coefficients between jump sizes and inter-arrival times at differentstages of the simulation.

In order to conclusively show independence between the jump sizes and the inter-arrivaltimes, in Figure 5, we provide a scatter plot for two-dimensional data at the different stagesof simulation. In addition, in Figure 6, we show that the joint distribution of jump sizesand inter-arrival times is essentially equal to the product of the corresponding marginals.Although not presented here, the separability hypothesis was also validated for other valuesof γ.

3.3. Nonhomogeneous Poisson process validation. First we show that the countingprocess Nt constitutes a nonhomogeneous Poisson process. According to Definition 2, Nt isa nonhomogeneous Poisson process if the following two conditions hold:

(a) given a countable, disjoint collection Ij of measurable subsets of R+, then NIjis a collection of independent random variables.

(b) if I ⊂ R+ is measurable, then NI has a Poisson distribution with arrival rate∫IλN (s)ds.


Figure 5. A scatter plot of the inter-arrival times against jump sizes at20%(top left), 40%(top right), 60%(bottom left), and 80%(bottom right) ofthe simulation for γ = 2.

Figure 6. Left: Joint distribution of the jump sizes and inter-arrival times.Middle: Product of the jump size marginal and the inter-arrival time mar-ginal. Right: Absolute error between the joint distribution and the productof marginals. All the statistics were computed at 40% of the simulation forγ = 2.

In order to test that Nt (or, equivalently, N0−Nt) is a nonhomogeneous Poisson process,we partition the time interval of the simulation into sub-intervals Ij. As evident fromFigure 7, the distribution of N0 −Nt on Ij resembles the Poisson distribution. The inde-pendence condition is validated in Figure 8. Although not presented here, both hypotheseswere also validated for other values of γ.

4. Self-similarity of solutions

4.1. Scaling laws. In this section, we show that following the relaxation stage, the processesMt and Vt become self-similar.


Figure 7. Comparing the distribution of N0 − NIj with the Poissondistribution for γ = 2. Left: At 30% of the simulation. Middle: At 50% ofthe simulation. Right: At 80% of the simulation.

Figure 8. Validating the independence condition for N0 −Nt for γ = 2.Left: Correlation values between N0 − NIi and N0 − NIj for i and j atdifferent stages of the simulation. The colorbar denotes the magnitude of thecorrelation. Right: p-values for the null hypothesis of no correlation betweenN0 −NIi and N0 −NIj for i and j at different stages of the simulation. Inthe gray area, there is no statistical evidence of correlation.

Definition 1. Let the process St be a stochastic process with probability density functions(x, t). Suppose there exists t∗ ≥ 0, representing the end of the relaxation stage, such that

∀t ≥ t∗ ∃b > 0 : Std= bSt∗ ⇔ ∀t ≥ t∗ ∃b > 0 : s(x, t) =

1

bs(xb, t∗).

Then we say that St is self-similar.

As seen in Figures 9 and 10, processes Vt and Mt indeed become self-similar, i.e.,

p(v, t) = b(t)p(b(t)v, t∗) = b(t)p0(b(t)v), ∀t ≥ t∗(4.1)

and

µ(∆v, t) = b(t)µ(b(t)∆v, t∗) = b(t)µ0(b(t)v), ∀t ≥ t∗(4.2)

after the relaxation stage corresponding to t < t∗. Here b : (t∗,∞)→ R+ is a time-dependentcoefficient of self-similarity the explicit form of which will be established next.


Figure 9. Velocity distribution at different stages of the simulation. Leftcolumn: Unscaled. Right column: Scaled to reveal self-similarity. The

scaling parameter is b(t) =(

E(Nt)E(Nt∗ )

)γ. Top row: γ = 2. Bottom row:

γ = 3.

4.2. Steady state equation. Now we are in a position to justify self-similar behavior ofthe one-dimensional coarsening model (1.2). As shown in Section 3, the dynamics of Nt isgoverned by equation (2.9). Further, we have the following theorem.

Theorem 2. Equation (2.9) admits a self-similar solution (p0, µ0) if the jump size and thevelocity processes Mt Vt, respectively, are self-similar. The scaling parameter b(t) satisfies

(4.3)

b′(t) = −βλ(t)b(t), t ≥ t∗b(t∗) = 1,

where β is a parameter independent of t but possibly dependent on γ. Moreover, the self-similar solution pair (p0, µ0) satistifes

(4.4) β[xp0(x)]′ = −∫ ∞−∞

µ0(y)[p0(x− y)− p0(x)]dy.

Proof. SinceMt and Vt are self-similar processes, then p(v, t) and µ(∆v, t) have self-similarforms as shown in (4.1) and (4.2) with scaling parameter b(t). Plugging (4.1) and (4.2) in(2.9),

(4.5) b(t)p′0(b(t)v)b′(t)v + b′(t)p0(b(t)v)

= λ(t)

∫ ∞−∞

b(t)µ0(b(t)∆v)b(t) [p0(b(t)(v −∆v))− p0(b(t)v)] d(∆v).


Figure 10. Velocity jump size distribution at different stages of the simula-tion. Left column: Unscaled. Right column: Scaled to reveal self-similarity.

The scaling parameter is b(t) =(

E(Nt)E(Nt∗)

)γ. Top row: γ = 2. Bottom row:

γ = 3.

Letting x = b(t)v and y = b(t)∆v and simplifying using (4.3),

−βxp′0(x)b(t)λ(t)− βb(t)λ(t)p0(x) = b(t)λ(t)

∫ ∞−∞

µ0(y)[p0(x− y)− p0(x)]dy

−βxp′0(x)− βp0(x) =

∫ ∞−∞

µ0(y)[p0(x− y)− p0(x)]dy

β[xp0(x)]′ = −∫ ∞−∞

µ0(y)[p0(x− y)− p0(x)]dy.

Remark 3. If p0 and µ0 satisfy the integro-differential equation (4.4), then (4.1) and (4.2)solve (2.9).

Corollary 2. We can represent b(t) as follows

b(t) =

(E(Nt)

E(Nt∗)

) β2

.(4.6)

Proof. According to (4.3),

b(t) = exp

(−β∫ t

0

λ(s)ds

).


Then

b(t) = exp

(−β∫ t

0

λ(s)ds

)= exp

(−β

2

∫ t

0

λN (s)

E(Ns)ds

)= exp

(−β

2

∫ t

0

− d

dsE(Ns)

1

E(Ns)ds

)= exp

(−β

2log

(E(Nt∗)

E(Nt)

))=

(E(Nt)

E(Nt∗)

) β2

.

4.3. Comparison with simulations. We find it convenient to write (4.4) in the form,

(4.7) βxp0(x) =

∫ x

0

p0(u)du−∫ ∞−∞

µ0(y)

(∫ x−y

0

p0(u)du

)dy

+

∫ ∞−∞

µ0(y)

(∫ −y0

p0(u)du

)dy.

We can show that the third integral on the right hand side of equation (4.7) vanishes becauseboth p0 and µ0 are even functions, as observed in Figures 9 and 10, i.e.,

(4.8)

∫ ∞−∞

µ0(y)

(∫ −y0

p0(u)du

)dy

=

∫ ∞0

µ0(y)

(∫ −|y|0

p0(u)du

)dy +

∫ 0

−∞µ0(y)

(∫ |y|0

p0(u)du

)dy

= −∫ ∞0

µ0(y)

(∫ 0

−|y|p0(u)du

)dy +

∫ 0

−∞µ0(y)

(∫ |y|0

p0(u)du

)dy = 0.

Hence, we can simplify (4.7),

βxp0(x) =

∫ x

0

p0(u)du−∫ ∞−∞

µ0(y)

(∫ x−y

0

p0(u)du

)dy.(4.9)

In Figure 11, we show that p0 and µ0 derived from our simulation satisfy (4.9) with the choiceof β = γ

2 for γ = 2 and γ = 3, where the integrals are computed using the trapezoidal rule.The two graphs in each figure correspond to the left hand side (LHS) and right hand side(RHS) of the equation (4.9), respectively. It must be noted that when testing other valuesof γ, we observe a very good agreement for γ > 2; this agreement progressively worsensas γ is decreased from 2 to 0. The agreement can be made almost exact by adjusting β;however, this will destroy self-similarity observed in Figures 9 and 10. The reasons behindthis discrepancy will be explored in a future work.


Figure 11. Comparing the right hand side and left hand side of (4.9).Left: γ = 2. Right: γ = 3.

4.4. General relationship between steady-state distributions. Taking the Fouriertransform of (4.4) with β = γ

2 , we have

−γ2kp′0(k) = −µ0(k)p0(k) + p0(k),(4.10)

where p0(k) = F(p0(x)) and µ0(k) = F(µ0(x)). Then (4.10) can be written as

1

2

p′0(k)

p0(k)=µ0(k)− 1

γk⇒ p0(k) = exp

(2

γ

∫ k

0

µ0(k)− 1

kdk

).(4.11)

Condition (4.11) indicates that the initial distribution of velocities p0 and the initial distri-bution of velocity jump sizes µ0 depend on each other. Hence it is sufficient to check (4.11)to know if (4.1) and (4.2) satisfy (2.9). Furthermore, we can use (4.11) to solve for thenon-trivial distribution µ0, i.e.,

µ0(k) =γkp′0(k)

2p0(k)+ 1.(4.12)

5. Summary

In this work we consider a simplified one-dimensional coarsening model inspired by earlierwork on modeling grain growth in polycrystals. We generalize the theory of continuous timerandom walks to the case of time-dependent jumps and derive the corresponding masterequation that is shown to admit self-similar solutions. Extensive numerical tests confirmthat this framework successfully describes the behavior of the velocity statistics harvestedfrom the one-dimensional simulation.

6. Acknowledgement

The authors express their deep gratitude to David Kinderlehrer for introducing them tothe mathematics of grain growth, his constant encouragement, numerous invaluable insights,and inspiration along the way.


7. Appendix. Superposition of nonhomogeneous Poisson processes

Here we develop a connection between the arrival rates for an ensemble of independentwalkers compared to that for a single walker.

Definition 2. [39] The counting process nt, t ≥ 0 is said to be a nonhomogeneous Poissonprocess with rate function λ(t), if

(1) n0 = 0.(2) nt, t ≥ 0 has independent increments.(3) P (nt+h − nt ≥ 2) = o(h).(4) P (nt+h − nt = 1) = λ(t)h+ o(h).

Theorem 3. Let ni(t)Ri=1 be independent nonhomogeneous Poisson processes with rate

λ(t). Define nt =∑Ri=1 ni(t). Then nt is a nonhomogeneous Poisson process with rate

λR(t) = Rλ(t).

Proof. We test conditions (1-4) in Definition 2. We start with condition (1),

n0 =

R∑i=1

ni(0) =

R∑i=1

0 = 0.

Condition (2) follows from the independence assumption of ni(t)Ri=1. Condition (4) followsfrom,

P [nt+h − nt = 1] =

R∑i=1

P [ni(t+ h)− ni(t) = 1]∏j 6=i

P [nj(t+ h)− nj(t) = 0]

=

R∑i=1

(λ(t)h+ o(h))∏j 6=i

(1− λ(t)h+ o(h))

=

R∑i=1

(λ(t)h+ o(h))(1− λ(t)h+ o(h))R−1

= R(λ(t)h+ o(h))(1− λ(t)h+ o(h))R−1

= Rλ(t)h+ o(h)

= λR(t)h+ o(h).

Lastly, condition (3) follows from an argument similar to (4). Hence nt is a nonhomogeneousPoisson process with rate λR(t) = Rλ(t).

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Department of Mathematical Sciences, George Mason University, Fairfax, VA 22030


Department of Mathematics, The University of Akron, Akron, OH 44325


JEPE Vol 2, 2016, p. 207-216

FINITE SPEED OF PROPAGATION AND WAITING TIME

FOR LOCAL SOLUTIONS OF DEGENERATE EQUATIONS IN

VISCOELASTIC MEDIA OR HEAT FLOWS WITH MEMORY

S.N. ANTONTSEV∗,1,2,3, J.I. DIAZ #,4

Abstract. The finite speed of propagation (FSP) was established for certain materialsin the 70’s by the American school (Gurtin, Dafermos, Nohel, etc.) for the special case of

the presence of memory effects. A different approach can be applied by the construction

of suitable super and sub-solutions (Crandall, Nohel, Dıaz and Gomez, etc.). In thispaper we present an alternative method to prove (FSP) which only uses some energy

estimates and without any information coming from the characteristics analysis. Thewaiting time property is proved for the first time in the literature for this class of non-

local equations.

Dedicated to Professor David Kinderlehrer on occasion of his 75th birthday.

1. Introduction

The main goal of this paper is to get some qualitative properties, such as finite speed ofpropagation and waiting time effect, for local in space solutions (i.e. independently of anypossible boundary conditions) of some nonlinear evolution equations involving a nonlocal intime term as in the following formulation:

(1.1)

∂u∂t = ∂(σ1(ux))

∂x + ∂∂x

(∫ t0γ(x, t, s)σ0(ux(x, s))ds

)+ f(x, t),

u(x, 0) = u0(x).

Formulations as (1.1) arise in many different contexts after making some easy transfor-mations. This, specially the case, when modelling different mechanical phenomena of vis-coelastic media. Indeed, if we introduce the displacement vector, as usual in ContinuumMechanics, by

u = x− ξ, x(0) = ξ,

2010 Mathematics Subject Classification. 35K92; 45K05.Key words and phrases. Nonlocal equation,non-linear viscoelastic equation,finite speed of propaga-

tion,waiting time property, heat flows with memory.Received 30/10/2016, accepted 07/12/2016.The research of SNA was partially supported by the Project UID/MAT/04561/2013 of the Portuguese

Foundation for Science and Techology (FCT), Portugal and by the Grant No.15-11-20019 of Russian Science

Foundation, Russia. The research of JID was partially supported by the project Ref.MTM2014-57113-Pof the DGISPI (Spain) and the Research Group MOMAT (Ref. 910480) supported by the UniversidadComplutense de Madrid.

207

208 S.N. ANTONTSEV∗,1,2,3, J.I. DIAZ #,4

the spatial velocity v is given by v =∂u

∂t(v(x,t) and x(t; ξ) are respectively the Euler and

the Lagrangian coordinates). Calculating the acceleration as the material derivative of thespeed

Dtv =∂v

∂t+ v · ∇v

and assuming that the second term on the right-hand side is sufficiently small (because vor |∇v| is small), we get

Dtv =∂2u

∂t2.

In the case of some one-dimensional motions with constant density (which can be re-scaledas to be the unit) we can assume that

u(x, t) = (u(x, t), 0, 0), x = (x, 0, 0), f = (f(x, t), 0, 0),

and that the components of the stress tensor S have the form

S11 = σ, Sij = 0 for i = 1, 2, 3, j = 2, 3.

With the above simplifications, the momentum balance law takes the form

(1.2)∂2u

∂t2=∂σ

∂x+ f(x, t).

Now we make the constitutive assumption typical of viscous-elastic media: the stress tensorS is a function of the deformation gradient of the displacement and the speed. In our case,this constitutive law can be written as

(1.3) σ = σ(ux, uxt).

Then, from (1.2) and (1.3) we get the equation

(1.4)∂2u

∂t2=∂σ(ux(x, t), uxt(x, t))

∂x+ f(x, t).

Obviously some initial conditions must be given:

(1.5) u(x, 0) = u0(x), ut(x, 0) = ϕ(x).

Equations of the type of (1.4) occur in various problems concerning the motions of viscous-elastic media and was intensively studied in the literature. For instance, in [1] it was provedexistence of solutions to the equation

utt = uxxt + σ(ux)x

and in [17] it was investigated the existence, uniqueness and stability of solutions of theequation

ρ0utt = λuxxt + σ′(ux)uxx.

The mixed initial-boundary value problem for the equations of nonlinear one-dimensionalviscoelasticity were considered in [10],[19].

Here, in this paper, we shall assume that the function σ(r, q) may include some (x, t)dependence but always under the following growth conditions:

FINITE SPEED OF PROPAGATION FOR A NONLOCAL VISCOELASTIC MEDIUM 209

(1.6)

σ ≡ σ(x, t, s, r, q) = γ(x, t, s)σ0(r) +

∂σ1(r)

∂rq, for any s ≤ t, r, q ∈ R,

C2|r|p ≤ σ1(r)r ≤ C1|r|p, 2 < p <∞, for any r ∈ R,

|σ0(r)| ≤ C3|r|p−1 for any r ∈ R-

The above given function γ(x, t, s) (which we assume to be bounded) may contain manydiverse information (as, for instance, some memory effects, etc.). Notice that under (1.6)equation (1.2) can be rewritten in the form

(1.7)∂2u

∂t2=∂2(σ1(ux))

∂x∂t+∂(γ(x, t, s)σ0(ux))

∂x+ f(x, t).

Hence, integrating in t we arrive to the formulation (1.1) with

(1.8) f(x, t) := ϕ(x)− ∂σ1(u0x(x))

∂x+

∫ t

0

f(x, s)ds.

Problem (1.1) also arises in the study of heat flows with memory. In that case a verygeneral starting point is the balance

(1.9)∂

∂t

(u(t, x) +

∫ t

0

b(t− s)u(s, x)x

)= d0σ(ux)x +

∫ t

0

a(t− s)Ψ(ux(s, x))xs

where the functions σ and Ψ are assumed to be increasing real-valued functions such thatσ(0) = Ψ(0) = 0. See, e.g., the expositions made in [29], [28] and [11] and their references.

We point out that in most of the papers in the previous literature it was assumed somesimilar conditions to (1.6) but for an uniformly elliptic diffusion term, p = 2. Our main goalin this paper is the consideration of the degenerate case p > 2. In the recent decades, theevolution equations with memory terms for p > 2 have been also considered. For instance,in [3], it was considered the Dirichlet problem for the evolution p-Laplace equation with anonlocal term

(1.10)

ut −∆pu =∫ t

0g(t− s)∆pu(x, s) ds+ Θ(x, t, u) + f(x, t) in Q = Ω× (0, T ),

u = 0 on ∂Ω× (0, T ),u(x, 0) = u0(x) in Ω,

where Ω ⊂ Rn is a bounded domain with Lipschitz-continuous boundary, g(s) is a givenmemory kernel, Θ is a given function and ∆p denotes the p-Laplace operator ∆pu :=

div(|∇u|p−2∇u

), 1 < p < ∞. Existence and uniqueness of solutions for this prob-

lem were proved. Also were established that the disturbances from the data propagate withfinite speed and the waiting time effect is possible. Once again, problem (1.10) appears inthe mathematical description of the heat propagation in materials with memory where theheat flux may depend on the past history of the process.

The question of the solvability, and the long time behavior of solutions of the abstractnonlinear Volterra equations of the type

ut(t)−Bu(t) +

∫ t

0

g(t− s)Au(s) ds = f(t),


and nonlocal equations of similar structure, were studied in many papers of the past lit-erature [5, 9, 13, 20, 23, 26, 31]. For instance, in [26, 27] the solution u(t) takes valuesin a reflexive Banach space W , ut is an element of the dual space W ′, and A and B arethe operators given by the subdifferentials of two convex, lower semicontinuous and properfunctions. An example of such an equation is furnished by the problem

ut(x, t)− uxx(x, t) =

∫ t

0

g(t− s)(σ(ux(x, s)))x ds+ f(x, t)

with a sufficiently smooth function σ - see [9]. For the semilinear equation (1.10) with p = 2and Θ 6= 0 the same questions were addressed by many authors, see, e.g., [6, 7, 8] andreferences therein. Nonexistence of global solutions (a finite time blow-up) for semilinearequations was studied in [22, 24, 25]. Similar results are also known for nonlocal parabolicequations and boundary conditions of other types, see, e.g., [14, 16, 21]. Doubly nonlinearnonlocal parabolic equations

∂tβ(u)− div σ(∇u) =

∫ t

0

g(t− s) div σ(∇u(s)) ds+ f(x, t, u)

were studied in [30] in an abstract setting. In [15] was investigated the existence of weak solu-tions of a class of quasilinear hyperbolic integro-differential equations describing viscoelasticmaterials.

In contrast to most of the previous studies on the derivation of the finite speed of pertur-bation for viscoelastic media we shall not use any characteristic argument but purely somesuitable energy arguments in the spirit of the monograph [4]. Our arguments are of a differ-ent nature to some other energy methods which need some information obtained trough thecharacteristics (see, e.g. [33] and [32]). As far as we know, the waiting time property wasnever before obtained in the literature for the class of nonlocal problems of the type (1.1)(see Remark 1 below).

2. Finite speed of propagation and the waiting time effect

Given, Ω = (−L,L), we consider (local in space) weak solutions to the equation

(2.1)∂u

∂t=∂(σ1(ux))

∂x+∂

∂x

(∫ t

0

γ(x, t, s)σ0(ux(x, s))ds

)+ f(x, t),

in the class of functions

u ∈W ≡ C([0, T ];L2

loc(Ω))∩ Lp

(0, T ;W 1,p

loc (Ω))

and satisfying the initial condition

(2.2) u(x, 0) = u0(x) x ∈ Ω.

Our main assumption on the initial condition u0 ∈ W 1,ploc (Ω) is that it represents a finite

propagation in the sense that

(2.3) u0(x) ≡ 0, when |x| ≤ ρ0 < L.

As mentioned before, we assume that ∀ r ∈ R

(2.4) C2|r|p ≤ σ1(r)r ≤ C1|r|p, 2 < p <∞,


(2.5) |σ0(r)| ≤ C3|r|p−1.

Thus, any local weak solution satisfies, for any ball Bρ ⊂ Ω, and for some M > 0,

(2.6) sup0≤t≤T

∫Bρ

u2dx+

∫ T

0

∫Bρ

|ux|pdxdτ ≤M.

Results on the existence of solutions in this class of functions (once we specify the bound-ary conditions) are well known in the literature (see for example [2, 4]). Our main result isthe following:

Theorem 1. Assume (2.4), (2.5) and also (2.3) and

f(x, t) ≡ 0, |x| ≤ ρ0 for a.e. t ∈ (0, T ).

Let u(x, t) be a local weak solution of problem (2.1), (2.2) satisfying (2.6). Then u(x, t)possesses the finite speed of propagation property (FSP) in the following sense:there exist t∗ ∈ (0, T ] and a function ρ(t), with 0 < ρ(t) < ρ0, ρ(0) = ρ0, such that

u(x, t) = 0 for |x| < ρ(t), 0 ≤ t ≤ t∗.The function ρ(t) satisfies

(2.7) ρ1+α(t) = ρ1+α0 − Ctκ

for some positive constants α, κ depending only on p, and C ≡ C(p, T,M). Moreover, if

(2.8)

∫Bρ

u20dx+

∫ T

0

∫Bρ

|f(x, τ)|2dxdτ ≤ C (ρ− ρ0)1/(1−ν)+

for any ρ ∈ (ρ0, L), with

ν = ν(p) =2p

3p− 2,

then u(x, t) possesses the waiting time property (WTP): there exists t∗ > 0 such that

u(x, t) = 0 for any |x| ≤ ρ0 and any t ∈ [0, t∗].

Remark 1. We point out that the growth estimate given by (2.7) is quite unusual in theliterature for this class of integro-differential equation. The main reason is that most ofthe authors assume p = 2 and then the application of the characteristics method leads toestimates on the interface involving expressions of the type σ′1(u0x(x±)), for the points x±defining the boundary of the support of u0. Notice that if we assume conditions (1.6) thenwe get σ′1(u0x(x±)) = 0 at least for initial data which are flat enough near x± (so thatu0x(x±) = 0) which explains (but it does not proves it!!) the possibility to get the waitingtime property.

Proof. We shall prove Theorem 1 by using an energy method similar to the ones presentedin the monographs [2, 4].

First of all we introduce the set Bρ and points Sρ by

Bρ = x, x0 ∈ Ω : |x− x0| < ρ ⊂ Ω, Sρ = ∂Bρ.

We define the energy functions


b(ρ, τ) =

∫Bρ

|u(·, τ)|2dx, b(ρ, t) = sup0≤τ≤tb(ρ, τ),

E(ρ, t) =

∫ t

0

∫Bρ

σ1(ux(x, τ))ux(x, τ)dxdτ,

E(ρ, t) = sup0≤τ≤tE(ρ, τ).

SinceC1|r|p ≤ σ1(r)r ≤ C2|r|p,

we get

C1

∫ t

0

∫Bρ

|ux|p ≤ E(ρ, t) ≤ C2

∫ t

0

∫Bρ

|ux|p,

and according to (1.6)

(2.9) b(ρ, τ) + E(ρ, t) ≤ CM.

Notice the following important properties on the energy functions:

sup0≤τ≤t∂E(ρ, τ)

∂ρ=∂E(ρ, t)

∂ρ=

=

∫ t

0

∫Sρ

σ1(ux(x, τ))ux(x, τ)dSdτ > 0,

Et =∂E(ρ, t)

∂t=

∫Bρ

σ1(ux(x, τ))ux(x, τ)dx > 0.

Since we are in the one dimensional case, we have

Eρ =∂E(ρ, t)

∂ρ=

=

∫ t

0

(σ1(ux)ux(−ρ, τ) + σ1(ux)ux(ρ, τ)) dτ > 0.

Multiplying equation (2.1) by u,integrating over the cylinder Bρ × (0, t) and applying theformula of integration by parts, we arrive to the energy relation

1

2

∫Bρ

u2(·, τ)dx

∣∣∣∣∣τ=t

τ=0

+

∫ t

0

∫Bρ

σ1(ux)uxdxdτ = I,

or, in the notation of the energy functions,

(2.10) b(ρ, τ)|τ=tτ=0 + E(ρ, t) =

4∑i=1

Ii ≡ I,

where

I1 =

∫ t

0

u(ξ, τ)σ1(ux)|ξ=ρξ=−ρ dτ,

I2 =

∫ t

0

u(ξ, τ)

∫ τ

0

γ(x, t, s)σ0(ux(ξ, s))ds|ξ=ρξ=−ρ dτ,


I3 = −∫ t

0

∫Bρ

ux(x, τ)

∫ τ

0

γ(x, t, s)σ0(ux(x, s))dsdxdτ,

I4 =

∫ t

0

∫Bρ

ufdxdτ.

First we shall prove the FSP property. Without lost of generality we can assume that f ≡ 0.In this case we use that b(ρ, 0) = 0 for ρ ≤ ρ0 since u0(x) = 0, if x ∈ [−ρ0, ρ0], and that

I4 =

∫ t

0

∫Bρ

ufdxdτ = 0.

The energy relation (2.10) takes now the form

(2.11) b(ρ, t) + E(ρ, t) =

3∑i=1

Ii ≡ I.

Next we use the multiplicative estimate (for any fixed t)

[u]p

:= (|u(−ρ, t) + u(ρ, t)|)p

≤ C(E

1p

t + ρ−δb12

)pθbp(1−θ)

2 ,

where δ and θ are some given positive parameters. We evaluate the terms Ii, i = 1, 2, 3 inthe following way:

|I1| ≤ C(∫ t

0

[u]p

) 1p(∫ t

0

(|ux(−ρ, ·)|p + ux(ρ, ·)|p)) p−1

p

≤ C

(∫ t

0

(E

1p

t + ρ−δb12

)pθbp(1−θ)

2

) 1p (Eρ) p−1

p

≤ Cb(1−θ)

2

(∫ t

0

(Et + ρ−δpb

p2

)θ) 1p (Eρ) p−1

p

≤ C max

(1, b

p−22

)b

(1−θ)2 t1−θρ−δθ

(E + b

) θp(Eρ) p−1

p

≤ Ct1−θρ−δθ(E + b

) θp+

(1−θ)2(Eρ) p−1

p ,

|I2| ≤ Ct1−θρ−δθ(E + b

) θp+

(1−θ)2(Eρ) p−1

p ,

|I3| ≤ CtκE(ρ, t).

Substituting last estimates to (2.11), taking the maximum with respect to t and applyingthe Young inequality, we arrive to the ordinary differential inequality, with respect to ρ,

E(ρ, t) ≤ b(ρ, t) + E(ρ, t) ≤ Cρ−αν tχν

(Eρ(ρ, t)

) 1ν ,

or equivalently

(2.12) Eν(ρ, t) ≤ Cρ−αtχEρ(ρ, t).


Here the time t is considered as a fixed parameter. Integrating the last inequality withrespect to ρ, over (ρ, ρ0), we obtain

E1−ν

(ρ, t) ≤ E1−ν(ρ0, t)− C

1− ν1 + α

(ρ1+α

0 − ρ1+α)t−χ.

Then defining ρ(t) by the formula

ρ1+α(t) = ρ1+α0 − E1−ν

(ρ0, t)1 + α

C(1− ν)tχ,

and assuming that

E1−ν

(ρ0, t) ≤ CM,

we arrive to the desired expression

ρ1+α(t) = ρ1+α0 − Ctκ,

and the first assertion of the theorem is proved.To prove the waiting time property we use the energy relation for ρ > ρ0 and evaluate

the additional terms in the following way:

I4 =

∫ t

0

∫Bρ

ufdxdτ

=

∫ t

0

∫Bρ

(ut(x, 0)− ∂σ1(u0x(x))

∂x+

∫ τ

0

f(x, s)ds

)u(x, τ)dxdτ,

and

I5 =

∫Bρ

u20dx,

which implies

|I4|+ |I5| ≤ δb(ρ, t) + C(δ)tη∫ t

0

∫Bρ

∣∣∣f ∣∣∣ dxdτ.Finally under conditions (1.6) we arrive to the ordinary non-homogeneous differential in-equality

(2.13) Eν(ρ, t) ≤ Ctχ

(ρ−αEρ(ρ, t) + (ρ− ρ0)

ν1−ν+

).

As in ([4]) we can prove that for a sufficiently small t∗ > 0 and 0 < t ≤ t∗ all solutions ofthe above inequality must satisfy

(2.14) E(ρ, t) ≤ C (ρ− ρ0)1

1−ν+ ,

and the result holds.

Remark 2. The localization properties can also be studied in a more general class of datain which the function σ = σ(r, q) is not subject to conditions (1.6). The study is performedin terms of the function

w(x, t) = ut(x, t), u(x, t) =

∫ t

0

w(x, τ)dτ + u0(x),

which satisfies the equation

∂w

∂t=

∂

∂x

(σ

(∫ t

0

wx(x, τ)dτ + u0x(x), wx(x, t)

))+ f(x, t).


The energy methods of [4] still apply and we can get similar properties to the (FSP) and(WTP) for the function w(x, t), but we shall not develop it here.

Remark 3. Some other qualitative properties, such as the finite extinction time, for manyother nonlocal problems can be obtained trough energy methods (see, e.g., [4] and [12] andits references).

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, 1M.A. Lavrentyev Institute of Hydrodynamics SB RAS, Novosibirsk, Russia, 2Novosibirskstate university, Novosibirsk, Russia, 3CMAF-CIO, University of Lisbon, Lisbon, Portugal, 4

Instituto de Matematica Interdisciplinar, Universidad Complutense de Madrid, Spain

E-mail address: ∗[email protected], #[email protected]

JEPE Vol 2, 2016, p. 217-233

A BOUNDARY VALUE PROBLEM FOR A NONLINEAR ELLIPTIC

SYSTEM RELEVANT IN GENERAL RELATIVITY

GIOVANNI CIMATTI

Abstract. A theorem of existence and non-existence of solutions for a boundary valueproblem for the equations of axially symmetric gravitational field in vacuum is given using

the inverse function theorem in Banach spaces and the method of functional solutions.

Conditions are given under which solutions exist or not exist.

1. Introduction

The Weyl’s metric

(1.1) ds2 = −e2ψdt2 + e2γ−2ψdρ2 + e2γ−2ψdz2 + ρ2e−2ψdϕ2

is one of the best studied in general relativity [9], [1]. The space-time is referred to the Weyl-Lewis-Papapetrou canonical coordinates [5], [6], [7] and [8] (t, ρ, z, ϕ), hereafter indexed as(0, 1, 2, 3). The potentials ψ and γ are assumed to depend only on ρ and z. Thus (1.1) iscompatible only with axially symmetric geometries. In this paper we study the so-calledelectrovac case. Therefore in the Einstein’s equations, corresponding to (1.1),

(1.2) Rik −1

2gikR = 8πTik, where

k

c4= 1 (k gravitational constant)

the stress-energy tensor Tik derives from the antisymmetric electro-magnetic tensor Fikwhich is taken of the special form

(1.3)

0 −φρ −φz 0φρ 0 0 0φz 0 0 00 0 0 0

,

where φ(ρ, z) is the electrostatic potential. In (1.2) Rik is the Ricci tensor, R its trace. Since[4]

(1.4) Tik =1

4π

(FikF

lk. −

1

2gikFmσF

mσ)

2010 Mathematics Subject Classification. 83C10, 83C05.Key words and phrases. Axially symmetric gravitational fields, boundary value problem, existence and

uniqueness of solutions.Received 04/11/2016, accepted 09/11/2016.

217

218 GIOVANNI CIMATTI

and F ik = e−2γFik, we have, recalling (1.3)

T00 =1

8πe2ψ−2γ

(φ2ρ + φ2

z), T01 = T02 = T03 = 0

T10 = 0, T11 =1

8πe−2ψ

(φ2ρ − φ2

z), T12 = − 1

4πe−2ψφρφz, T13 = 0

T20 = 0, T21 = − 1

4πe−2ψφρφz, T22 =

1

8πe−2ψ

(φ2ρ − φ2

z), T23 = 0

T30 = 0, T31 = 0, T32 = 0, T33 = 0.

Since T = gikTik the Einstein equation (1.2) reduces to

Rik = 8πTik.

For the components of the Ricci tensor we have, in the case of the Weyl’s metric,

R00 = A(ψ)e4ψ−2γ , R01 = 0, R02 = 0, R03 = 0

R10 = 0, R11 = A(ψ)− γρρ − γzz +1

ργρ − 2ψ2

ρ, R12 = −2ψρψz +1

ργz, R13 = 0

R20 = 0, R21 = −2ψρψz +1

ργz, R22 = A(ψ)− γρρ − γzz −

1

ργρ − 2ψ2

z , R23 = 0

R30 = 0, R31 = 0, R32 = 0, R33 = ρ2A(ψ) + e−2γ ,

where

A(ψ) = ψρρ +1

ρψρ + ψzz.

The operator A is singular for ρ = 0, i.e. on the z axis, and expressed in Cartesian coordi-nates reads

A(ψ) =1

x2 + y2

(x2ψxx + 2xyψxy + y2ψyy + xψx + yψy

)+ψzz.

A has the immediately seen, but important, property of being “a piece” of the Laplaceoperator. For, we have

(1.5) ∆ψ = A(ψ) +1

ρ2ψϕϕ.

From the Einstein’s equation R33 = 8πT33 or R00 = 8πT00 we obtain the equation

(1.6) A(ψ) = e−2ψ(φ2ρ + φ2

z).

and from R11 = 8πT11

(1.7) A(ψ)− γρρ − γzz +1

ργρ − 2ψ2

ρ = e−2ψ(φ2z − φ2

ρ

).

A BOUNDARY VALUE PROBLEM 219

By difference (1.6) and (1.7) give

γρ = ρ[ψ2ρ − ψ2

z − e−2ψ(φ2ρ − φ2

z

)].

Finally, from R12 = 8πT12 we have

γz = ρ(2ψρψz − 2e−2ψφρφz

).

All the others Einstein’s equations are automatically satisfied. The Maxwell’s equations, inthe present electrovac case, reduce to the vector equation

(1.8)(√−gF ik

),k

= 0,

where g = det(gik). By direct computation we find that the only non-vanishing component

in (1.8) is (ρe−2ψφρ

)ρ+(ρe−2ψφz

)z= 0

which, if ρ > 0, can be equivalently written as

A(φ) = 2(φρψρ + φzψz

).

In the end, we have for the determination of ψ, φ and γ the system of partial differentialequations

(1.9) A(ψ) = e−2ψ(φ2ρ + φ2

z)

(1.10) A(φ) = 2(φρψρ + φzψz

)(1.11) γρ = ρ

[ψ2ρ − ψ2

z − e−2ψ(φ2ρ − φ2

z

)](1.12) γz = ρ

(2ψρψz − 2e−2ψφρφz

).

If φ and ψ are obtained from the first two equations (1.9), (1.10) γ can be computed fromthe remaining equations (1.11), (1.12) by simple integration, since we have

(1.13)[ρ(ψ2ρ − ψ2

z − e−2ψφ2ρ + e−2ψφ2

z

)]z=[ρ(2ψρψz − 2e−2ψφρφz

)]ρ.

Thus γ is determined apart for an arbitrary constant (which can be determined if γ is knownin one point). In this paper we study the system

(1.14) A(ψ) = e−2ψ(φ2ρ + φ2

z)

(1.15) A(φ) = 2(φρψρ + φzψz

)in an axially symmetric domain Ω of R3 with suitable prescribed boundary conditions.Usually it is assumed that Ω is unbounded and that at infinity the metric of the flat space,


i.e ds2 = −dt2 + dρ2 + dz2 + ρ2dϕ2 holds. In the last Section we show that this formulationleads in certain cases to a not well-posed boundary value problem. Therefore, we prefer tostate the boundary value problem in a bounded subset of R3. In part of the boundary of Ωwe could assume the conditions which correspond to the flat space solution. In a differentpart of the boundary we prescribe the values of the potentials which are determined byexternal masses and electric charges.

In Section 2 we prove, using the implicit function theorem in Banach spaces, that forarbitrary sufficiently small axially symmetric boundary data the corresponding boundaryvalue problem has one and only one axially symmetric solution. Section 3 gives a theoremof existence and uniqueness for large, but special data. Section 4 deals with the class offunctional solutions.

2. Existence and uniqueness of ”small” solutions

Even if the basic equations (1.14) (1.15) have been derived on the assumption that thepotentials ψ and φ do not depend on the axial variable ϕ, the problem in itself is three-dimensional and therefore we state it in a bounded, open and axially symmetric subset Ω ofR3 not containing the z axis with a regular boundary Γ. We suppose Ω to be homeomorphicto the region G between two coaxial cylinders of radii R1 and R2 with R2 > R1 and of finitelength t 2L. The part of Γ corresponding to the bottom and upper part of the boundaryof G shall be denoted Γ4 and Γ3 respectively. Whereas the parts of Γ corresponding to theinternal and external parts of Γ are denoted Γ1 and Γ2.

2Γ

Γ3

Γ4

Γ1

Figure 1. The domain G

We assume that inside the cylinder of radius R1 a distribution of masses and electriccharges exists which determines the axially symmetric value of ψ and φ on Γ1. The spacebetween the two cylinders is free from masses and charges. Taking R2 much greater of R1


it is not unreasonable to assume on the lateral surface Γ2 of the external cylinder the valuesof ψ pertaining to the flat space solution or, more generally, to arbitrary axially symmetricvalue of ψ and φ determined by an external distributions of masses and charges. On bothbases Γ4 and Γ3 we assume the vanishing on the normal derivatives of ψ and φ in accordancewith the expected axial symmetry of the solutions. Therefore, we study the boundary valueproblem

(2.1) A(ψ) = e−2ψ(φ2ρ + φ2

z) in Ω

(2.2) ψ = ψ1 on Γ1, ψ = ψ2 on Γ2

(2.3) ψρν1 + ψzν2 on Γ3 ∪ Γ4

(2.4) A(φ) = 2(φρψρ + φzψz

)in Ω

(2.5) φ = φ1 on Γ1, ψ = φ2 on Γ2

(2.6) φρν1 + φzν2 on Γ3 ∪ Γ4,

where ν = (ν1, ν2, ν3) is the exterior pointing normal unit vector to Γ3 ∪ Γ4 and ψ1, ψ2, φ1

and φ2 are given axially symmetric C2,λ functions. We have

Lemma 1. If in the problem (2.1)-(2.6) we take φ1 = φ2 = 0 1 the solution is unique andit is given by (ψ, φ) = (ψ, 0), where ψ is the unique solution of the problem

(2.7) A(ψ) = 0 in Ω

(2.8) ψ = ψ1 on Γ1, ψ = ψ2 on Γ2

(2.9) ψρν1 + ψzν2 on Γ3 ∪ Γ4.

Proof. We prove, as a first step, that the solution of problem (2.7)-(2.9) exists and is unique.Let ψ′ and ψ′′ be two solutions and define w = ψ′ − ψ′′. We have

(2.10)1

ρ

(ρwρ

)ρ+wzz = 0 in Ω

(2.11) w = 0 on Γ1 ∪ Γ2

(2.12) wρν1 + wzν2 on Γ3 ∪ Γ4.

1We could with minor changes assume φ1 = φ2 = constant.


Multiplying (2.10) by ρw and integrating by parts over an arbitrary invariant cross-sectionD of Ω we obtain

(2.13)

∫D

(w2ρ + w2

z

)ρdρdz = 0.

Since D is arbitrary, we have wρ(ρ, z, ϕ) = 0, wz(ρ, z, ϕ) = 0 in Ω. Thus w may dependonly on ϕ, but this dependence is excluded in view of the axial symmetry of all the data.Moreover, by (2.11) we have w = 0 in Ω. To prove that (2.7)-(2.9) has a solution, weconsider the auxiliary problem involving the ”full” Laplace operator

(2.14) Uρρ +1

ρUρ + Uzz +

1

ρ2Uϕϕ = 0 in Ω

(2.15) U = ψ1 on Γ1, U = ψ2 on Γ2,∂U

∂ν= 0 on Γ3 ∪ Γ4.

By standard results on the theory of elliptic equation [2] the solution U exists and is unique.On the other hand, if we define U (k)(ρ, z, ϕ) = U(ρ, z, ϕ + k) in view of the uniqueness ofthe solution of problem (2.14), (2.15) and of the axial symmetry of all the data we haveU (k) = U (0) = U . Hence U does not depend on ϕ and solves (2.7)-(2.9).

To prove that problem (2.1)-(2.6) has one and only one solution if φ1 = φ2 = 0, we rewrite(2.4) in the equivalent form 2

(2.16)(ρe−2ψφρ

)ρ+(ρe−2ψφz

)z= 0.

We multiply (2.16) by ρ and integrate by parts over an invariant section D of Ω. Takinginto account the boundary conditions we have∫

D

ρe−2ψ(φ2ρ + φ2

z

)dρdz = 0.

Hence φρ(ρ, z, ϕ) = 0, φz(ρ, z, φ) = 0 in Ω. The possible dependence on φ is excluded sinceφ = 0 on Γ1 ∪ Γ2. Hence φ = 0 in Ω. Thus from (2.1) we have

A(ψ) = 0 in Ω

ψ = ψ1 on Γ1, ψ = ψ2 on Γ2

ψρν1 + ψzν2 on Γ3 ∪ Γ4.

Hence, by uniqueness, ψ = ψ.

In the remaining part of this Section we show that from every solution of the form (ψ, φ) =(ψ, 0) of problem (2.1)-(2.6) originates a branch of solutions of ”small” electric potential ofthe same problem if the boundary data for φ is sufficiently small and the boundary datafor ψ is sufficiently close to that of ψ. To this end we use the following form of the implicittheorem in Banach spaces.

2Note that (2.15) is fully equivalent to (2.4) only if ρ > 0.


Theorem 1. Let X and Y be Banach spaces, N a given subset of X , u∗ ∈ N and F : N → Ywith F ∈ C1. Assume the Frechet’s differential F ′(u∗) to be invertible. Then there exists aneighbourhood U of u∗ in X and a neighbourhood V of v∗ = F (u∗) in X such that F is adiffeomorphism from U to V.

In order to overcome the difficulties inherent in the singular character of the operator Awe shall consider, as suggested by (1.5), also the ”full” problem

(2.17) ∆ψ = e−2ψ|∇φ|2 in Ω, |∇φ|2 = φ2ρ + φ2

z +1

ρ2φ2ϕ

(2.18) ψ = ψ1 on Γ1, ψ = ψ2 on Γ2

(2.19)∂ψ

∂ν= 0 on Γ3 ∪ Γ4

(2.20) ∆φ = 2∇φ · ∇ψ in Ω

(2.21) φ = φ1 on Γ1, φ = φ2 on Γ2

(2.22)∂φ

∂ν= 0 on Γ3 ∪ Γ4.

We have the following

Lemma 2. If the problem (2.17)-(2.22) has a unique solution (ψ, φ) this solution is axiallysymmetric and it gives the unique axially symmetric solution of problem (2.1)-(2.6).

Proof. Let (ψ(ρ, z, ϕ), φ(ρ, z, ϕ)) be the unique solution of problem (2.17)-(2.22). All thegeometric data Ω, Γ1, Γ2, Γ3, Γ4 are axially symmetric and also the boundary data ψ1,ψ2, φ1, φ2 do not depend on ϕ by assumption. Define ψ(k)(ρ, z, ϕ) = ψ(ρ, z, ϕ + k) andφ(k)(ρ, z, ϕ) = φ(ρ, z, ϕ + k). In view of the axial symmetry of all the data (ψ(k), φ(k)) isalso a solution of problem (2.17)-(2.22) for every k ∈ R1. Therefore (ψ, φ) does not dependon ϕ and is also the unique solution of problem (2.1)-(2.6).

Theorem 2. Let ψ be the unique solution of the problem

A(ψ) = 0 in Ω

ψ = ψ1 on Γ1, ψ = ψ2 on Γ2

ψρν1 + ψzν2 on Γ3 ∪ Γ4.

There exists a constant δ > 0 such that if

‖ψ1 − ψ1‖C2,α(Γ1) ≤ δ, ‖ψ2 − ψ2‖C2,α(Γ2) ≤ δ


‖φ1‖C2,α(Γ1) ≤ δ, ‖φ2‖C2,α(Γ2) ≤ δthe problem

A(ψ) = e−2ψ(φ2ρ + φ2

z) in Ω

ψ = ψ1 on Γ1, ψ = ψ2 on Γ2

ψρν1 + ψzν2 on Γ3 ∪ Γ4

A(φ) = 2(φρψρ + φzψz

)in Ω

φ = φ1 on Γ1, ψ = φ2 on Γ2

φρν1 + φzν2 on Γ3 ∪ Γ4

has one and only one axially symmetric solution.

Proof. We start by studying the ”full” problem (2.17)-(2.22) which, by Lemma 2.1, hasonly the trivial solution (ψ, φ) = (ψ, 0) if φ1 = φ2 = 0. Referring to the notations of

Theorem 2.2 we define X = A × A where A =η ∈ C2,α(Ω), ∂η

∂ν = 0 on Γ3 ∪ Γ4

and

Y =(B × C1 × C2

)×(B × C1 × C2

)where B = C0,α(Ω), C1 = C2,α(Γ1), C2 = C2,α(Γ2) and

u∗ = (ψ, 0). Let F : X → Y be defined by

F ((ψ, φ)) =((

∆ψ − e−2ψ|∇φ|2, ψ|Γ1, ψ|Γ2

),(∆φ− 2∇φ · ∇ψ, φ|Γ1

, φ|Γ2

)).3

The differential of F in (ψ, 0) is easily computed and it is given by

F ′((ψ, 0))[Ψ,Φ] =((

∆Ψ, Ψ|Γ1, Ψ|Γ2

),(∆Φ− 2∇ψ · ∇Φ, Φ|Γ1

, Φ|Γ2

)).

To prove that F ′, a linear operator from X to Y, is invertible, as required by Theorem 2.2,we simply note that the linear elliptic boundary value problem

∆Ψ = a in Ω, Ψ = b on Γ1, Ψ = c on Γ2,∂Ψ

∂ν= 0 on Γ3 ∪ Γ4

∆Φ− 2∇ψ · ∇Φ = e in Ω, Φ = f on Γ1, Φ = g on Γ2,∂Φ

∂ν= 0 on Γ3 ∪ Γ4

has, by standard results [2], one and only one solution if ((a, b, c), (e, f, g)) ∈ Y. Thus the”full” problem has a unique small solution which, by uniqueness and by the axial symmetryof all the data, is axially symmetric. By Lemma 2.3 this solution is also the unique axiallysymmetric solution of the ”truncated” problem.

3ψΓ1denotes the restriction of ψ to Γ1


3. Existence and uniqueness of “large” axially symmetric solutions

The results of the previous Section are purely local in nature. Here we prove a theoremof existence, uniqueness and of non-existence of axially symmetric solutions, not necessarilysmall, but assuming constant boundary data. More precisely we consider the problem

(3.1) A(ψ) = e−2ψ(φ2ρ + φ2

z) in Ω

(3.2) ψ = ψ1 on Γ1, ψ = ψ2 on Γ2

(3.3) ψρν1 + ψzν2 on Γ3 ∪ Γ4

(3.4)(ρe−2ψφρ

)ρ+(ρe−2ψφz

)z= 0 in Ω

(3.5) φ = φ1 on Γ1, ψ = φ2 on Γ2

(3.6) φρν1 + φzν2 on Γ3 ∪ Γ4,

where φ1, φ2, ψ1 and ψ2 are given constants. Together with the ”truncated” problem (3.1)-(3.6) we consider the corresponding ”full” problem i.e

(3.7) ∇ ·(e−2ψ∇φ

)= 0 in Ω

(3.8) φ = φ1 on Γ1, φ = φ2 on Γ2

(3.9)∂φ

∂ν= 0 on Γ3 ∪ Γ4

(3.10) ∆ψ = e−2ψ|∇φ|2 in Ω 4

(3.11) ψ = ψ1 on Γ1, ψ = ψ2 on Γ2

(3.12)∂ψ

∂ν= 0 on Γ3 ∪ Γ4,

4It is interesting to note that the equations (3.7) and (3.10) are the Euler’s system of the functionalJ =

∫Ω

(e−2ψ |∇φ|2 + |∇ψ|2

)dV .


again assuming φ1, φ2, ψ1 and ψ2 as constants.5 Since the case φ1 = φ2 is analogous to thecase φ1 = φ2 = 0 which as been treated in Section 2, we assume

(3.13) ψ1 < ψ2, φ1 < φ2.

Two ”a priori” estimates for the solutions of problem (3.7)-(3.12) follow from the maximumprinciple which gives

(3.14) φ1 ≤ φ ≤ φ2 in Ω

(3.15) ψ ≤ ψ2 in Ω.

We want to show that the solutions of problem (3.7)-(3.12) can be expressed in terms of thesolutions of the problem

(3.16) ∆w = 0 in Ω, w = 0 on Γ1, w = w2 on Γ2,∂w

∂ν= 0 on Γ3 ∪ Γ4

where w2 is a constant suitably chosen. To see heuristically how this reduction of problem(3.7)-(3.12) to problem (3.16) is possible, let us define

(3.17) F (ψ) =1

2

(e2ψ − e2ψ1

)and

(3.18) θ =φ2

2− F (ψ).

If (ψ, φ) is a solution of (3.7)-(3.12), we have, from (3.18)

(3.19) e−2ψ∇θ = φe−2ψ∇φ−∇ψ.Taking into account (3.7) and (3.10) we obtain, from (3.19),

(3.20) ∇ ·(e−2ψ∇θ

)= 0 in Ω

and by (3.8), (3.9), (3.11), (3.12) and (3.18)

(3.21) θ =φ2

1

2on Γ1, θ =

φ22

2− F (ψ2) on Γ2,

∂θ

∂ν= 0 on Γ3 ∪ Γ4.

The equations (3.8) and (3.21) imply the existence, between θ and φ, of a functional relationof the form

5There is a curious similarity between problem (3.7)-(3.12) and a problem of electrical heating of con-

ductor. For, if we interpret ψ as the temperature inside the conductor Ω and σ(ψ) = e−2ψ as temperaturedependent electrical conductivity, the problem of finding the electric potential and the temperature inside the

conductor is modelled by the equations ∇ · (σ(ψ)∇φ) = 0, −∆ψ = σ(ψ)|∇φ|2. The crucial difference from

(3.7), (3.10) is in the sign in the second equation. Thus our original problem would be modeled assuming adensity flow of potential of the form q = e−2ψ∇φ.


(3.22) θ = k1φ+ k2

since θ and φ satisfy the same equation with different, but constant, boundary conditions.The constants k1 and k2 are easily computed from the conditions (3.8) and (3.21). We find

(3.23) θ =[φ1 + φ2

2− F (ψ2

φ1 − φ2

]φ− φ1φ2

2+F (ψ2)φ1

φ2 − φ1.

If we define

(3.24) H(φ) =φ2

2− 1

2

(φ1 + φ2

)φ− F (ψ2)

φ1 − φφ2 − φ1

+φ1φ2

2

we have from (3.23), (3.18) and (3.24)

(3.25) F (ψ) = H(φ).

If (3.25) can be solved with respect to ψ we can write

(3.26) ψ = F−1(H(φ)).

Hence the equation ∇ · (e−2ψ∇φ) = 0 becomes

∇ ·(e−2F−1(H(φ))∇φ

)= 0

to which we add the boundary conditions

φ = φ1 on Γ1, φ = φ2 on Γ2,∂φ

∂ν= 0 on Γ3 ∪ Γ4.

Thus we succeeded in reducing the original problem to a form to which the Kirchhoff’stransformation is applicable. For, if we define

(3.27) w = L(φ) =

∫ φ

φ1

e−2F−1(H(t))dt

we have by (3.27) and (3.26),

(3.28) ∇w = e−2F−1(H(φ))∇φ = e−2ψ∇φ

and, by (3.7)

(3.29) ∆w = 0.

To this equation we add the boundary conditions

(3.30) w = 0 on Γ1, w = L(φ2) on Γ2,∂w

∂ν= 0 on Γ3 ∪ Γ4.


Thus, if (3.26) and (3.27) are invertible, we obtain as solution of problem (3.7)-(3.12)

(3.31) (ψ(x), φ(x)) = (F−1(H(φ(x)), L−1(w(x)))).

To validate this procedure we must ascertain under which conditions the inverse functionsinvolved do really exist. This is done in the following

Theorem 3. Let ψ1, ψ2, φ1 and φ2 be given constants with ψ2 > ψ1 and φ2 > φ1. Define

(3.32) F (ψ) =1

2

(e2ψ − e2ψ1

).

If

(3.33) F (ψ2) ≥ 1

2

(φ2 − φ1

)2the problem (3.1)-(3.6) has one and only one solution. If

(3.34) F (ψ2) <1

2

(φ2 − φ1

)2the problem (3.1)-(3.6) has no solution.

Proof. We prove, as a first step, the result for the ”full” problem associated with (3.1)-(3.6),i.e.

(3.35) ∇ ·(e−2ψ∇φ

)= 0 in Ω

(3.36) φ = φ1 on Γ1, φ = φ2 on Γ2

(3.37)∂φ

∂ν= 0 on Γ3 ∪ Γ4

(3.38) ∆ψ = e−2ψ|∇φ|2 in Ω

(3.39) ψ = ψ1 on Γ1, ψ = ψ2 on Γ2

(3.40)∂φ

∂ν= 0 on Γ3 ∪ Γ4.

Define

(3.41) θ =φ2

2− F (ψ)

and the parabola

(3.42) H(φ) =1

2φ2 − 1

2

(φ1 + φ2

)φ− F (ψ2)

φ1 − φφ2 − φ1

+1

2φ1φ2.


We have H(φ1) = 0, H(φ2) = F (ψ2). Since dHdφ (φ1) = 0 if F (ψ2) = 1

2

(φ2−φ1

)2, we conclude

that, if

(3.43) F (ψ2) ≥ 1

2

(φ2 − φ1

)2,

we have H(φ) > 0 in(φ1, φ2] and 0 ≤ H(φ) ≤ F (ψ2) in [φ1, φ2]. Therefore, the inverse

function ψ = F−1(H(φ)), when φ ∈ [φ1, φ2], is well-defined and we have the functionalrelation

(3.44) F (ψ) = H(φ).

Let us consider the mixed problem for the laplacian

(3.45) ∆w = 0, w = 0 on Γ1, w = L(φ2) on Γ2,∂w

∂ν= 0 on Γ3 ∪ Γ4,

where

(3.46) L(φ) =

∫ φ

φ1

e−2F−1(H(t))dt.

By the maximum principle applied to (3.45) we have

(3.47) 0 ≤ w(x) ≤ L(φ2) in Ω.

Therefore, w = L(φ) defines a one-to-one mapping from [φ1, φ2] onto [0, L(φ2]. By (3.47)the functions

(3.48) φ(x) = L−1(w(x)), ψ(x) = F−1(H(φ(x)))

are well-defined. We prove that they give a solution to problem (3.35)-(3.40). Since ∇w =e−2ψ∇φ, by (3.45) we have

(3.49) ∇ ·(e−2ψ∇φ

)= 0.

Moreover, φ(x) satisfies the required boundary conditions. For,

φ = L−1(0) = φ1 on Γ1, φ = L−1(L(φ2) = φ2 on Γ2,∂φ

∂ν= 0 on Γ3 ∪ Γ4.

By (3.44) and (3.42), recalling (3.41), we obtain the following functional relation between φand θ

θ =[φ1 + φ2

2− F (ψ2

φ2 − φ1

]φ+

F (ψ2)φ1

φ2 − φ1− φ1φ2

2.

Hence

(3.50) ∇θ =[φ1 + φ2

2− F (ψ2

φ2 − φ1

]∇φ.


Multiplying (3.50) by e−2ψ and recalling (3.49) we obtain

(3.51) ∇ ·(e−2ψ∇θ

)= 0.

On the other hand, from (3.41) we have, after multiplication by e−2ψ

(3.52) e−2ψ∇θ = e−2ψφ∇φ−∇ψand, by (3.51) and (3.49), finally we get

(3.53) ∆ψ = e−2ψ|∇φ|2 in Ω.

Moreover, ψ satisfies the correct boundary conditions, in fact

ψ = F−1(H(φ1)) = F−1(0) = ψ1 on Γ1

ψ = F−1(H(φ2)) = F−1(F (ψ2)) = ψ2 on Γ2

∂ψ

∂ν= 0 on Γ3 ∪ Γ4.

We conclude that (3.48) gives a solution to the ”full” problem. We claim that this solutionis in fact the only solution. By contradiction, let (ψ′, φ′) be a second solution. Proceedingexactly as before, we find between ψ′ and φ′ the functional relation

F (ψ′) = H(φ′).

Thus, if w(x) is the (unique) solution of problem (3.45) we have

φ′(x) = L−1(w(x)) = φ(x)

and

ψ′(x) = F−1(H(φ′(x)) = F−1(H(L−1(w(x)))) = ψ(x).

Thus the solution of the ”full” problem exists and is unique and in view of the axial symmetryof all data it is axially symmetric. Hence by Lemma 2.3 it is also the unique axially symmetricsolution of the truncated problem.

We prove now that if (3.34) holds the ”full” problem has no solution. Suppose the contraryand let (ψ, φ) be a solution. We have again the functional relation

F (ψ) = H(φ),

but this time the point of minimum φ of the parabola H(φ) belongs to the interval (φ1, φ2)

and H(φ) < 0. Let x1 and x2 be arbitrary points of Γ1 and Γ2 respectively. If x = x(t) isa curve connecting x1 and x2, the function φ(x(t)) takes all the value between φ1 and φ2.

Thus there exists t∗ such that φ(x(t∗)) = φ and

0 ≤ F (ψ(x(t∗))) = H(φ(x(t∗))) = H(φ) < 0.

A contradiction. Therefore the ”full” problem and, by the usual argument, also the truncatedproblem have no solution if (3.34) holds.


Remark 1. Theorem 3.1 gives also a further estimate on ψ(x) in addition to (3.15) i.e.

(3.54) ψ ≥ ψ1 in Ω,

which does not follow from the maximum principle.

4. the ordinary differential equations of the functional solutions

The functional relation (3.25) can also be seen as the first integral of two ordinary dif-ferential equations which form the object of the present Section, where the point of view ofthe functional solutions is adopted in the sense of the following definition.

Definition 1. We say that (ψ(ρ, z, ϕ), φ(ρ, z, ϕ)) is a functional solution of the system ofpartial equations

(4.1) ψρρ +1

ρψρ + ψzz = e−2ψ

(φ2ρ + φ2

z

)(4.2) φρρ +

1

ρφρ + φzz = 2

(φrψρ + φzψz

)if a regular function ψ = Ψ(φ) exists such that

(4.3) ψ(ρ, z, φ) = Ψ(φ(ρ, z, ϕ))

or, as an alternative, if there is a function φ = Φ(ψ) such that

(4.4) φ(ρ, z, φ) = Φ(ψ(ρ, z, ϕ)).

Since the (4.3) or (4.4) hold up to the boundary, functional solutions are useful to studythe system (4.1), (4.2) only with special boundary conditions. This is what has been donein the previous Section. On the positive side, we have the fact that the functions Φ(ψ) orΨ(φ) entering in the definition above can be explicitly computed as solutions of two ordinarydifferential equations. In fact, we have

Lemma 3. If (ψ, φ) is a functional solution of (4.1), (4.2) such that

(4.5) φ2ρ + φ2

z 6= 0

the function Ψ(φ) of Definition 4.1 is a solution of the autonomous differential equation

(4.6)d2Ψ

dφ2+ 2(dΨ

dφ

)2= e−2Ψ.

If ψ2ρ + ψ2

z 6= 0 the function Φ(ψ) is a solution of the Riccati equation

(4.7)d2Φ

dψ2+ e−2ψ

(dΦ

dψ

)3

−2dΦ

dψ= 0.


Proof. We consider the first case. We have ψρ = Ψ′(φ)φρ, ψz = Ψ′(φ)φz, ψρρ = Ψ′′(φ)φ2ρ +

Ψ′(φ)φρρ, ψzz = Ψ′′(φ)φ2z + Ψ′(φ)φzz. Substituting in (4.1) we have

Ψ′′(φ2ρ + φ2

z

)+Ψ′

(φρρ +

1

ρφρ + φzz

)= e−2ψ

(φ2ρ + φ2

z

).

Using (4.2) and recalling (4.5) we obtain (4.6). The Riccati equation for the determinationof Φ(ψ) is obtained in a similar manner.

The equations (4.6) and (4.7) have the same first integral, see [3]

(4.8) e2ψ = φ2 − 2Cφ+B.

If we solve (4.8) with respect to ψ we find the solutions of (4.6) and if we solve (4.8) withrespect to φ we find the solutions of (4.7).

5. Remark on the Exterior Boundary Value Problem

At first sight it may appear more natural to state the problem (2.1)-(2.6) not in a boundeddomain, but as an exterior Dirichlet’s problem, prescribing at infinity the condition on ψpertaining to the flat space solution which corresponds to the metric of flat space.This wouldimply the boundary condition

limρ→∞

ψ = 0 uniformly with respect to z.

Similarly one would like to assume on the electric potential

limρ→∞

φ = 0 uniformly with respect to z.

The corresponding boundary value problem, however, in general has no solutions, at leastin the class of functional solutions as shown in the following example. Let us take

Ω = (ρ, z, ϕ); ρ > 1, |z| <∞, 0 < ϕ ≤ 2πand

Γ1 = (ρ, z, ϕ); ρ = 1, |z| <∞, 0 < ϕ ≤ 2π.Given the special geometry we search for solutions depending only on ρ on which we prescribethe boundary conditions

ψ(1) = ψ1, φ(1) = φ1

with φ1 and ψ1 given constants, φ1 6= ψ1 and

(5.1) limρ→∞

ψ(ρ) = 0, limρ→∞

φ(ρ) = 0.

If we search functional solutions (4.8) becomes, by (5.1),

(5.2) e2ψ = φ2 − 2Cφ+ 1.

Moreover, if we assume in (5.2) ψ as a function of φ i.e. ψ = Ψ(φ), we have

(5.3)dΨ

dφ=

φ− Cφ2 − 2Cφ+ 1


and from (2.4), since ψρ = Ψ′(φ)φρ, ψz = Ψ′(φ)φz, we have

ρd2φ

dρ2+dφ

dρ= 2Ψ′(φ)

(dφdρ

)2.

Hence, from (5.3) we arrive to the equation

(5.4) ρd2φ

dρ2+dφ

dρ=

2(φ− C)(dφdρ

)2φ2 − 2Cφ+ 1

.

The solution of (5.4) is given by

(5.5) φ(ρ;C,C1, C2) = C −√C2 − 1 tanh

(C1

√C2 − 1 ln ρ+ C2

√C2 − 1

).

Now, whatever the choice of the constants C, C1 and C2, we can never satisfy the secondcondition in (5.1). For, if C2 − 1 = 0 we have φ(ρ) = C which is only compatible with thetrivial solution φ(ρ) = 0. If C2 − 1 6= 0 and C1 > 0 we have

limρ→∞

φ(ρ) = C −√C2 − 1.

However, the equation C −√C2 − 1 = 0 has no real solutions. If C2− 1 6= 0 and C1 < 0 we

have

limρ→∞

φ(ρ) = C +√C2 − 1

and again C +√C2 − 1 = 0 has no real solutions.

References

[1] J. Bicak, Selected solutions of Einstein’s field equations: their role in general relativity and astrophysics,

in “Einstein’s field equations and their physical implications”, Bern G. Schmidt (Ed.), Lecture Notes in

Physics, Springer-Verlag, Berlin, 2000.[2] D. Gilbarg and N.S. Trudinger, Elliptic Partial Differential Equations of the second Order, Springer-

Verlag, New York, 1983.

[3] G.A. Gonzalez, A.C. Gutierrez-Pineros and P.A. Ospina, Finite axisymmetric charged dust disks inconformastatic spacetimes, arXiv:0806.4285v1.

[4] L. Landau and E. Lifchitz, Thorie des Champs, Editions MIR Moscow, 1970.[5] T. Lewis, Solutions of the equations of axially symmetric gravitational fields, Proc. Roy. Soc. London A,

136, (1932), 176-192.[6] A. Papapetrou, Eine rotationssymmetrische Losung in der Allgemeinen Relativitats Theorie, Ann.

Physik, 12, (1953), 309-315.[7] A. Papapetrou, Champs gravitationnels stationnaires a symetrie axiale, Ann. Inst. Henry Poincar 4,

(1966), 83-105.[8] H. Weyl, Zur Gravitationstheorie, Ann. Phys., 54, (1917), 117-145.

[9] H. Stephani, D. Kramer, M. McCallun, C. Hoenselaers and E. Herlt, Exact Solutions to Einstein’s FieldEquations, University Press, Cambridge, 2003.

Department of Mathematics, University of Pisa, Largo Bruno Pontecorvo 5, 56127 Pisa ItalyE-mail address: [email protected]

JEPE Vol 2, 2016, p. 235-266

POSITIVE SOLUTIONS FOR A CLASS OF NONLOCAL PROBLEMS

INVOLVING LEBESGUE GENERALIZED SPACES: SCALAR AND

SYSTEM CASES

GELSON C. G. DOS SANTOS AND GIOVANY M. FIGUEIREDO

Abstract. In this work we prove the existence of positive solutions for a class of scalar

nonlocal problems and systems of such equations. We use sub-supersolution method

combined with fixed point arguments and apply the results to some concrete problems.

1. Introduction

In this work we employ the sub-supersolution method in order to prove the existence ofpositive solutions for a class of nonlocal problems. More precisely, we study the class ofnonlocal problems given by

(P )

−A(x, |u|Lr(x))∆u = f1(x, u)|u|α(x)

Lq(x)+ f2(x, u)|u|γ(x)

Ls(x)in Ω,

u = 0 on ∂Ω

and

(S)

−A(x, |v|Lr1(x))∆u = f1(x, u, v)|v|α1(x)

Lq1(x) + f2(x, u, v)|v|γ1(x)

Ls1(x) in Ω,

−A(x, |u|Lr2(x))∆v = g1(x, u, v)|u|α2(x)

Lq2(x) + g2(x, u, v)|u|γ2(x)

Ls2(x) in Ω,

u = v = 0 on ∂Ω,

where Ω is a bounded domain in RN , N ≥ 1, | · |Lm(x) is the Luxemburg norm of Lebesguegeneralized space Lm(x)(Ω), A : Ω×R→ R and r, ri, q, qi, s, si, α, αi, γ, γi : Ω→ R are contin-uous functions. In the scalar case, we consider f1, f2 : Ω×R→ R continuous functions and inthe system case f1, f2, g1, g2 : Ω×R×R→ R are also continuous functions. This class of prob-

lems is called nonlocal because of the presence of the terms A(., |u|Lr(x)), |u|αi(.)

Lqi(x), |u|γi(.)

Lsi(x),

which imply that equations in (P ) and in (S) are no longer pontwise equalities. It is im-portant to stress that this kind of problem comes from important applications of biology,physics and chemistry, as can be seen in [2], [3], [4], [5], [6], [7], [10], [11], [12], [13], [15], [16],[17], [22], [23], [25], [27], [28] and references therein.

2010 Mathematics Subject Classification. Primary: 35J60 ; Secondary: 3 35Q53.

Key words and phrases. nonlocal problems, sub-supersolution, fixed point arguments.Received 09/11/2016, accepted 08/12/2016.Supported by CNPq/PQ, Capes and Fapesp.

235

236 GELSON C. G. DOS SANTOS AND GIOVANY M. FIGUEIREDO

In the last decades, many authors have studied such classes of problems. For example, in[18], the author studied the following problem −a(

∫Ω

|u|q)∆u = H(x)f(u) in Ω,

u = 0 on ∂Ω

using the Krasnoselskii and Schaefer Fixed Point Theorem in order to prove the existenceof positive solutions. In [21], using the Galerkin method, the authors proved the existenceof positive solutions to the problem

−∆u = a(x, u)|u|pLq in Ω,u = 0 on ∂Ω.

The problem −∆pu = |u|α(x)

Lq(x)in Ω,

u = 0 on ∂Ω

was studied in [19]. The authors used the sub-supersolution method with monotonic iterationand also proved the existence of a positive solution.

In [1], using the sub-supersolution method and a version of Minty-Browder’s Theorem forpseudomonotonic operators, the authors proved the existence of a positive solutions of theproblem

−a(∫

Ω|u|q)∆u = h1(x, u)f(

∫Ω|u|p) + h2(x, u)g(

∫Ω|u|r) in Ω,

u = 0 on ∂Ω,

where hi : Ω × R+ → R are continuous functions, q, p, r ∈ [1,∞) and the functions a, f, g :[0,∞)→ R+ satisfy f, g ∈ L∞([0,∞)) and

a(t), f(t), g(t) ≥ a0 > 0, ∀ t ∈ [0,∞).

Another problem that we would like to comment on is

(P )λ

−A(x, u)∆u = λf(u) in Ω,

u = 0 on ∂Ω,

that was studied in [9] with f ∈ C1([0, θ],R), f(0) = 0 = f(θ), f ′(0) > 0, f(t) > 0 in (0, θ).The function A : Ω × Lp(Ω) → R is such that x 7→ A(x, u) is measurable, u 7→ A(x, u) iscontinuous and there are constants a0, a∞ > 0, such that

a0 ≤ A(x, u) ≤ a∞ a.e in Ω, ∀ u ∈ Lp(Ω).

In that work, the authors used Schauder’s Fixed Point Theorem to prove the existence of apositive solution. The multiplicity of solutions to this problem was studied in [14], where theauthors proved that if f has n different roots, then (P )λ has n different positive solutions.They used Schauder’s Fixed Point Theorem and Variational Methods. Moreover, in bothworks, the authors have studied the asymptotic behavior of solutions when λ approaches to∞.

Motivated by the results found in [1], [9], [14] and [19], in this work we study problem(P ) and we prove the following result:

POSITIVE SOLUTIONS FOR... 237

Theorem 1. Suppose that r(x), q(x), s(x) ∈ C+(Ω), 0 ≤ α(x), γ(x) ∈ C0(Ω), (u, u) is a pairof sub-supersolution for (P ) with u > 0 a.e in Ω and f1(x, t), f2(x, t) ≥ 0 in Ω × [0, |u|L∞ ].Suppose also that A : Ω× (0,∞)→ R is a continuous function and

A(x, t) > 0 in Ω×[|u|Lr(x) , |u|Lr(x)

].

Then, problem (P ) has a weak positive solution u with

u ≤ u ≤ u.

In the next sections, we define sub-supersolutions and weak solutions for (P ) and thespace C+(Ω).

With respect to system (S), in [20] the authors studied the system−∆um = a|v|αLp in Ω,

−∆vn = b|u|βLq in Ω,

u = v = 0 on ∂Ω

and proved the existence of a solution using Rabinowitz’s result [26] concerning connectedcomponents of solutions. In [8], the authors studied

−∆u = f1(x, u)|v|p1Lα1 in Ω,−∆v = f2(x, v)|u|p2Lα2 in Ω,

u = v = 0 on ∂Ω

using the Galerkin method combined with the sub-supersolution method and monotoniciteration. The system

−∆p1u = |v|α1(x)

Lq1(x) in Ω,

−∆p2v = |u|α2(x)Lq2 (x) in Ω,

u = v = 0 on ∂Ω

was studied in [19], where the authors have also used Rabinowitz’s result [26] concerningconnected components of solutions. In this article, motivated by results that can be foundin [8], [19] and [20], we prove the following result:

Theorem 2. Suppose ri(x), qi(x), si(x) ∈ C+(Ω), 0 ≤ αi(x), γi(x) ∈ C0(Ω). Assume thatthe pairs (u, v), (u, v) are sub-supersolution to (S) with u, v > 0 a.e in Ω, fi(x, t, s), gi(x, t, s) ≥0 in Ω× [0, |u|L∞ ]× [0, |v|L∞ ]. Suppose also that A : Ω×(0,∞)→ R is a continuous functionand

A(x, t) > 0 in Ω×[σ, σ

],

where σ := min|w|Lr1(x) , |w|Lr2(x)

and σ := max

|w|Lr1(x) , |w|Lr2(x)

. Then, system (S)

has a weak positive solution (u, v) with

u ≤ u ≤ u e v ≤ v ≤ v.

In the next sections, we define subsolutions, supersolutions and weak solutions for (S).Now we make some comparisons concerning our results and some previously published re-sults. For example:i) We study a more general problem than the problems studied in [9], [14] and [19].ii) In contrast with [9], [14] and [19], we do not need that A be bounded in all its domain.


iii) In contrast with [1], we discarded the hypotheses f, g ∈ L∞([0,∞)) and f(t), g(t) ≥ a0 >0. In this paper we deal with the functions f(t) = tα(x) and g(t) = tβ(x) that, even in thecase where α(x) = α and β(x) = β, do not satisfy the hypotheses of the main theorem in[1]. Moreover, in [1] the authors consider A(x, t) = a(t) ≥ a0 > 0. Our result includes thiscase and other cases like A(x, 0) = 0 and lim

t→0A(x, t) = ±∞.

iv) We also provide three applications of Theorem 1. More precisely, we make applicationsof this theorem to a problem of sublinear type, to a problem of concave and convex type andto a problem that is a generalization of the classic logistic equation.v) We study a more general system than the systems studied in [8], [19] and [20].vi) We also provide three applications of Theorem 2. More precisely, we apply this theoremto a system of sublinear type, to a concave and a convex system and to a system that is ageneralization of the classic logistic system.

2. Basic results involving Generalized Lebesgue space

In this section we state some basic properties of the space Lp(x)(Ω). Firstly, we recallthat thoughout this work we are always considering Ω ⊂ IRN a bounded smooth domain.Let

C+(Ω) =h;h ∈ C(Ω), h(x) > 1 for all x ∈ Ω

and

h+ = maxΩ

h(x), h− = minΩh(x).

For p ∈ C+(Ω), we define

Lp(x)(Ω) =

u;u is a measurable real-valued function,

∫Ω

|u(x)|p(x) <∞.

We introduce the norm on Lp(x)(Ω) defined by

|u|p(x) = inf

λ > 0;

∫Ω

∣∣∣∣u(x)

λ

∣∣∣∣p(x)

≤ 1

and (Lp(x)(Ω), | · |p(x)) becomes a Banach space. We call it a generalized Lebesgue space.

Proposition 1. (i) The space (Lp(x)(Ω), | · |p(x)) is a separable, uniformly convex Banach

space, and its conjugate space is (Lq(x)(Ω), | · |q(x)), where q ∈ C+(Ω) and 1q(x) + 1

p(x) = 1.

For any u ∈ Lp(x)(Ω) and v ∈ Lq(x)(Ω), we have∣∣∣∣∫Ω

uv

∣∣∣∣ ≤ ( 1

p−+

1

q−

)|u|p(x)|v|q(x).

(ii) If p1, p2 ∈ C+(Ω), p1(x) ≤ p2(x), for any x ∈ Ω, then

Lp2(x)(Ω) → Lp1(x)(Ω)

and the embedding is continuous.

Proposition 2. For ρ(u) =∫

Ω|u|p(x) and for all u, un ∈ Lp(x)(Ω), we have

(i) If u 6= 0, the |u|Lp(x) = λ is equivalent to ρ(uλ ) = 1.(ii) If |u|Lp(x) < 1 (= 1; > 1), then ρ(u) < 1 (= 1; > 1).

(iii) If |u|Lp(x) > 1, then |u|p−

Lp(x)≤ ρ(u) ≤ |u|p

+

Lp(x).


(iv) If |u|Lp(x) < 1, then |u|p+

Lp(x)≤ ρ(u) ≤ |u|p

−

Lp(x).

(v) |un|Lp(x) → 0 is equivalent to ρ(un)→ 0.(vi) |un|Lp(x) →∞ is equivalent to ρ(un)→∞.

More information on these spaces may be found in Fan-Zhang [24] and in its references.

3. The scalar case

From now on we use in H10 (Ω) the usual norm ‖u‖2 =

∫Ω

|∇u|2. We start by defining a

weak solution and a pair of sub-supersolution to problem (P ).

Definition 1. The function u ∈ H10 (Ω)

⋂L∞(Ω) is a weak solution of problem (P ) if u > 0

a.e in Ω and∫Ω

∇u∇ϕ =

∫Ω

(f1(x, u)|u|α(x)

Lq(x)

A(x, |u|Lr(x))+f2(x, u)|u|γ(x)

Ls(x)

A(x, |u|Lr(x))

)ϕ, ∀ ϕ H1

0 (Ω).

Definition 2. Given z, w ∈ L∞(Ω), with z ≤ w a.e in Ω, we define

[z, w] :=u ∈ L∞(Ω) : z(x) ≤ u(x) ≤ w(x) a.e in Ω

.

Definition 3. The pair (u, u) is a sub-supersolution to problem (P ) if u ∈ H10 (Ω)

⋂L∞(Ω),

u ∈ H1(Ω)⋂L∞(Ω) with u ≤ u a.e in Ω, u = 0 ≤ u a.e in ∂Ω and for each ϕ ∈ H1

0 (Ω)with ϕ ≥ 0, we have

(3.1)

∫Ω

∇u∇ϕ ≤∫

Ω

(f1(x, u)|u|α(x)

Lq(x)

A(x, |w|Lr(x))+f2(x, u)|u|γ(x)

Ls(x)

A(x, |w|Lr(x))

)ϕ, ∀ w ∈ [u, u]

and

(3.2)

∫Ω

∇u∇ϕ ≥∫

Ω

(f1(x, u)|u|α(x)

Lq(x)

A(x, |w|Lr(x))+f2(x, u)|u|γ(x)

Ls(x)

A(x, |w|Lr(x))

)ϕ, ∀ w ∈ [u, u].


Proof. Consider the truncation operator associated with u and u, that is,

T : L2(Ω)→ L∞(Ω)

Tu(x) =

u(x) if u(x) ≤ u(x)u(x) if u(x) ≤ u(x) ≤ u(x)

u(x) if u(x) ≥ u(x).

Since u, u ∈ L∞(Ω) and u ≤ Tu ≤ u, T is well-defined. Now consider the operator

H : [u, u]→ L2(Ω)

given by

H(v)(x) =f1(x, v)|v|α(x)

Lq(x)

A(x, |v|Lr(x))+f2(x, v)|v|γ(x)

Ls(x)

A(x, |v|Lr(x)).

We are going to prove that H is well-defined because, since Tu ∈ [u, u] ⊂ L∞(Ω) and u > 0a.e in Ω, we have

|u|L∞ ≤ |Tu|L∞ ≤ |u|L∞ , ∀ u ∈ L2(Ω).


Moreover,

|u|Lm(x) ≤ |Tu|Lm(x) ≤ |u|Lm(x) , ∀ u ∈ L2(Ω), m(x) ∈ C+(Ω).

Therefore, since A(x, t) is positive and continuous in the compact set Ω×[|u|Lr(x) , |u|Lr(x)

],

there are constants k,K > 0 such that

0 < k ≤ A(x, |Tu|Lr(x)) ≤ K in Ω, ∀ u ∈ L2(Ω).

By the continuity of f1(x, t) and f2(x, t) in Ω ×[0, |u|L∞

], there are constants c1, c2 > 0

such that

|H(v)| ≤c1(|u|α−

Lq(x)+ |u|α+

Lq(x)) + c2(|u|γ

−

Ls(x)+ |u|γ

+

Ls(x))

kin Ω, ∀ v ∈ [u, u].

Then, H is well-defined. Now, we prove that u 7→ H(Tu) of L2(Ω) in L2(Ω) is continuous.Let (un) ⊂ L2(Ω) be a sequence such that un → u in L2(Ω), for some u ∈ L2(Ω). Then, upto subsequences,

un(x)→ u(x) a.e in Ω,

Tun(x)→ Tu(x) a.e in Ω,

|Tun(x)− Tu(x)|m(x) → 0 a.e in Ω

and

|Tun(x)− Tu(x)|m(x) ≤ 2m(x)|u|m(x)L∞ ≤ 2m+(|u|m

−

L∞ + |u|m+

L∞) a.e Ω,

for m(x) ∈ C+(Ω). From Lebesgue’s Dominated Convergence Theorem we obtain∫Ω

|Tun − Tu|m(x) → 0.

By Proposition 2, item v), we derive

Tun → Tu in Lm(x)(Ω), m(x) ∈ C+(Ω).

Using the continuity of f1(x, t), f2(x, t), A(x, t) and Lebesgue’s Dominated ConvergenceTheorem, we get

H(Tun)→ H(Tu) in L2(Ω).

Given v ∈ L2(Ω), let u = S(v) be the unique solution of the problem

(PL)

−∆u = H(Tv) in Ω,

u = 0 on ∂Ω,

where S is the solution operator. We are going to prove the existence of a fixed point of Susing Schaefer’s Fixed Point Theorem. Note that S is compact because, for (vn) ⊂ L2(Ω)bounded and un = S(vn), by the definition of S we get∫

Ω

∇un∇ϕ =

∫Ω

H(Tvn)ϕ, ∀ ϕ ∈ H10 (Ω).

Taking ϕ = un and using the definition of T and H, we have

‖un‖2 ≤ K0|un|L1 , ∀ n ∈ N, and for some K0 > 0

and, hence, (un) ⊂ H10 (Ω) is bounded. Now, up to a subsequence,

un u in H10 (Ω).


From Sobolev’s embedding, we derive

S(vn) = un → u in L2(Ω).

Now we prove that S is continuous. In fact, for vn → v in L2(Ω), un = S(vn) and u = S(v),by the definition of S we obtain∫

Ω

∇un∇ϕ =

∫Ω

H(Tvn)ϕ, ∀ ϕ ∈ H10 (Ω)

and ∫Ω

∇u∇ϕ =

∫Ω

H(Tv)ϕ, ∀ ϕ ∈ H10 (Ω).

Taking ϕ = un, by Holder’s inequality, we get∣∣∣ ∫Ω

∇un∇(un − u)∣∣∣ ≤ ∫

Ω

|H(Tvn)−H(Tv)||un|

≤ |H(Tvn)−H(Tv)|L2 |un|L2 .

Since, (un) is bounded in H10 (Ω) and by the continuity of u 7→ H(Tu), we conclude∫

Ω

∇un∇(un − u)→ 0,

which implies

‖un − u‖2 → 0.

Hence,

S(vn)→ S(v) in L2(Ω)

and S is continuous. Now, we are going to prove that there is R > 0 such that if u = θS(u)with θ ∈ [0, 1], then |u|L2 < R. Indeed, if θ = 0, we have u = 0. If θ 6= 0, we get

S(u) =u

θ.

From the definition of S, we obtain∫Ω

∇(u

θ

)∇ϕ =

∫Ω

H(Tv)ϕ, ∀ ϕ ∈ H10 (Ω).

Taking ϕ = u and using the definition of T and H once again, for some K0 > 0 we derive

‖u‖2 ≤ θK0|u|L1 ,

which implies

|u|L2 < R.

Hence, from Schaefer’s Fixed Point Theorem, there is u ∈ L2(Ω) with u = S(u). Then,∫Ω

∇u∇ϕ =

∫Ω

H(Tu)ϕ, ∀ ϕ ∈ H10 (Ω),

that is,

(3.3)

∫Ω

∇u∇ϕ =

∫Ω

(f1(x, Tu)|Tu|α(x)

Lq(x)

A(x, |Tu|Lr(x))+f2(x, Tu)|Tu|γ(x)

Ls(x)

A(x, |Tu|Lr(x))

)ϕ, ∀ ϕ ∈ H1

0 (Ω).

Now we are going to prove that

u ≤ u ≤ u a.e in Ω.


In fact, since Tu ∈ [u, u], using w = Tu in (3.1) and considering (3.3), we have, for eachϕ ∈ H1

0 (Ω) with ϕ ≥ 0 that

∫Ω

∇(u− u)∇ϕ ≤∫

Ω

(f1(x, u)|u|α(x)

Lq(x)− f1(x, Tu(x))|Tu|α(x)

Lq(x)

A(x, |Tu|Lr(x))

)ϕ

+

∫Ω

(f2(x, u)|u|γ(x)

Ls(x)− f2(x, Tu(x))|Tu|γ(x)

Ls(x)

A(x, |Tu|Lr(x))

)ϕ.

Taking ϕ = (u− u)+ = max(u− u), 0 and recalling that fi(x, t) ≥ 0 in [0, |u|L∞ ], Tu = uinx ∈ Ω : u(x) ≥ u(x)

and Tu ∈ [u, u], we get

∫Ω

∇(u− u)∇(u− u)+ ≤∫

Ω

(f1(x, u)|u|α(x)

Lq(x)− f1(x, Tu(x))|Tu|α(x)

Lq(x)

A(x, |Tu|Lr(x))

)(u− u)+

+

∫Ω

(f2(x, u)|u|γ(x)

Ls(x)− f2(x, Tu(x))|Tu|γ(x)

Ls(x)

A(x, |Tu|Lr(x))

)(u− u)+

=

∫x∈Ω : u(x)≥u(x)

f1(x, u)(|u|α(x)

Lq(x)− |Tu|α(x)

Lq(x))

A(x, |Tu|Lr(x))(u− u)

+

∫x∈Ω : u(x)≥u(x)

f2(x, u)(|u|γ(x)

Ls(x)− |Tu|γ(x)

Ls(x))


≤ 0,

which implies

‖(u− u)+‖2 ≤ 0,

and conclude,

u ≤ u, a.e in Ω.

Now, using w = Tu in (3.2) and considering (3.3), we have, for each ϕ ∈ H10 (Ω) with ϕ ≥ 0,

∫Ω

∇(u− u)∇ϕ ≤∫

Ω

(f1(x, Tu(x))|Tu|α(x)

Lq(x)− f1(x, u(x))|u|α(x)

Lq(x)

A(x, |Tu|Lr(x))

)ϕ

+

∫Ω

(f2(x, Tu(x))|Tu|γ(x)

Ls(x)− f2(x, u(x))|u|γ(x)

Ls(x)

A(x, |Tu|Lr(x))

)ϕ.


Taking ϕ = (u− u)+ = max(u− u), 0 and recalling that fi(x, t) ≥ 0 in [0, |u|L∞ ], Tu = uinx ∈ Ω : u(x) ≥ u(x)

and Tu ∈ [u, u], we have

‖(u− u)+‖2 ≤∫

Ω

(f1(x, Tu(x))|Tu|α(x)

Lq(x)− f1(x, u(x))|u|α(x)

Lq(x)

A(x, |Tu|Lr(x))

)(u− u)+

+

∫Ω

(f2(x, Tu(x))|Tu|γ(x)

Ls(x)− f2(x, u(x))|u|γ(x)

Ls(x)

A(x, |Tu|Lr(x))

)(u− u)+

=

∫x∈Ω : u(x)≥u(x)

f1(x, u(x))(|Tu|α(x)

Lq(x)− |u|α(x)

Lq(x))


+

∫x∈Ω : u(x)≥u(x)

f2(x, u(x))(|Tu|γ(x)

Ls(x)− |u|γ(x)

Ls(x))


≤ 0,

which implies

‖(u− u)+‖2 ≤ 0

and conclude

u ≤ u a.e in Ω.

Then, from definition of T, we derive Tu = u. Since u satisfies (3.3), we conclude the proofof Theorem 1.

Remark 1. The weak positive solution u ∈ H10 (Ω) ∩ L∞(Ω) of problem (P ) is a strong

solution. In fact, since H(Tu) ∈ L∞(Ω) and Tu = u, we have H(u) ∈ L∞(Ω) and u is aweak solution of −∆u = H(u) in Ω with u = 0 on ∂Ω. Then, by regularity results, we getu ∈ H1

0 (Ω) ∩W 2,p(Ω), ∀ p ≥ 1, such that−A(x, |u|Lr(x))∆u = f1(x, u)|u|α(x)

Lq(x)+ f2(x, u)|u|γ(x)

Ls(x)a.e in Ω,

u > 0 in Ω,

u = 0 on ∂Ω.

4. Applications of Theorem 1

In this section we make three applications of Theorem 1. From now on we denote bye ∈ H1

0 (Ω) ∩C2,τ (Ω), for some 0 < τ < 1, the unique positive solution of problem −∆e = 1in Ω, e = 0 on ∂Ω and by ϕ1 ∈ H1

0 (Ω)⋂C2,α(Ω) a positive eigenfunction associated with

the first eigenvalue λ1 of (−∆, H10 (Ω)).

4.1. Sublinear problem. First of all, we study the nonlocal sublinear problem given by

(Ps)

−A(x, |u|Lr(x))∆u = uβ(x)|u|α(x)

Lq(x)in Ω,

u = 0 on ∂Ω.

Problem (Ps) is related with the problem studied in [19]. Let us study the above problemconsidering two cases: the first case is A(x, t) ≥ a0 > 0 in Ω× [0,∞) and the second caseis A(x, 0) = 0.

The main result in this subsection is:


Theorem 3. Suppose that r(x), q(x) ∈ C+(Ω) and 0 ≤ α(x), β(x) ∈ C0(Ω) are such that

0 < α+ + β+ < 1.

Suppose also that A : Ω× [0,∞)→ R is continuous and satisfies one of these two conditions:(A1) There is a positive constant a0 > 0 such that

A(x, t) ≥ a0 > 0 in Ω× [0,∞).

(A2) There are positive constants a1, a∞ > 0 such that

A(x, 0) = 0 < A(x, t) ≤ a1 in Ω× (0,∞)

andlimt→∞

A(x, t) = a∞ uniformly in Ω.

Then, problem (Ps) has a weak positive solution.

Proof. We start with (A1). Let us construct u. Since α(x), β(x) ∈ C0(Ω), we have

|e|α(.)

Lq(x), |e|β(.)

L∞ Ω : → R+

x 7→ |e|α(x)

Lq(x), |e|β(x)

L∞ ,

are continuous functions. Then, there are positive constants C1, C2 > 0, such that

|e|α(x)

Lq(x)≤ C1 and |e|β(x)

L∞ ≤ C2, ∀ x ∈ Ω.

Since 0 < α+ + β+ < 1, we choose R > 0, such that

R ≥ max

(C1C2

a0

) 1

1−(α++β+)

, 1

.

For each w ∈ L∞(Ω) and setting u = Re, by some straight forward algebraic manipulationswe get

−∆u ≥ 1

A(x, |w|Lr(x))uβ(x)|u|α(x)

Lq(x)in Ω,

u > 0 in Ω,

u = 0 on ∂Ω.

Now let us set u. Considering K = maxA(x, t) : (x, t) ∈ Ω ×

[0, |u|Lr(x)

], given

w ∈[0, u], then |w|Lr(x) ≤ |u|Lr(x) , which implies

a0 ≤ A(x, |w|Lr(x)) ≤ K in Ω, ∀ w ∈ [0, u].

Choosing

0 < ε ≤ min

(|ϕ1|α

+

Lq(x)

λ1|ϕ1|1−β+

L∞ K

) 1

1−(α++β+)

, 1

,

u = εϕ1, for each w ∈ [0, u] and by some straight forward algebraic manipulations we have−∆u ≤ 1


Lq(x)in Ω,

u > 0 in Ω,

u = 0 on ∂Ω.


Now we are going to prove that u ≤ u. For ε > 0 sufficiently small we have

λ1ε|ϕ1|L∞ ≤ R.Then,

−∆(εϕ1) ≤ −∆(Re) in Ω,

and by the Comparison Principle we get

u := εϕ1 ≤ Re =: u.

Hence, (u, u) is a pair of sub-supersolution to problem (Ps). By Theorem 1, problem (Ps)has a weak positive solution u ∈ H1

0 (Ω)⋂L∞(Ω) with

u ≤ u ≤ u,which proves that the result holds with (A1).

Suppose now that (A2) holds. We are going to set u. We take ε > 0 such that

0 < ε ≤ min

(

|ϕ1|α+

Lq(x)

λ1|ϕ1|1−β+

L∞ a1

) 1

1−(α++β+)

, 1

.

Then, defining u = εϕ1 for ε > 0 small, we get−∆u ≤ 1


Lq(x)in Ω,

u > 0 in Ω,

u = 0 on ∂Ω.

Now we are going to find u. Since limt→∞

A(x, t) = a∞ uniformly in Ω, there exists M > 0

sufficiently large such that

A(x, t) ≥ a∞2

in Ω× [M,∞).

Letm = min

A(x, t) : (x, t) ∈ Ω×

[|u|Lr(x) ,M

]> 0.

Taking k = minm, a∞2

, we have

A(x, t) ≥ k > 0 in Ω×[|u|Lr(x) ,∞

).

Now consider u = Re, with R > 0 such that

R ≥ max

(C1C2

k

) 1

1−(α++β+)

, 1

,

where|e|α(x)

Lq(x)≤ C1 and |e|β(x)

L∞ ≤ C2, ∀ x ∈ Ω.

Then, for each w ∈ L∞(Ω); u ≤ w, we have−∆u ≥ 1


Lq(x)in Ω,

u > 0 in Ω,

u = 0 on ∂Ω.


We are going to prove that u ≤ u. Considering R > 0 such that

λ1ε|ϕ1|L∞ ≤ R,

we get

−∆(εϕ1) ≤ −∆(Re) in Ω.

By the Comparison Principle, we conclude that

u := εϕ1 ≤ Re =: u.

Then, (u, u) is a pair of sub-supersolution to problem (Ps). By Theorem 1, problem (Ps)has a weak positive solution u ∈ H1

0 (Ω)⋂L∞(Ω) with

u ≤ u ≤ u,

and the proof is finished.

4.2. Concave and convex problem. In this subsection we study the concave and convexproblem

(P )λ,µ

−A(x, |u|Lr(x))∆u = λ|u|β(x)−1u|u|α(x)

Lq(x)+ µ|u|η(x)−1u|u|γ(x)

Ls(x)in Ω,

u = 0 on ∂Ω.

Let us study problem (P )λ,µ for two classes of functions A.

Theorem 4. Suppose that r(x), q(x), s(x) ∈ C+(Ω) and 0 ≤ α(x), γ(x), β(x), η(x) ∈C0(Ω) such that

0 < α− + β− ≤ α+ + β+ < 1.

Suppose also that A : Ω × [0,∞) → R is a continuous function satisfying the followinghypotheses:(A1) Suppose that 1 < η− + γ− and there are constants a0, b0 > 0 such that

A(x, t) ≥ a0 > 0 in Ω× [0, b0].

Then, given µ > 0, there is λ0 > 0 such that for each λ ∈ (0, λ0), problem (P )λ,µ has a weakpositive solution uλ,µ.(A2) Suppose that 1 < η+ + γ+ and that there are constants a1, a∞ > 0, such that

A(x, 0) = 0 < A(x, t) ≤ a1 in Ω× (0,∞)

and

limt→∞


Then, given λ > 0, there is µ0 > 0 such that, for each µ ∈ (0, µ0), problem (P )λ,µ has aweak positive solution uλ,µ.

Proof. First of all, let us consider condition (A1). We begin by constructing u. Note that,for each M > 0, we get

−∆(Me) = M in Ω.

Then, we want to get M > 0 such that

(4.1) M ≥ 1

a0

(λ(Me)β(x)|Me|α(x)

Lq(x)+ µ(Me)η(x)|Me|γ(x)

Ls(x)

)in Ω.


Note that, for 0 < M ≤ 1, inequality (4.1) is true when

(4.2) M ≥ 1

a0

(λMβ−+α− |e|β(x)

L∞ |e|α(x)

Lq(x)+ µMη−+γ− |e|η(x)

L∞ |e|γ(x)

Ls(x)

)in Ω.

Considering

(4.3) R = max|e|L∞ , |e|Lq(x) , |e|Ls(x) , 1

,

inequality (4.2) is true when

(4.4) 1 ≥ 1

a0

(λMβ−+α−−1Rη

++γ+

+ µMη−+γ−−1Rη++γ+

).

We are going to prove that (4.4) holds. Considering ϕ : (0,∞)→ R given by

ϕ(M) =Rη

++γ+

λ

a0Mβ−+α−−1 +

Rη++γ+

µ

a0Mη−+γ−−1,

since ϕ is a C1 class function and 0 < α− + β− < 1 < η− + γ−, we get

limM→0+

ϕ(M) = limM→∞

ϕ(M) =∞.

Then, ϕ has a global minimum point Mλ,µ = M(λ, µ) > 0. Hence,

ϕ′(Mλ,µ) = 0⇔Mλ,µ = c1

(λ

µ

) 1

(η−+γ−)−(β−+α−)

,

where c1 =(

1−(β−+α−)(η−+γ−)−1

) 1

(η−+γ−)−(β−+α−).

Note also that

ϕ(Mλ,µ) =Rη

++γ+

c2a0

[λ(η−+γ−)−1

µ(β−+α−)−1

] 1

(η−+γ−)−(β−+α−)

,

where c2 = c(β−+α−)−11 + c

(η−+γ−)−11 . Hence, given µ > 0, there is λ0 = λ0(µ) > 0, such

that, for each λ ∈ (0, λ0], the pair (λ, µ) satisfies

0 < Mλ,µ ≤ 1 and ϕ(Mλ,µ) ≤ 1, ∀ 0 < λ ≤ λ0.

Note that Mλ,µ also satisfies (4.1). Since

A(x, t) ≥ a0 > 0 in Ω× [0, b0],

Mλ,µ → 0 as λ → 0 and λ 7→ Mλ,µ is increasing, we can choose λ0 > 0 such that, for eachλ ∈ (0, λ0), we have

A(x, |w|Lr(x)) ≥ a0 > 0, ∀ w ∈ [0,Mλ,µe],

where[0,Mλ,µe

]:=w ∈ L∞(Ω) : 0 ≤ w(x) ≤ Mλ,µe a.e in Ω

. Now, defining u =

u(λ, µ) := Mλ,µe, for each w ∈ [0, u] we have−∆u ≥ 1

A(x, |w|Lr(x))λuβ(x)|u|α(x)

Lq(x)+

1

A(x, |w|Lr(x))µuη(x)|u|γ(x)

Ls(x)in Ω,

u > 0 in Ω,

u = 0 on ∂Ω.


We are going to construct u. Considering K = maxA(x, t) : (x, t) ∈ Ω×

[0, |u|Lr(x)

],

given w ∈ [0, u], we have |w|Lr(x) ≤ |u|Lr(x) . Then,

a0 ≤ A(x, |w|Lr(x)) ≤ K in Ω, ∀ w ∈ [0, u].

Now consider ϕ1 > 0 in Ω, |ϕ1|L∞ ≤ 1 and |ϕ1|Lq(x) ≤ 1. Since 0 < α+ + β+ < 1, takeε = ε(λ) > 0, such that

0 < ε ≤ min

(

λ|ϕ1|α+

Lq(x)

λ1|ϕ1|1−β+

L∞ K

) 1

1−(α++β+)

, 1

.

If u = u(λ) = εϕ1, for each w ∈ [0, u], we get−∆u ≤ 1


Lq(x)+

1


Lq(x)in Ω,

u > 0 in Ω,

u = 0 on ∂Ω.

We are going to prove that u ≤ u. For λ ∈ (0, λ0], taking ε > 0 sufficiently small such that

λ1ε|ϕ1|L∞ ≤Mλ,µ,

we have,−∆(εϕ1) ≤ −∆(Mλ,µe).

Then, by the Comparison Principle,

u := εϕ1 ≤Mλ,µe =: u,

that implies, (u, u) is a pair of sub-supersolution to problem (P )λ,µ. From Theorem 1, foreach 0 < λ ≤ λ0, there is a weak positive solution uλ,µ ∈ H1

0 (Ω)⋂L∞(Ω) to (P )λ,µ such

thatu ≤ uλ,µ ≤ u,

which finishes the prove for the case (A1).We consider the case (A2) in order to construct u. Given λ > 0 and making ε = ε(λ) > 0,

such that

ε ≤ min

(

λ|ϕ1|α+

Lq(x)

λ1|ϕ1|1−β+

L∞ a1

) 1

1−(α++β+)

, 1

and considering u = u(λ) := εϕ1, for each w ∈ L∞(Ω), we have

−∆u ≤ 1


Lq(x)+

1


Lq(x)in Ω,

u > 0 in Ω,

u = 0 on ∂Ω.

We construct u. Since limt→∞

A(x, t) = a∞ uniformly in Ω, there is M > 0 sufficiently large

such thatA(x, t) ≥ a∞

2, in Ω× [M,∞).

Considermλ = m(λ) = min

A(x, t) : (x, t) ∈ Ω×

[|u|Lr(x) ,M

]> 0.


Taking kλ = k(λ) = minmλ,

a∞2

, we have

A(x, t) ≥ kλ > 0 in Ω×[|u|Lr(x) ,∞

).

We want to obtain a positive constant T > 0 such that, for each w ∈ L∞(Ω) with u ≤ w,we have

T ≥ 1

A(x, |w|Lr(x))

(λ(Te)β(x)|Te|α(x)

Lq(x)+ µ(Te)η(x)|Te|γ(x)

Ls(x)

)in Ω.

But this relation is true if T ≥ 1 and

(4.5) 1 ≥ 1

kλ

(λT β

++α+−1Rη++γ+

+ µT η++γ+−1Rη

++γ+),

where R = max|e|L∞ , |e|Lq(x) , |e|Ls(x) , 1

. Consider ψ : (0,∞)→ R given by

ψ(t) =Rη

++γ+

λ

kλtβ

++α+−1 +Rη

++γ+

µ

kλtη

++γ+−1.

Since 0 < α+ + β+ < 1 < η+ + γ+, ψ has a minimum point Tλ,µ given by

Tλ,µ = c3

(λ

µ

) 1

(η++γ+)−(β++α+)

,

where c3 =(

1−(β++α+)(η++γ+)−1

) 1

(η++γ+)−(β++α+).

Note that

(4.6) Tλ,µ ≥ 1⇔ λ ≥ µ

[(η+ + γ+)− 1

1− (β+ + α+)

]> 0

and

(4.7) ψ(Tλ,µ) ≤ 1⇔ Rη++γ+

c4kλ

[λ(η++γ+)−1

µ(β++α+)−1

] 1

(η++γ+)−(β++α+)

≤ 1.

Then, given λ > 0, there exists µ0 = µ0(λ) > 0 such that, for each µ ∈ (0, µ0), the pair(λ, µ) satisfies

Tλ,µ ≥ 1 and ψ(Tλ,µ) ≤ 1.

Hence, considering u = u(λ, µ) := Tλ,µe, for each w ∈ L∞(Ω) with uλ ≤ w we have−∆u ≥ 1


Lq(x)+

1


Lq(x)in Ω,

u > 0 in Ω,

u = 0 on ∂Ω.

We are going to prove that u ≤ u. Since Tλ,µ →∞ as µ→ 0+, we can choose µ0 = µ0(λ) > 0,such that

λ1ε|ϕ1|L∞ ≤ Tλ,µ0 .

Then,

−∆(εϕ1) ≤ −∆(Tλ,µ0e),


and by the Comparison Principle u := εϕ1 ≤ Tλ,µ0e. Since the function µ→ Tλ,µ is decreas-

ing, we get

u ≤ Tλ,µ0e ≤ Tλ,µe := u, ∀ µ ∈ (0, µ0).

Hence, (u, u) is a pair of sub-supersolution to (P )λ,µ and from Theorem 1, there is a weakpositive solution uλ,µ ∈ H1

0 (Ω)⋂L∞(Ω) of (P )λ,µ, such that

u ≤ uλ,µ ≤ u,which finishes the proof for the case (A2).

4.3. A generalized classical logistic equation. In this application we consider the fol-lowing class of problems given by

(P ′)λ

−A(x, |u|Lr(x))∆u = λf(u)|u|α(x)

Lq(x)in Ω,

u = 0 on ∂Ω.

We assume that there is θ > 0 such that f : [0,∞)→ R satisfies

(f1) f ∈ C0[0, θ].(f2) f(0) = f(θ) = 0, f ′(0) > 0 (f ′(0) ∈ R or f ′(0) =∞)

and f(s) > 0 ∀s ∈ (0, θ).

Some prototypes of functions f that satisfy the above hypotheses are given by f1(t) = t(γ−t)and f2(t) = µtq − tp; 0 < q < 1 < p e γ, µ > 0. The main result in this subsection is:

Theorem 5. Suppose that r(x), q(x) ∈ C+(Ω), 0 ≤ α(x) ∈ C0(Ω) and f : [0,∞) → Rsatisfies (f1) and (f2). Suppose also that A : Ω× (0,∞)→ R is a continuous function suchthat

A(x, t) > 0 in Ω×(0, |θ|Lr(x)

].

Then, there is λ0 > 0 such that, for each λ ≥ λ0, problem (P ′)λ has a weak positive solutionuλ with

0 < uλ ≤ θ.

Proof. We are going to construct u. Given δ > 0 define λ :=λ1

f ′(0)+ δ when f ′(0) ∈ R and

λ := δ when f ′(0) =∞. We are going to prove that the solution of problem

(P2)λ

−∆u = λf(u) in Ω,

u = 0 on ∂Ω,

is a subsolution of problem (P ′)λ. Note that θ > 0 is a supersolution to (P2)λ, because

f(θ) = 0. Now, since f ′(0) >λ1

λand f(0) = 0, there is τ > 0 such that

f(t)

t≥ λ1

λ, ∀ t ∈ (0, τ ].

Considering ε > 0 such that ε|ϕ1|L∞ ≤ τ, we have

f(εϕ1)

εϕ1≥ λ1

λin Ω.

Then,

−∆(εϕ1) = λ1(εϕ1) ≤ λf(εϕ1) in Ω.


Hence, εϕ1 is a subsolution of (P2)λ. Making ε > 0 small if necessary, we get εϕ1 ≤ θ. Then,using Theorem 1 with A(x, t) ≡ 1, α(x) ≡ 0, f2 ≡ 0 and f1 ≡ λf , we obtain a weak positivesolution ϕ = ϕλ of (P2)λ verifying

εϕ1 ≤ ϕ ≤ θ.(4.8)

Note that

|ϕ|α(.)

Lq(x): Ω → R+

x 7→ |ϕ|α(x)

Lq(x)

is a continuous function. Then, there is a constant C > 0, such that

|ϕ|α(x)

Lq(x)≥ C in Ω.

Considering K = maxA(x, t) : (x, t) ∈ Ω × [|ϕ|Lr(x) , |θ|Lr(x) ]

, µ = K

C and ϕ = ϕλ thesolution of (P2)λ, we have

−∆ϕ = λf(ϕ) =λµf(ϕ)|ϕ|α(x)

Lq(x)

K

K

µ|ϕ|α(x)

Lq(x)

.

Since K

µ|ϕ|α(x)

Lq(x)

≤ 1, we get

−∆ϕ ≤ λµf(ϕ)|ϕ|α(x)

Lq(x)

K.

Then, for each λ ≥ λµ, we conclude that

−∆ϕ ≤ λf(ϕ)|ϕ|α(x)

Lq(x)

A(x, |w|Lr(x)), ∀ w ∈ [ϕ, θ].

Defining u := ϕ = ϕλ we have proved that u is a subsolution of (P ′)λ, for each λ ≥ λµ.Since f(θ) = 0, the function u := θ is a supersolution to (P ′)λ, because

−∆u = 0 = λ f(u)

A(x,|w|Lr(x)

) |u|α(x)

Lq(x)in Ω, ∀ w ∈ [ϕ, θ],

u > 0 in Ω,

u > 0 on ∂Ω.

Note that from (4.8) we get

u := ϕλ ≤ θ =: u.

Hence, (u, u) is a pair of sub-supersolution to (P ′)λ and by Theorem 1, for each λ ≥ λ0 := λµ,there is a weak positive solution uλ ∈ H1

0 (Ω)⋂L∞(Ω) of (P ′)λ such that

ϕλ ≤ uλ ≤ θ.


5. The system case

We start with the definition of weak solutions and sub-supersolutions to (S).

Definition 4. The pair (u, v) is a weak positive solution of (S) if u, v ∈ H10 (Ω)

⋂L∞(Ω)

with u, v > 0 a.e in Ω,∫Ω

∇u∇ϕ =

∫Ω

(f1(x, u, v)|v|α1(x)

Lq1(x)

A(x, |v|Lr1(x))+f2(x, u, v)|v|γ1(x)

Ls1(x)

A(x, |v|Lr1(x))

)ϕ, ∀ ϕ ∈ H1

0 (Ω)

and ∫Ω

∇v∇ψ =

∫Ω

(g1(x, u, v)|u|α2(x)

Lq2(x)

A(x, |u|Lr2(x))+g2(x, u, v)|u|γ2(x)

Ls2(x)

A(x, |u|Lr2(x))

)ψ, ∀ ψ ∈ H1

0 (Ω).

Definition 5. Given z, θ ∈ L∞(Ω), with z ≤ θ a.e in Ω, we define

[z, θ] :=w ∈ L∞(Ω) : z(x) ≤ w(x) ≤ θ(x) a.e in Ω

and

[z,∞) :=w ∈ L∞(Ω) : z(x) ≤ w(x) a.e in Ω

.

Definition 6. The pairs (u, v), (u, v) are sub-supersolution to (S), if u, v ∈ H10 (Ω)∩L∞(Ω),

u, v ∈ H1(Ω)⋂L∞(Ω) with u ≤ u, v ≤ v a.e in Ω and u = 0 ≤ u, v = 0 ≤ v a.e on ∂Ω

and given ϕ,ψ ∈ H10 (Ω) with ϕ,ψ ≥ 0, we have

(5.1)

∫

Ω∇u∇ϕ ≤

∫Ω

(f1(x,u,w)|v|α1(x)

Lq1(x)

A(x,|w|Lr1(x) ) +

f2(x,u,w)|v|γ1(x)

Ls1(x)

A(x,|w|Lr1(x) )

)ϕ ∀ w ∈ [v, v],

∫Ω∇v∇ψ ≤

∫Ω

(g1(x,w,v)|u|α2(x)

Lq2(x)

A(x,|w|Lr2(x) ) +

g2(x,w,v)|u|γ2(x)

Ls2(x)

A(x,|w|Lr2(x) )

)ψ ∀ w ∈ [u, u]

and

(5.2)

∫

Ω∇u∇ϕ ≥

∫Ω

(f1(x,u,w)|v|α1(x)

Lq1(x)

A(x,|w|Lr1(x) ) +

f2(x,u,w)|v|γ1(x)

Ls1(x)

A(x,|w|Lr1(x) )

)ϕ ∀ w ∈ [v, v],

∫Ω∇v∇ψ ≥

∫Ω

(g1(x,w,v)|u|α2(x)

Lq2(x)

A(x,|w|Lr2(x) ) +

g2(x,w,v)|u|γ2(x)

Ls2(x)

A(x,|w|Lr2(x) )

)ψ ∀ w ∈ [u, u].


Proof. Consider the truncation operators given by

T, S : L2(Ω)→ L∞(Ω)

Tz(x) =

u(x) if z(x) ≤ u(x),z(x) if u(x) ≤ z(x) ≤ u(x),

u(x) if z(x) ≥ u(x)

and

Sw(x) =

v(x) if w(x) ≤ v(x),w(x) if v(x) ≤ w(x) ≤ v(x),

v(x) if w(x) ≥ v(x).


By definition of T and S, we have

u ≤ Tz ≤ u and v ≤ Sw ≤ v in Ω, ∀ z, w ∈ L2(Ω).

Since w = minu, v and w = maxu, v, then

w ≤ Tz, Sw ≤ w in Ω, ∀ z, w ∈ L2(Ω)

which implies Tz, Sw ∈ L∞(Ω). Hence T and S are well-defined,

|w|L∞ ≤ |Tz|L∞ , |Sw|L∞ ≤ |w|L∞ , ∀ z, w ∈ L2(Ω).

and

|w|Lm(x) ≤ |Tz|Lm(x) , |Sw|Lm(x) ≤ |w|Lm(x) , ∀ z, w ∈ L2(Ω), m(x) ∈ C+(Ω).

Let us consider the operators H1, H2 : [u, u]× [v, v]→ L2(Ω) given by

H1(u, v)(x) :=f1(x, u(x), v(x))|v|α1(x)

Lq1(x)

A(x, |v|Lr1(x))+f2(x, u(x), v(x))|v|γ1(x)

Ls1(x)

A(x, |v|Lr1(x))

and

H2(u, v)(x) :=g1(x, u(x), v(x))|u|α2(x)

Lq2(x)

A(x, |u|Lr2(x))+g2(x, u(x), v(x))|u|γ2(x)

Ls2(x)

A(x, |u|Lr2(x)).

We are proving that Hi, i = 1, 2 are well-defined and (z, w) 7→ Hi(Tz, Sw); (z, w) ∈ L2(Ω)×L2(Ω) are continuous from L2(Ω)× L2(Ω) in L2(Ω). In fact, since A(x, t) > 0 in Ω× [σ, σ]there are positive constants k,K > 0, such that

k ≤ A(x, t) ≤ K, ∀ (x, t) ∈ Ω× [σ, σ].

Given (u, v) ∈ [u, u]× [v, v], we have u, v ∈ [w,w]. Then, |u|Lri(x) , |v|Lri(x) ∈ [σ, σ] and

0 < k ≤ A(x, |u|Lri(x)),A(x, |v|Lri(x)) ≤ K in Ω, ∀ (u, v) ∈ [u, u]× [v, v], i = 1, 2.

Since fi(x, t, s), gi(x, t, s) are continuous in Ω×[0, |u|L∞ ]×[0, |v|L∞ ], there are c1, c2, c3, c4 > 0such that

|H1(u, v)| ≤c1(|w|α

−1

Lq1(x) + |w|α+1

Lq1(x)) + c2(|w|γ−1

Ls1(x) + |w|γ+1

Ls1(x))

kin Ω,

∀ (u, v) ∈ [u, u]× [v, v] and

|H2(u, v)| ≤c3(|w|α

−2

Lq2(x) + |w|α+2

Lq2(x)) + c4(|w|γ−2

Ls2(x) + |w|γ+2

Ls2(x))

kin Ω,

∀ (u, v) ∈ [u, u]× [v, v] that implies that Hi are well-defined. Now we prove the continuity ofthe functions (z, w) 7→ Hi(Tz, Sw) from L2(Ω)× L2(Ω) to L2(Ω). Let us consider L2(Ω)×L2(Ω) with the norm

|(u, v)|L2×L2 = |u|L2 + |v|L2

and (zn, wn)→ (z, w) in L2(Ω)× L2(Ω). Then, up to a subsequence,

zn(x)→ z(x) and wn(x)→ w(x) a.e in Ω,

T zn(x)→ Tz(x) and Swn(x)→ Sw(x) a.e in Ω.

Hence,

|Tzn(x)− Tz(x)|m(x), |Swn(x)− Sw(x)|m(x) → 0 a.e in Ω


and

|Tzn(x)− Tz(x)|m(x) ≤ 2|u|m(x)L∞ ≤ C a.e in Ω,

|Swn(x)− Sw(x)|m(x) ≤ 2|v|m(x)L∞ ≤ C a.e in Ω.

From Lebesgue’s Dominated Convergence Theorem∫Ω

|Tzn − Tz|m(x) → 0 and

∫Ω

|Swn − Sw|m(x) → 0,

we conclude

Tzn → Tz and Swn → Sw in Lm(x)(Ω).

Since fi(x, t), gi(x, t) and A(x, t), i = 1, 2, are continuous, we have

Hi(Tzn, Swn)→ Hi(Tz, Sw) a.e in Ω.

By Lebesgue’s Dominated Convergence Theorem we get

Hi(Tun, Swn)→ Hi(Tu, Sw) in L2(Ω),

which proves that Hi(Tu, Sw) are continuous from L2(Ω) × L2(Ω) to L2(Ω). Now, given(z, w) ∈ L2(Ω)× L2(Ω), consider the linear system

(SL)

−∆u = H1(Tz, Sw) in Ω,−∆v = H2(Tz, Sw) in Ω,

u = v = 0 on ∂Ω.

Then, each equation of (SL) has a unique weak solution in H10 (Ω) and we can define the

operator

Φ : L2(Ω)× L2(Ω)→ L2(Ω)× L2(Ω)

(z, w) 7→ Φ(z, w) = (u, v),

where (u, v) ∈ H10 (Ω)×H1

0 (Ω) is a weak solution of (SL). We are going to prove that Φ hasa fixed point using Schaefer’s Fixed Point Theorem. Arguing as in the proof of Theorem 1,we can prove that the operator Φ is compact and continuous. Now we are going to provethat there is R > 0 such that, if

(u, v) = θΦ(u, v) with θ ∈ [0, 1],

then we have

|(u, v)|L2×L2 < R.

Indeed, if θ = 0, we get (u, v) = (0, 0). If θ 6= 0, we obtain

Φ(u, v) =(uθ,v

θ

),

which implies ∫Ω

∇(uθ

)∇ϕ =

∫Ω

H1(Su, Tv)ϕ, ∀ ϕ ∈ H10 (Ω)

and ∫Ω

∇(vθ

)∇ψ =

∫Ω

H2(Su, Tv)ψ, ∀ ψ ∈ H10 (Ω).

Taking ϕ = u and ψ = v, we conclude that

‖u‖2 ≤ θK1|u|L1 and ‖v‖2 ≤ θK1|v|L1 ,


and using Poincare’s inequality we obtain R > 0, such that

|u|L2 + |v|L2 < R.

From Schaefer’s Fixed Point Theorem, there exists (u, v) ∈ L2(Ω)× L2(Ω), such that

Φ(u, v) = (u, v) and |(u, v)|L2×L2 < R.

By the definition of Φ, we have proved that the pair (u, v) satisfies(5.3)∫

Ω

∇u∇ϕ =

∫Ω

(f1(x, Tu, Sv)|Sv|α1(x)

Lq1(x)

A(x, |Sv|Lr1(x))+f2(x, Tu, Sv)|Sv|γ1(x)

Ls1(x)

A(x, |Sv|Lr1(x))

)ϕ, ∀ ϕ ∈ H1

0 (Ω)

and(5.4)∫

Ω

∇v∇ψ =

∫Ω

(g1(x, Tu, Sv)|Tu|α2(x)

Lq2(x)

A(x, |Tu|Lr2(x))+g2(x, Tu, Sv)|Tu|γ2(x)

Ls2(x)

A(x, |Tu|Lr2(x))

)ψ, ∀ ψ ∈ H1

0 (Ω).

We are going to prove that

u ≤ u ≤ u and v ≤ v ≤ v.

Making w = Sv ∈ [v, v] in the definition of subsolution, for each ϕ ∈ H10 (Ω), ϕ ≥ 0, we have∫

Ω

∇(u− u)∇ϕ ≤∫

Ω

(f1(x, u(x), Sv(x))|v|α1(x)

Lq1(x) − f1(x, Tu(x), Sv(x))|Sv|α1(x)

Lq1(x)

A(x, |Sv|Lr1(x))

)ϕ

+

∫Ω

(f2(x, u(x), Sv(x))|v|γ1(x)

Ls1(x) − f2(x, Tu(x), Sv(x))|Sv|γ1(x)

Ls1(x)

A(x, |Sv|Lr1(x))

)ϕ.

Taking ϕ = (u − u)+ := max(u − u), 0, since fi(x, t, s) ≥ 0 in Ω × [0, |u|L∞ ] × [0, |v|L∞ ],Tu = u in x ∈ Ω : u(x) ≥ u(x) and Sv ∈ [v, v], we get

‖(u− u)+‖2 ≤∫x∈Ω: u(x)≥u(x)

f1(x, u(x), Sv(x))(|v|α1(x)

Lq1(x) − |Sv|α1(x)

Lq1(x))

A(x, |Sv|Lr1(x))(u− u)

+

∫x∈Ω: u(x)≥u(x)

f2(x, u(x), Sv(x))(|v|γ1(x)

Ls1(x) − |Sv|γ1(x)

Ls1(x))


≤ 0.

Then, (u − u)+ = 0, that implies u ≤ u. Considering again Sw = v ∈ [v, v], for eachϕ ∈ H1

0 (Ω);ϕ ≥ 0, we have∫Ω

∇(u− u)∇ϕ ≤∫

Ω

(f1(x, Tu(x), Sv(x))|Sv|α1(x)

Lq1(x) − f1(x, u(x), Sv(x))|v|α1(x)

Lq1(x)

A(x, |Sv|Lr1(x))

)ϕ

+

∫Ω

(f2(x, Tu(x), Sv(x))|Sv|γ1(x)

Ls1(x) − f2(x, u(x), Sv(x))|v|γ1(x)

Ls1(x)

A(x, |Sv|Lr1(x))

)ϕ.


Taking ϕ = (u−u)+ := max(u−u), 0, since que fi(x, t, s) ≥ 0 in Ω× [0, |u|L∞ ]× [0, |v|L∞ ],Tu = u in x ∈ Ω : u(x) ≥ u(x) and Sv ∈ [v, v], we get

‖(u− u)+‖2 ≤∫x∈Ω: u(x)≥u(x)

f1(x, u(x), Sv(x))(|Sv|α1(x)

Lq1(x) − |v|α1(x)

Lq1(x))


+

∫x∈Ω: u(x)≥u(x)

f2(x, u(x), Sv(x))(|Sv|γ1(x)

Ls1(x) − |v|γ1(x)

Ls1(x))


≤ 0.

Then, (u− u)+ = 0 which implies u ≤ u. Using the same arguments we can prove

v ≤ v ≤ v.By the definition of T and S, we conclude Tu = u and Sv = v. Then the pair (u, v) is aweak positive solution of (S) with

u ≤ u ≤ u and v ≤ v ≤ v.

Remark 2. As in the scalar case, the weak solution (u, v) found is, in fact, a strong solutionand satisfies

−A(x, |v|Lr1(x))∆u = f1(x, u, v)|v|α1(x)

Lq1(x) + f2(x, u, v)|v|γ1(x)

Ls1(x) a.e in Ω,

−A(x, |u|Lr2(x))∆v = g1(x, u, v)|u|α2(x)

Lq2(x) + g2(x, u, v)|u|γ2(x)

Ls2(x) a.e in Ω,

u, v > 0 in Ω,

u = v = 0 on ∂Ω.

6. Applications of Theorem 2

In this section we make three applications of Theorem 2. From now on we denote bye ∈ H1

0 (Ω) ∩ C2,τ (Ω) for some 0 < τ < 1, the unique positive solution of problem −∆e = 1in Ω, e = 0 on ∂Ω and by ϕ1 ∈ H1

0 (Ω)⋂C2,α(Ω) a positive eigenfunction associated to the

first eigenvalue λ1 of (−∆, H10 (Ω)).

6.1. The sublinear system. In this subsection we study

(Ss)

−A(x, |v|Lr1(x))∆u = (uβ1(x) + vγ1(x))|v|α1(x)

Lq1(x) in Ω,

−A(x, |u|Lr2(x))∆v = (uβ2(x) + vγ2(x))|u|α2(x)

Lq2(x) in Ω,

u = v = 0 on ∂Ω.

The main result is:

Theorem 6. Suppose that ri(x), qi(x) ∈ C+(Ω) and 0 ≤ αi(x), βi(x), γi(x) ∈ C0(Ω) suchthat

0 < α+i + β+

i , α+i + γ+

i < 1, i = 1, 2.

Suppose also A : Ω× [0,∞)→ R is continuous and satisfies one of these two conditions:(A1) There is a constant a0 > 0, such that

A(x, t) ≥ a0 > 0 in Ω× [0,∞).


(A2) There are constants a1, a∞ > 0, such that

A(x, 0) = 0 < A(x, t) ≤ a1 in Ω× (0,∞)

and

limt→∞


Then (Ss) has a weak positive solution (u, v).

Proof. Let us assume first that (A1) is true. We are going to construct (u, v). Since thefunctions

|e|αi(.)Lqi(x)

, |e|βi(.)L∞ , |e|γi(.)L∞ : Ω → R+

x 7→ |e|αi(x)

Lqi(x), |e|βi(x)

L∞ , |e|γi(.)L∞

are continuous, there are C1, C2, C3 > 0, such that

|e|αi(x)

Lqi(x)≤ C1, |e|βi(x)

L∞ ≤ C2 and |e|γi(x)L∞ ≤ C3, ∀ x ∈ Ω, i = 1, 2.

Recalling that 0 < α+i + β+

i , α+i + γ+

i < 1, with i = 1, 2, we have

limR→∞

Rαi+β+i −1 = 0 = lim

R→∞Rαi+γ

+i −1, i = 1, 2.

Hence, we can choose R > 0, sufficiently large such that1 ≥ C1C2

a0Rα1+β+

1 −1 +C1C3

a0Rα1+γ+

1 −1,

1 ≥ C1C2

a0Rα2+β+

2 −1 +C1C3

a0Rα2+γ+

2 −1.

Now, for each w ∈ L∞(Ω), we getR ≥ 1

A(x, |w|Lr1(x))

((Re)β1(x) + (Re)γ1(x)

)|Re|α1(x)

Lq1(x) in Ω,

R ≥ 1

A(x, |w|Lr2(x))

((Re)β2(x) + (Re)γ2(x)

)|Re|α2(x)

Lq2(x) in Ω.

Considering u = Re and v = Re, we derive

−∆u ≥ 1

A(x, |w|Lr1(x))(uβ1(x) + wγ1(x))|v|α1(x)

Lq1(x) in Ω, ∀ w ∈ [0, v],

−∆v ≥ 1

A(x, |w|Lr2(x))(wβ2(x) + vγ2(x))|u|α2(x)

Lq2(x) in Ω, ∀ w ∈ [0, u],

u, v > 0 in Ω,u = v = 0 on ∂Ω.

We are going to construct (u, v). Using the notation of Theorem 2, we have, w = Re, wherew = maxu, v. Let K = max

A(x, t) : (x, t) ∈ Ω× [0, σ]

, where σ = max|w|Lri(x) : i =

1, 2. Given w ∈ [0, w := Re], we get |w|Lri(x) ≤ σ and conclude,

a0 ≤ A(x, |w|Lri(x)) ≤ K in Ω, ∀ w ∈ [0, w], i = 1, 2.


Let ϕ1 > 0 in Ω, |ϕ1|L∞ ≤ 1 and |ϕ1|Lq(x) ≤ 1. Considering

0 < ε ≤ min

|ϕ1|

α+1

Lq1(x)

λ1|ϕ1|1−β+

1

L∞ K

1

1−(α+1 +β

+1 )

,

|ϕ1|α+

2

Lq2(x)

λ1|ϕ1|1−γ+

2

L∞ K

1

1−(α+2 +γ

+2 )

, 1

; i = 1, 2,

for each w ∈ [0, w], we haveλ1εϕ1(x) ≤ 1

A(x, |w|Lr1(x))(εϕ1(x))β1(x)|εϕ1|α1(x)

Lq1(x) in Ω,

λ1εϕ1(x) ≤ 1

A(x, |w|Lr2(x))(εϕ1(x))γ2(x)|εϕ1|α2(x)

Lq2(x) in Ω.

Taking u = εϕ1 and v = εϕ1, we get

−∆u ≤ 1


Lq1(x) in Ω, ∀ w ∈ [0, v],

−∆v ≤ 1


Lq2(x) in Ω, ∀ w ∈ [0, u],

u, v > 0 in Ω,u, v = 0 on ∂Ω.

We are going to prove u ≤ u and v ≤ v. For ε > 0 sufficiently small, we derive λ1ε|ϕ1|L∞ ≤R. Then,

−∆(εϕ1) ≤ −∆(Re)

and by the Comparison Principle, we obtain

u := εϕ1 ≤ Re =: u and v := εϕ1 ≤ Re =: v.

Hence, (u, v) and (u, v) are a sub-supersolution to (Ss). From Theorem 2, there is a weakpositive solution (u, v) to (Ss) with

u ≤ u ≤ u and v ≤ v ≤ v,

with finishes the proof of the case (A1). Now we consider the hypothesis (A2) and we aregoing to construct (u, v). Considering

0 < ε ≤ min

|ϕ1|

α+1

Lq1(x)

λ1|ϕ1|1−β+

1

L∞ a1

1

1−(α+1 +β

+1 )

,

|ϕ1|α+

2

Lq2(x)

λ1|ϕ1|1−γ+

2

L∞ a1

1

1−(α+2 +γ

+2 )

, 1

; i = 1, 2

and defining u = εϕ1 and v = εϕ1, we have

−∆u ≤ 1


Lq1(x) in Ω, ∀ w ∈ [v,∞),

−∆v ≤ 1


Lq2(x) in Ω, ∀ w ∈ [u,∞),

u, v > 0 in Ω,u, v = 0 on ∂Ω.


We are going to construct (u, v). Since limt→∞

A(x, t) = a∞ uniformly in Ω, we have

k := minA(x, t) : (x, t) ∈ Ω× [σ,∞)

> 0,

where σ := min|w|ri : i = 1, 2 and w = εϕ1. Choosing R > 0 such that1 ≥ C1C2

kRα1+β+

1 −1 +C1C3

kRα1+γ+

1 −1,

1 ≥ C1C2

kRα2+β+

2 −1 +C1C3

kRα2+γ+

2 −1,

and defining u = v := Re, where the constants C1, C2 and C3 are as in the first case, we get

−∆u ≥ 1


Lq1(x) in Ω, ∀ w ∈ [v, v]

−∆v ≥ 1


Lq2(x) in Ω, ∀ w ∈ [u, u]

u, v > 0 in Ω,u, v = 0 on ∂Ω.

Now we are going to prove that u ≤ u and v ≤ v. In fact, taking R > 0 sufficiently large,such that λ1ε|ϕ1|L∞ ≤ R, we have

−∆(εϕ1) ≤ −∆(Re).

By the Comparison Principle,

u := εϕ1 ≤ Re =: u and v := εϕ1 ≤ Re =: v.

Then, (u, v) and (u, v) are a sub-supersolution to (Ss). From Theorem 2, there is a weakpositive solution (u, v) to (Ss) with

u ≤ u ≤ u and v ≤ v ≤ v,

which finishes the proof.

6.2. Concave and convex system. In this subsection we study

(S)λ,µ

−A(x, |v|Lr1(x))∆u = λuβ1(x)−1u|v|α1(x)

Lq1(x) + µvη1(x)−1v|v|γ1(x)

Ls1(x) in Ω,

−A(x, |u|Lr2(x))∆v = λvβ2(x)−1v|u|α2(x)

Lq2(x) + µuη2(x)−1u|u|γ2(x)

Ls2(x) in Ω,

u, v = 0 on ∂Ω.

The main result is:

Theorem 7. Suppose ri(x), qi(x), si(x) ∈ C+(Ω) and 0 ≤ αi(x), γi(x), βi(x), ηi(x) ∈C0(Ω), such that

0 < α−i + β−i ≤ α+i + β+

i < 1.

Suppose also that A : Ω× [0,∞)→ R is continuous and satisfies one of these two conditions:(A1) Suppose that 1 < η−i + γ−i and that there are constants a0, b0 > 0 such that

A(x, t) ≥ a0 > 0 in Ω× [0, b0].


Then, given µ > 0, there exists λ0 > 0 such that, for each λ ∈ (0, λ0), (S)λ,µ has a weakpositive solution (uλ,µ, vλ,µ).(A2) Suppose that 1 < (η+

1 + γ+1 )(η+

2 + γ+2 ) and there are constants a1, a∞ > 0 such that

A(x, 0) = 0 < A(x, t) ≤ a1 in Ω× (0,∞)

and

limt→∞


Then, given λ > 0, there exists µ0 > 0 such that, for each, µ ∈ (0, µ0), (S)λ,µ has a weakpositive solution (uλ,µ, vλ,µ).

Proof. Firstly, we assume (A1). We are going to construct (u, v). Note that, for each M > 0,we have

−∆(Me) = M in Ω.

We are going to prove that there exists M > 0 such that

(S)a0

M ≥ 1

a0

(λ(Me)β1(x)|Me|α1(x)

Lq1(x) + µ(Me)η1(x)|Me|γ1(x)

Ls1(x)

)in Ω,

M ≥ 1

a0

(λ(Me)β2(x)|Me|α2(x)

Lq2(x) + µ(Me)η2(x)|Me|γ2(x)

Ls2(x)

)in Ω.

Considering

R = max|e|L∞ , |e|Lqi(x) , |e|Lsi(x) , 1

, i = 1, 2,

% = min(β−i +α−i ) : i = 1, 2, τ = min(η−i + γ−i ) : i = 1, 2 and p = max(η+i + γ+

i ) : i =1, 2, note that, for

M ≥ 1

a0(λM%Rp + µMτRp) and 0 < M ≤ 1,

the constant M is a solution of (S)a0 . But this inequality is equivalent to

1 ≥ 1

a0

(λM%−1Rp + µMτ−1Rp

)and 0 < M ≤ 1.

Since 0 < % < 1 < τ, given µ > 0, there exists λ0 > 0 such that, for each λ ∈ (0, λ0), thereexists Mλ,µ > 0, given by

Mλ,µ = c1

(λµ

) 1τ−%

,

such that

0 < Mλ,µ ≤ 1 and 1 ≥ 1

a0

(λM%−1

λ,µ Rp + µMτ−1

λ,µ Rp).

Then, Mλ,µ is a solution of (S)a0 . Since Mλ,µ → 0 as λ → 0, we can choose λ0 > 0 suchthat

σ0 := max|Mλ0,µe|Lri(x) : i = 1, 2. ≤ b0.Note also that λ 7→Mλ,µ is increasing, then

|Mλ,µe|Lri(x) ≤ σ0 ≤ b0, ∀ λ ∈ (0, λ0).

Using

A(x, t) ≥ a0 > 0 in Ω× [0, b0],


we have

A(x, |w|Lri(x)) ≥ a0 > 0, ∀ w ∈ [0,Mλ,µe].

From this relation, the fact that Mλ,µ is a solution of (S)a0 , and considering u = v := Mλ,µe,we get

−∆u ≥ 1

A(x, |w|Lr1(x))

(λuβ1(x)|v|α1(x)

Lq1(x) + µwη1(x)|v|γ1(x)

Ls1(x)

)in Ω, ∀ w ∈ [0, v],

−∆v ≥ 1

A(x, |w|Lr2(x))

(λvβ2(x)|u|α2(x)

Lq2(x) + µwη2(x)|u|γ2(x)

Ls2(x)

)in Ω, ∀ w ∈ [0, u],

u, v > 0 in Ω,u, v = 0 on ∂Ω.

We are going to construct (u, v). Let K = maxA(x, t) : (x, t) ∈ Ω× [0, σ]

be a constant,

where σ = max|w|Lri(x) : i = 1, 2 and w = Mλ,µe. We can suppose ϕ1 > 0 in Ω,|ϕ1|L∞ ≤ 1 and |ϕ1|Lq(x) ≤ 1. Considering

0 < ε ≤ min

λ|ϕ1|

α+i

Lqi(x)

λ1|ϕ1|1−β+

i

L∞ K

1

1−(α+i

+β+i

)

, 1

; i = 1, 2,

we have, for each w ∈ [0,Mλ,µe],λ1εϕ1(x) ≤ 1


Lq1(x) in Ω,

λ1εϕ1(x) ≤ 1


Lq2(x) in Ω.

Taking u = εϕ1 and v = εϕ1, we get

−∆u ≤ 1

A(x, |w|Lr1(x))

(λuβ1(x)|v|α1(x)


Ls1(x)

)in Ω, ∀ w ∈ [0, v]

−∆v ≤ 1

A(x, |w|Lr2(x))

(λvβ2(x)|u|α2(x)


Ls2(x)

)in Ω, ∀ w ∈ [0, u]

u, v > 0 in Ω,u, v = 0 on ∂Ω.

We are going to prove u ≤ u e v ≤ v. Given λ ∈ (0, λ0], considering ε sufficiently small, weget

λ1ε|ϕ1|L∞ ≤Mλ,µ.

Then,

−∆(εϕ1) ≤ −∆(Mλ,µe)

and by the Comparison Principle we get

u = v := εϕ1 ≤Mλ,µe =: v = u.

Hence, (u, v) and (u, v) are sub-supersolution to (S)λ,µ. By Theorem 2, for each λ ∈ (0, λ0),there is (uλ,µ, vλ,µ) a weak positive solution of (S)λ,µ with

u ≤ uλ,µ ≤ u and v ≤ vλ,µ ≤ v.


We consider the hypothesis (A2). We are going to construct (u, v). Consider

0 < ε ≤ min

λ|ϕ1|

α+i

Lqi(x)

λ1|ϕ1|1−β+

i

L∞ a1

1

1−(α++β+)

, 1

; i = 1, 2.

Taking u = u(λ) := εϕ1 and v = v(λ) =: εϕ1, we have

−∆u ≤ 1

A(x, |w|Lr1(x))

(λuβ1(x)|v|α1(x)


Ls1(x)

)in Ω, ∀ w ∈ [v,∞),

−∆v ≤ 1

A(x, |w|Lr2(x))

(λvβ2(x)|u|α2(x)


Ls2(x)

)in Ω, ∀ w ∈ [u,∞),

u, v > 0 in Ω,u = v = 0 on ∂Ω.

We are going to construct (u, v). Since A(x, t) > 0 in Ω × [σ,∞) and limt→∞

A(x, t) =

a∞ uniformly in Ω, we have

kλ := minA(x, t) : (x, t) ∈ Ω× [σ,∞)

> 0,

where σ := min|w|ri : i = 1, 2 and w = εϕ1. We are going to prove that, for eachw ∈ [εϕ,∞), there exists T ≥ 1, such that

(S)a∞

T ≥ 1

A(x, |w|Lr1(x))

(λ(Te)β1(x)|Te|α1(x)

Lq1(x) + µ(Te)η1(x)|Te|γ1(x)

Ls1(x)

)in Ω,

T ≥ 1

A(x, |w|Lr2(x))

(λ(Te)β2(x)|Te|α2(x)

Lq2(x) + µ(Te)η2(x)|Te|γ2(x)

Ls2(x)

)in Ω.

Considering

R = max|e|L∞ , |e|Lqi(x) , |e|Lsi(x) , 1

, i = 1, 2,

ζ = max(β+i + α+

i ) : i = 1, 2 and p = max(η+i + γ+

i ) : i = 1, 2, for T sufficiently large,system ((S)a∞) has a solution when T satisfies

T ≥ 1

kλ

(λT ζRp + µT pRp

)and T ≥ 1,

which is equivalent to

1 ≥ 1

kλ

(λT ζ−1Rp + µT p−1Rp

)and T ≥ 1.

Since 0 < ζ < 1 < p, given λ > 0, there exists µ0 > 0, such that, for each µ ∈ (0, µ0), thereexits

Tλ,µ = c3

(λµ

) 1p−ζ

,

such that

Tλ,µ ≥ 1 and 1 ≥ 1

kλ

(λT ζ−1

λ,µ Rp + µT p−1

λ,µ Rp).


Moreover, µ 7→ Tλ,µ, µ ∈ (0, µ0] is a decreasing function with Tλ,µ →∞ as µ→ 0+. Then,Tλ,µ is a solution of (S)a∞ and we can consider u = v := Tλ,µe because

−∆u ≥ 1

A(x, |w|Lr1(x))

(λuβ1(x)|v|α1(x)


Ls1(x)

)in Ω, ∀ w ∈ [v, v],

−∆v ≥ 1

A(x, |w|Lr2(x))

(λvβ2(x)|u|α2(x)

Lq2(x) + µwη2(x)|u|γ2(x)

Ls2(x)

)in Ω, ∀ w ∈ [u, u],

u, v > 0 in Ω,u, v = 0 on ∂Ω.

We are going to prove that u ≤ u and v ≤ v. Since Tλ,µ → ∞ as µ → 0+, we can chooseµ0 > 0, such that

−∆(εϕ1) ≤ λ1ε|ϕ1|L∞ ≤ Tλ,µ0= −∆(Tλ,µ0

e).

By the Comparison Principle εϕ1 ≤ Tλ,µ0e. Since µ→ Tλ,µ is decreasing, we get

εϕ1 ≤ Tλ,µ0e ≤ Tλ,µe, ∀ µ ∈ (0, µ0),

which implies,

u := εϕ1 ≤ Tλ,µe =: u e v := εϕ1 ≤ Tλ,µe =: v.

Hence, (u, v) and (u, v) is a sub-supersolution to (S)λ,µ and from Theorem 2, there exists(uλ,µ, vλ,µ), a weak positive solution of (S)λ,µ with

u ≤ uλ,µ ≤ u and v ≤ vλ,µ ≤ v,

which finishes the proof in the case that (A2) holds.

6.3. A generalized classical logistic system. In this subsection we study the system

(S′)λ,µ

−A(x, |v|Lr1(x))∆u = λf1(u)|v|α1(x)

Lq1(x) in Ω,

−A(x, |u|Lr2(x))∆v = µf2(v)|u|α2(x)

Lq2(x) in Ω,

u = v = 0 on ∂Ω.

We assume there are constants θi > 0, such that fi : [0,∞)→ R, i = 1, 2 satisfy

(f1) fi ∈ C0[0, θi].(f2) fi(0) = fi(θ) = 0, f ′i(0) > 0 (f ′i(0) ∈ R or f ′i(0) =∞) and

fi(s) > 0 ∀s ∈ (0, θi), where i=1,2.

The main result in this subsection is:

Theorem 8. Suppose that ri(x), qi(x) ∈ C+(Ω) 0 ≤ αi(x) ∈ C0(Ω) and fi : [0,∞) → Rsatisfy (f1) and (f2). Suppose also that A : Ω× (0,∞)→ R is continuous and

A(x, t) > 0 in Ω× (0, σ],

where σ = max|θi|Lri(x) : i = 1, 2. Then, there are constants λ0, µ0 > 0 such that, for eachλ ≥ λ0 and µ ≥ µ0, system (S′)λ,µ has a weak positive solution (uλ,µ, vλ,µ), such that

0 < uλ,µ ≤ θ1 and 0 < vλ,µ ≤ θ2 in Ω.


Proof. We are going to construct (u, v). Since fi(0) = fi(θi) = 0 and f ′i(0) > 0, by Theorem5, there are η0, ν0 > 0 such that, for each η ≥ η0 and ν ≥ ν0, problems

(P )η

−∆u = ηf1(u) in Ω,

u = 0 on ∂Ω,

and

(P )ν

−∆v = νf2(v) in Ω,

v = 0 on ∂Ω,

have solutions ϕη and ϕν , respectively, such that

(6.1) 0 < ϕη ≤ θ1 and 0 < ϕν ≤ θ2 in Ω.

Taking ϕ1 := ϕη0 and ϕ2 := ϕν0 as solutions of (P )η0 and (P )ν0 , respectively, we consider

w := maxθi : i = 1, 2, w := minϕi : i = 1, 2 and K := maxA(x, t) : (x, t) ∈ Ω×[σ, σ]

,

where σ = min|w|Lri(x) : i = 1, 2 and σ = max|w|Lri(x) : i = 1, 2. Note that

|ϕ1|α2(.)

Lq2(x) , |ϕ2|α1(.)

Lq1(x) : Ω → R+

x 7→ |ϕ1|α2(x)

Lq2(x) , |ϕ2|α1(x)

Lq1(x)

are continuous. Then, there is a constant C > 0, such that

|ϕ1|α2(x)

Lq2(x) , |ϕ2|α1(x)

Lq1(x) ≥ C in Ω.

Now consider τ = KC and note that

−∆ϕ1 = η0f1(ϕ1) =η0τf1(ϕ1)|ϕ2|α1(x)

Lq1(x)

K

K

τ |ϕ2|α1(x)

Lq1(x)

.

Since τ = KC , we derive

K

τ |ϕ2|α1(x)

Lq1(x)

≤ 1, that implies,

−∆ϕ1 ≤η0τf1(ϕ1)|ϕ2|α1(x)

Lq1(x)

K.

By the definition of K, we have

−∆ϕ1 ≤η0τf1(ϕ1)|ϕ2|α1(x)

Lq1(x)

A(x, |w|Lr1(x)), ∀ w ∈ [w,w]

and

−∆ϕ2 ≤ν0τf2(ϕ2)|ϕ1|α2(x)

Lq2(x)

A(x, |w|Lr2(x)), ∀ w ∈ [w,w].


Defining u := ϕ1 and u := ϕ2, we get, for each λ ≥ λ0 and for each µ ≥ µ0,

−∆u ≤ λf1(u)|v|α1(x)

Lq1(x)

A(x, |w|Lr1(x))in Ω, ∀ w ∈ [w,w],

−∆v ≤ µf2(v)|u|α2(x)

Lq2(x)

A(x, |w|Lr2(x))in Ω, ∀ w ∈ [w,w],

u, v > 0 in Ω,u, v = 0 on ∂Ω.

Now we are going to construct (u, v). Since fi(θi) = 0, the pair (u, v), where u := θ1 andv := θ2, is a supersolution to (S′)λ,µ, because

−∆u = 0 = λf1(u)|v|α1(x)

Lq1(x)

A(x,|w|Lr1(x) ) in Ω, ∀ w ∈ [w,w],

−∆v = 0 = µf2(v)|u|α2(x)

Lq2(x)

A(x,|w|Lr2(x) ) in Ω, ∀ w ∈ [w,w],

u, v > 0 in Ω,

u, v > 0 on ∂Ω.

We are going to prove that u ≤ u and v ≤ v. But

u := ϕ1 ≤ θ1 =: u and v := ϕ2 ≤ θ2 =: v.

Then, (u, v) and (u, v) are sub-supersolution to (S′)λ,µ and by Theorem 2, for each λ ≥λ0 := η0τ and for each µ ≥ µ0 := ν0τ, there is a weak positive solution (uλ,µ, vλ,µ) to (S′)λ,µwith

ϕ1 ≤ uλ,µ ≤ θ1 and ϕ2 ≤ vλ,µ ≤ θ2 in Ω.

Acknowledgments: The authors thank the referee for his/her useful suggestions andcomments.

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[25] J. Furter & M. Grinfeld Local vs. nonlocal interactions in population dynamics, J. Math. Biol. 27 (1989).

65-80.[26] P. H. Rabinowitz, Some global results for nonlinear eigenvalue problems, J. Funct. Anal. 7(1971), 487-

513.

[27] P. Souplet, Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal non-linear source, J. Differential Equations, 153(1999), 374-406.

[28] Zheng, Songmu & M. Chipot, Asymptotic behavior of solutions to nonlinear parabolic equations withnonlocal terms, Asymptot. Anal., 45(2005), no. 3-4, 301-312.

Universidade Federal do Para, Faculdade de Matematica, CEP: 66075-110 Belem - Pa , Brazil.E-mail address: [email protected] and [email protected]

JEPE Vol 2, 2016, p. 267-295

INTERPOLATION INEQUALITIES, NONLINEAR FLOWS, BOUNDARY

TERMS, OPTIMALITY AND LINEARIZATION

JEAN DOLBEAULT, MARIA J. ESTEBAN, AND MICHAEL LOSS

Abstract. This paper is devoted to the computation of the asymptotic boundary termsin entropy methods applied to a fast diffusion equation with weights associated with

Caffarelli-Kohn-Nirenberg interpolation inequalities. So far, only elliptic equations have

been considered and our goal is to justify, at least partially, an extension of the carredu champ / Bakry-Emery / Renyi entropy methods to parabolic equations. This makes

sense because evolution equations are at the core of the heuristics of the method evenwhen only elliptic equations are considered, but this also raises difficult questions on the

regularity and on the growth of the solutions in presence of weights.

We also investigate the relations between the optimal constant in the entropy – en-tropy production inequality, the optimal constant in the information – information pro-

duction inequality, the asymptotic growth rate of generalized Renyi entropy powers under

the action of the evolution equation and the optimal range of parameters for symmetrybreaking issues in Caffarelli-Kohn-Nirenberg inequalities, under the assumption that the

weights do not introduce singular boundary terms at x = 0. These considerations are

new even in the case without weights. For instance, we establish the equivalence of carredu champ and Renyi entropy methods and explain why entropy methods produce opti-

mal constants in entropy – entropy production and Gagliardo-Nirenberg inequalities in

absence of weights, or optimal symmetry ranges when weights are present.

1. Introduction

In this paper we consider the Gagliardo-Nirenberg inequality

(1.1) ‖∇w‖θ2 ‖w‖1−θq+1 ≥ CGN ‖w‖2q ∀w ∈ C∞0 (Rd)

in relation with the nonlinear diffusion equation in Rd, d ≥ 1,

(1.2)∂v

∂t= ∆vm , (t, x) ∈ R+ × Rd ,

2010 Mathematics Subject Classification. Primary: 35K55, 35B06; Secondary: 49K30, 35J60, 35J20.Key words and phrases. Caffarelli-Kohn-Nirenberg inequalities; Gagliardo-Nirenberg inequalities;

weights; optimal functions; symmetry; symmetry breaking; optimal constants; improved inequalities; par-abolic flows; fast diffusion equation; self-similar solutions; asymptotic behavior; intermediate asymptotics;

rate of convergence; entropy methods; carre du champ; Renyi entropy powers; entropy – entropy productioninequality; self-similar variables; bifurcation; instability; rigidity results; linearization; spectral estimates;spectral gap; Hardy-Poincare inequality.

Received 11/11/2016, accepted 14/11/2016.Research partially supported by the Projects STAB and Kibord (J.D.) of the French National Research

Agency (ANR), and by NSF Grant DMS- 1600560 and the Humboldt Foundation (M.L.). Part of this workwas done at the Institute Mittag-Leffler during the fall program Interactions between Partial DifferentialEquations & Functional Inequalities.

268 J. DOLBEAULT, M.J. ESTEBAN, AND M. LOSS

in the fast diffusion regime m ∈ [1− 1/d, 1).We also consider more general interpolation inequalities with weights. With the norm

defined by

‖w‖q,γ :=

(∫Rd

|w|q |x|−γ dx)1/q

,

which extends the case without weight ‖w‖q = ‖w‖q,0, let us consider the family of Caffarelli-Kohn-Nirenberg interpolation inequalities given by

(1.3) ‖w‖2p,γ ≤ Cβ,γ,p ‖∇w‖ϑ2,β ‖w‖1−ϑp+1,γ

in a suitable functional space Hpβ,γ(Rd) obtained by completion of smooth functions with

compact support in Rd \0, w.r.t. the norm given by ‖w‖2 := (p?−p) ‖w‖2p+1,γ +‖∇w‖22,β .Here Cβ,γ,p denotes the optimal constant, the parameters β, γ and p are subject to therestrictions

(1.4) d ≥ 2 , γ − 2 < β <d− 2

dγ , γ ∈ (−∞, d) , p ∈ (1, p?] with p? :=

d− γd− β − 2

and the exponent ϑ is determined by the scaling invariance, i.e.,

ϑ =(d− γ) (p− 1)

p(d+ β + 2− 2 γ − p (d− β − 2)

) .These inequalities have been introduced, among other inequalities, by L. Caffarelli, R. Kohnand L. Nirenberg in [10]. The evolution equation associated with (1.3) is the weightednonlinear diffusion equation

(1.5) vt = |x|γ ∇ ·(|x|−β ∇vm

), (t, x) ∈ R+ × Rd ,

with exponent m = p+12 p ∈ [m1, 1) where

m1 :=2 d− 2− β − γ

2 (d− γ).

Details about the existence of solutions for the above evolution equation and their propertiescan be found in [7].

Our first goal is to give a proof of (1.3) with an integral remainder term using (1.5)whenever the optimal function in (1.3) is radially symmetric. This requires some parabolicestimates. As in the elliptic proof of (1.3) given in [21] and [22], the main difficulty arisesfrom the justification of the integrations by parts. We also investigate why the methodprovides the optimal constant in (1.1) and the optimal range of symmetry in (1.3).

1.1. The symmetry breaking issue. Equality in (1.3) is achieved by Aubin-Talenti typefunctions

w?(x) =(1 + |x|2+β−γ

)−1/(p−1) ∀x ∈ Rd

if we know that symmetry holds, that is, if we know that the equality is achieved amongradial functions. In this case it is not very difficult to check that w? is the unique radialcritical point, up to the transformations associated with the invariances of the equation. Ofcourse, any element of the set of functions generated by the dilations and the multiplicationby an arbitrary constant is also optimal if w? is optimal. Conversely, there is symmetrybreaking if equality in (1.3) is not achieved among radial functions.

NONLINEAR FLOWS, BOUNDARY TERMS & LINEARIZATION 269

Deciding whether symmetry or symmetry breaking holds is a central problem in physics,and it is also a difficult mathematical question. It is well known that symmetric energyfunctionals may have states of lowest energy that may or may not have these symmetries.In our example (1.3) the weights are radial and the functional is invariant under rotation. Inthe language of physics, a broken symmetry means that the symmetry group of the minimizeris strictly smaller than the symmetry group of the functional. For computing the optimalvalue of the functional it is of great advantage that an optimizer is symmetric. The optimalconstant Cβ,γ,p can then be explicitly computed in terms of the Γ function. Otherwise,this is a difficult question which has only numerical solutions so far and involves a delicateenergy minimization as shown in [18, 19]. In other contexts the breaking of symmetry leadsto various interesting phenomena and this is why it is important to decide what symmetrytypes, if any, an optimizer has. Our problem is a model case, in which homogeneity andscaling properties are essential to obtain a clear-cut answer to these symmetry issues.

To show that symmetry is broken in (1.3), one can minimize the associated functionalin the class of symmetric functions and then check whether the value of the functionalcan be lowered by perturbing the minimizer away from the symmetric situation. This is thestandard method, and it has been used to establish that symmetry breaking occurs in (1.3) if

(1.6) γ < 0 and βFS(γ) < β <d− 2

dγ

where

βFS(γ) := d− 2−√

(γ − d)2 − 4 (d− 1) .

In the critical case p = p?, the method was implemented by F. Catrina and Z.-Q. Wangin [14], and the sharp result was obtained by V. Felli and M. Schneider in [28]. The samecondition was recently obtained in the sub-critical case p < p?, in [7]. Here by critical wesimply mean that ‖w‖2p,γ scales like ‖∇w‖2,β . One has to observe that proving symmetrybreaking by establishing the linear instability is a local method, which is based on a painfulbut rather straightforward linearization around the special function w?.

When the minimizer in the symmetric class is stable, i.e., all local perturbations thatbreak the symmetry (in our case, non-radial perturbations) increase the energy, the problemto decide whether the optimizer is symmetric, is much more difficult. It is obvious that, ingeneral, one cannot conclude that the minimizer is symmetric by using a local perturbation,because the minimizer in the symmetric class and the actual minimizer might not be closeeven in any notion of distance adapted to the functional space Hp

β,γ(Rd). In general it isextremely difficult to decide, assuming stability, wether the minimizer is symmetric or not.This is a global problem and not amenable to linear methods.

One general technique for establishing symmetry of optimizers are rearrangement in-equalities and the moving plane method. These methods, however, can only be applied forfunctionals that are in one way or another related to the isoperimetric problem. Outsidethis context there are no general techniques available for understanding the symmetry ofminimizers. This is quite obvious when the weights and the nonlinearity do not cooper-ate to decrease the energy under symmetrization and in these cases moving planes andrelated comparison techniques fail. As usual in nonlinear analysis, advances have alwaysbeen made by studying relevant and non-trivial examples, such as finding the sharp con-stant in Sobolev’s inequality [1, 36], the Hardy-Littlewood-Sobolev inequality [33] or the


logarithmic Sobolev inequality [31], to mention classical examples. In all these cases sym-metrization and the moving plane methods work. Likewise, these techniques can be applied,in the case of Caffarelli-Kohn-Nirenberg inequalities, to prove that symmetry holds if p = p?and β > 0. In fact using symmetrization methods, fairly good ranges have been achievedin [4]. The results, however, are not optimal and can be improved by direct energy andspectral estimates as in [20]. Various perturbation techniques have also been implemented,as in [23, 24], to extend the region of the parameters for which symmetry is known but themethod, at least in [23] and related papers, is not constructive. To establish the optimalsymmetry range in (1.3), and thus determine the sharp constant in the Caffarelli-Kohn-Nirenberg inequalities, a new method had to be designed. What has been proved in [21] inthe critical case p = p?, and extended in [22] to the sub-critical case 1 < p < p?, is thatthe symmetry breaking range given in (1.6) is optimal, i.e., symmetry holds in the regionof admissible parameters that is complementary to the region in which symmetry breakingwas established.

The strategy used in [21, 22] to prove symmetry in the desired parameter range consistsof perturbing the functional about the (unknown) critical point in a particular direction.Notwithstanding what has been said before about perturbations being local, the directiondepends in a non-linear fashion on the critical point. It turns out that this perturbationvanishes precisely if the critical point is a radial optimizer. Of course, this begs the questionhow this direction can be found. In the case at hand it turns out that the functionalis monotone under the action of a particular non-linear flow, and the derivative of thefunctional at a critical point turns out to be strictly negative unless the critical point is aradial optimizer. In carrying out this program one has to perform integration by parts and agood deal of work enters in proving the necessary regularity properties of the critical pointsthat justify these computations.

A more appealing possibility is to use the fact that the non-linear flow, written in suitablevariables, converges to a Barenblatt profile. Starting with any reasonable initial conditionone would, as above, differentiate the functional along the flow and, in a formal fashion,see that that the functional decreases as time tends to infinity towards its minimal value.In addition to having an intuitive approach, one would potentially obtain correction termsto the inequality. This can be carried out, but so far the corresponding computations areformal because they rely on various integrations by parts that have to be justified. It is thefirst purpose of this paper to (partially) fill this gap and establish the optimal symmetryrange using the full picture of entropy methods, at least as far as integration by parts onunbounded domains is concerned. In the case of non constant coefficients, the problemsthat might arise when dealing with the singularities of the weights at x = 0 poses additionaldifficulties which are not studied in this paper, so that our results are still formal in theweighted case. But at least we make what we think is a significant step towards a completeparabolic proof.

Additionally a method based on a parabolic flow provides for free an integral remainderterm, and sheds a fresh light on the method used in [21, 22]. The results in [21, 22] aresurprising in the sense that the locally stable radial optimizers are precisely the global op-timizers. From the flow perspective, however, this can be understood, because stationarityunder the flow characterizes all critical points. The flow monotonously decreases the func-tional associated with (1.3): this also explains why we are able to extend a local property


(the linear stability of radial solutions) to a global stability result (the uniqueness, up to theinvariances, of the critical point).

The parabolic approach is based on an inequality between the Fisher information andits time derivative, i.e., the production of Fisher information, and provides us with theoptimal range of symmetry. This is a remarkable fact, common to various nonlinear diffusionequations, that can be explained as follows. When there are no weights, the optimality inthe entropy – entropy production inequality is achieved through a linearization which alsoprovides us with large-time asymptotic rates of convergence. As a consequence the bestconstant in the inequality is equal to the optimal constant which arises from the computationof Fisher information – production of Fisher information (see [3, 32]), and which is alsoreached in the large-time asymptotics. With general weights, the picture is actually slightlymore complex, as discussed in [7, 8], but by studying the large time asymptotics, one can atleast understand why the optimal symmetry range is achieved in our flow approach. Thiswill be detailed in the last section of this paper. One more comment has to be done atthis point. Quite generally computations based on the second derivative of an entropy withrespect to the time, along a flow, are known as Bakry-Emery or carre du champ methods.The geometry or the presence of an external potential usually allows us to relate a positivityestimate of the curvature or a uniform convexity bound of the potential with a rate of decayof the Fisher information. In the Renyi entropy powers approach, as can be seen from [27],there is no such bound neither on the curvature nor on the potential: what matters isonly the fact that we apply a nonlinear flow to some nonlinear quantities. The interplayof the various quantities that are generated by integrations by parts and hitting powers offunctions when taking derivatives delivers nontrivial coefficients that allows to relate theFisher information with its time derivative, i.e., the production of Fisher information. Asexplained below, in order to control the boundary terms, we are dealing with the moreclassical setting of relative entropies and self-similar variables. By making the link withRenyi entropy powers, we finally get rid of any geometry or convexity requirements on anexternal potential. Although this is a side remark of our paper, we believe that it is ofinterest by itself.

In [21, 22], we analyzed the symmetry properties not only of the extremal functionsof (1.3), but also of all positive solutions in Hp

β,γ(Rd) of the corresponding Euler-Lagrangeequations, i.e., up to a multiplication by a constant and a dilation, of

(1.7) −div(|x|−β ∇w

)= |x|−γ

(w2p−1 − wp

)in Rd \ 0 .

Theorem 1. [21, 22] Under Condition (1.4) assume that

(1.8) either β ≤ βFS(γ) ∀ γ < 0 , or γ ≥ 0 .

Assume that d ≥ 2 and (β, γ) 6= (0, 0). Then all positive solutions of (1.7) in Hpβ,γ(Rd) are

radially symmetric and, up to a scaling and a multiplication by a constant, equal to w?.

Theorem 1 determines the optimal symmetry range, as shown by (1.6). Our first result isactually a more precise version of Theorem 1, under a regularity assumption at x = 0 thatstill has to be proved.

1.2. Main result. The Renyi entropy power functional relates (1.2) with (1.1). We adopta similar approach in the weighted case. Let us consider the derivative of the generalized


Renyi entropy power functional, defined up to a multiplicative constant as

G[v] :=

(∫Rd

vm |x|−γ dx)σ−1 ∫

Rd

v |∇P|2 |x|−β dx

where P := m1−m vm−1 is the pressure variable, σ := m−mc

1−m and mc := d−2−βd−γ . The exponent

m is the one which appears in (1.5), and it is such that m ∈ [m1, 1). With

n := 2d− γ

β + 2− γ,

we observe that m1 = 1−1/n, so that, when (β, γ) = (0, 0), we have n = d and m1 = 1−1/d.As we will see later using scalings, n plays the role of a dimension. With µ? := (d− γ) (m−mc), µ := n (m−mc), κ :=

(2m1−m

)1/µ, the function

(1.9) v?(t, x) :=1

κn(µ t)n/µB?(

x

κµ/µ? (µ t)1/µ?

)where B?(x) :=

(1 + |x|2+β−γ

)−1/(1−m) ∀x ∈ Rd ,∀ t > 0

is a self-similar solution of (1.5). Here B? is the Barenblatt profile with mass M? :=∫Rd B? |x|−γ dx = ‖B?‖1,γ . A simple computation shows that G[v?] does not depend on

t > 0 and ‖v?‖1,γ = M?. Theorem 1 is equivalent to prove that

G[v] ≥ G[v?]

for any nonnegative function v such that ‖v‖1,γ = ‖v?‖1,γ = M?, if (1.4) and (1.8) hold.

In [21, 22] we proved Theorem 1 using elliptic methods and well chosen multipliers inspiredby the heuristics arising from the parabolic equation (1.5). However, so far, we were notable to deal with the time-dependent solution by lack of estimates for justifying integrationsby parts and this is why we only worked with the elliptic equation. In this paper westudy the evolution problem. When (β, γ) 6= (0, 0), we are not yet able to deal with thepossible singularities of the solutions to (1.5) at the origin, but otherwise we can handle allintegrations by parts. The method is based on the approximation of the solution on a ballin self-similar variables, with no-flux boundary conditions on the boundary, and then byletting the radius of the ball go to infinity. By using parabolic methods to prove Theorem 1,we obtain improvements of (1.1) and (1.3), with a remainder term computed as an integralterm along the flow.

Let us define

(1.10) h(t) :=

(1 +

2m

1−mµ t

)1/µ

∀ t ≥ 0 , with µ = 22 + β − d+m (d− γ)

2 + β − γ.

Theorem 2. Let d ≥ 2. Under Condition (1.4), if

either β < βFS(γ) ∀ γ ≤ 0 , or γ > 0 ,

then there exists a positive constant C depending only on β, γ and d such that the followingproperty holds. If v0 satisfies ‖v0‖1,γ = M? and if there exist two positive constants C1 andC2 such that

(1.11)(C1 + |x|2+β−γ

)−1/(1−m) ≤ v0(x) ≤(C2 + |x|2+β−γ

)−1/(1−m) ∀x ∈ Rd ,


then for any positive solution of (1.5) with initial datum v0 we have

G[v(t, ·)] ≥ G[v?] + C∫ ∞t

h(s)3µ−2∫Rd

vm(s, x)|∇ωvm−1(s, x)|2

|x|4|x|γ−2β dx ds ∀ t ≥ 0 ,

if v is smooth at x = 0 for any t ≥ 0. Here h and µ are defined by (1.10).

Here ω = x|x| , and ∇ω denotes the gradient with respect to angular derivatives. The

explicit expression of C and further remainder terms will be given in Theorem 3, in Section 3.The condition that v is smooth at x = 0 means that integrations by parts can be done withoutpaying attention to the weight in a neighborhood of x = 0. Condition (1.11) may seem veryrestrictive, but it is probably not since, as explained in the introduction of [7], it is expectedthat for any smooth initial datum v0 with finite mass, condition (1.11) will be satisfied byany solution after some finite time t > 0. At least this is known from [9] when (β, γ) = (0, 0).

The mass normalization ‖v0‖1,γ = M? simplifies the computations but the result caneasily be generalized to any positive mass. The smoothness condition at x = 0 simplymeans that all computations can be carried in Rd \Bε where Bε is the ball of radius ε > 0centered at the origin, and that the boundary terms on ∂Bε vanish as ε → 0. Up to thesmoothness assumption, the result of Theorem 2 is stronger than the result of Theorem 1.Indeed, if m and p are related by p = 1

2m−1 and if w solves (1.7), then we have that

d

dtG[v(t, ·)] = 0

at t = 0, where v is the solution of (1.5) with initial datum v0 = w2p. This is enough toconclude that R[v0] = 0 which, as shown in [22], implies the result of Theorem 1.

1.3. Outline of the paper. Our goals are:

(1) To give a proof of the monotonicity of G for the solution to the evolution equationand establish the remainder term of Theorem 3 under the smoothness assumptionof the solutions of (1.5) at x = 0.

(2) To study the outer boundary terms by using self-similar variables and an approx-imation scheme on large balls. The novelty here is that we are able to justify theintegrations by parts away from the origin for the solution to the evolution prob-lem (1.5).

(3) To study the role of large time asymptotics and of the linearized problem, andconsequently explain why the method in [21, 22] determines the optimal range forsymmetry breaking. Corresponding results are stated in Section 4.

Weights induce various technicalities, so that, in order to emphasize the strategy, wealso consider the case without weights. In that case Theorem 3 is rigorous without anysmoothness assumption on the solution at x = 0. This is not by itself new, but at least twoobservations are new:• The equivalence of the Renyi entropy powers introduced by G. Savare and G. Toscani in [35]and the computation based on the relative Fisher information in self-similar variables,• The characterization of the optimality case in the Fisher information – production of Fisherinformation inequality, which explains why computations based on flows provide us withthe optimal constant in the entropy – entropy production inequality and, as a consequence,in the Gagliardo-Nirenberg inequality (1.1), when there are no weights.


2. Gagliardo-Nirenberg inequalities, fast diffusion and boundary terms

To start with and clarify our strategy, let us consider the case of the Gagliardo-Nirenberginequality (1.1), without weights, i.e., (β, γ) = (0, 0) on the Euclidean space Rd. In thissection, integrations by parts will be fully justified. We recall that, when (β, γ) = (0, 0),q = 1/(2m − 1) is in the range 1 < q ≤ d/(d − 2), which means that m ∈ [m1, 1) withm1 = 1− 1/d and n = d. Here we implicitly assume that d ≥ 3. The cases d = 1 and d = 2can also be covered, with q ∈ (1,+∞), and the additional restriction that m > 0 if d = 1.

2.1. A proof of Gagliardo-Nirenberg inequalities based on the Renyi entropypowers.

2.1.1. Variation of the Fisher information along the flow. Let v be a smooth function onRd and define the Fisher information as

I[v] :=

∫Rd

v |∇P|2 dx with P =m

1−mvm−1 .

Here P is the pressure variable. If v solves (1.2), in order to compute I′ := ddt I[v(t, ·)], we

will use the fact that

(2.1)∂P

∂t= (1−m)P∆P− |∇P|2 .

Using (1.2) and (2.1), we can compute

I′ =d

dt

∫Rd

v |∇P|2 dx =

∫Rd

∂v

∂t|∇P|2 dx+ 2

∫Rd

v∇P · ∇∂P∂t

dx

=

∫Rd

∆(vm) |∇P|2 dx+ 2

∫Rd

v∇P · ∇(

(m− 1)P∆P + |∇P|2)dx .

The key computation relies on integrations by parts and requires a sufficient decay of thesolutions as |x| → +∞ to ensure that all integrals are finite, including the boundary integrals.In the next result, we focus on the algebra of the integrations by parts used to deal withthe r.h.s. of the above equality and time plays no role. How to apply this computation to asolution of the parabolic problem will be explained afterwards.

Lemma 1. Assume that v is a smooth and rapidly decaying function on Rd, as well as itsderivatives. If we let P := m

1−m vm−1, then we have

(2.2)

∫Rd

∆(vm) |∇P|2 dx+ 2

∫Rd

v∇P · ∇(

(m− 1)P∆P + |∇P|2)dx

= − 2

∫Rd

vm(‖D2P‖2 − (1−m) (∆P)2

)dx .


Proof. We follow the computation of [27] or [35, Appendix B].∫Rd

∆(vm) |∇P|2 dx+ 2

∫Rd

v∇P · ∇(

(1−m)P∆P− |∇P|2)dx

=

∫Rd

vm ∆ |∇P|2 dx+ 2 (1−m)

∫Rd

v P∇P · ∇∆P dx

+ 2 (1−m)

∫Rd

v |∇P|2 ∆P dx− 2

∫Rd

v∇P · ∇ |∇P|2 dx

= −∫Rd

vm ∆ |∇P|2 dx+ 2 (1−m)

∫Rd

v P∇P · ∇∆P dx+ 2 (1−m)

∫Rd

v |∇P|2 ∆P dx

where the last line is given by the observation that v∇P = −∇(vm) and an integration byparts:

−∫Rd

v∇P · ∇ |∇P|2 dx =

∫Rd

∇(vm) · ∇ |∇P|2 dx = −∫Rd

vm ∆ |∇P|2 dx .

1) Using the elementary identity

1

2∆ |∇P|2 = ‖D2P‖2 +∇P · ∇∆P ,

we get that ∫Rd

vm ∆ |∇P|2 dx = 2

∫Rd

vm ‖D2P‖2 dx+ 2

∫Rd

vm∇P · ∇∆P dx .

2) Since v∇P = −∇(vm), an integration by parts gives∫Rd

v |∇P|2 ∆P dx = −∫Rd

∇(vm) · ∇P∆P dx =

∫Rd

vm (∆P)2 dx+

∫Rd

vm∇P · ∇∆P dx

and with v P = m1−m vm we find that

2 (1−m)

∫Rd

v P∇P · ∇∆P dx+ 2 (1−m)

∫Rd

v |∇P|2 ∆P dx

= 2 (1−m)

∫Rd

vm (∆P)2 dx+ 2

∫Rd

vm∇P · ∇∆P dx .

Collecting terms establishes (2.2).

The result of Lemma 1 can be applied to a solution of (1.2).

Corollary 1. If v solves (1.2) with initial datum v(x, t = 0) = v0(x) ≥ 0 such that∫Rd v0 dx = M?,

∫Rd v

m0 dx < +∞ and

∫Rd |x|2 v0 dx < +∞, then P = m

1−m vm−1 solves (2.1)and

I′ = − 2

∫Rd

vm(‖D2P‖2 − (1−m) (∆P)2

)dx .


Proof. If we perform the same computations as in the proof of Lemma 1 in a ball BR (insteadof R2), we find additional boundary terms

I′ =d

dt

∫BR

v |∇P|2 dx

= − 2

∫BR

vm(‖D2P‖2 + (m− 1) (∆P)2

)dx

+

∫∂BR

ω ·(∇(vm) |∇P|2 + vm∇ |∇P|2 + 2 (m− 1) vm∇P∆P

)dσ .

Here dσ denotes the measure induced on ∂BR by Lebesgue’s measure. It follows from [5,Theorem 2, (iii)] that these boundary terms vanish as R → +∞, which completes theproof.

2.1.2. Concavity of the Renyi entropy powers and consequences. Lemma 1 establishes thatI′ ≤ 0 if m ≥ m1 with m1 = 1− 1/d. Indeed we have the identity

‖D2P‖2 = 1d (∆P)2 +

∥∥D2P− 1d ∆P Id

∥∥2and, as a consequence, we obtain

I′ = − 2

∫Rd

vm(∥∥D2P− 1

d ∆P Id∥∥2 dx− 2 (m−m1)

∫Rd

vm (∆P)2 dx .

In the sub-critical range m1 < m < 1, let us define the entropy as E =∫Rd v

m dx and observethat, if v solves (1.2),

E′ = (1−m) I .

Next we introduce the Renyi entropy power given by F = Eσ with

σ :=2

d

1

1−m− 1 .

Using Lemma 1, we find that F′′ = (Eσ)′′

can be computed as

1

σ (1−m)E2−σ F′′ = (1−m) (σ − 1)

(∫Rd

v |∇P|2 dx)2

− 2

(1

d+m− 1

)∫Rd

vm dx

∫Rd

vm (∆P)2 dx

− 2

∫Rd

vm dx

∫Rd

vm∥∥D2P− 1

d ∆P Id∥∥2 dx .

Using v∇P = −∇(vm), we know that∫Rd

v |∇P|2 dx = −∫Rd

∇(vm) · ∇P dx =

∫Rd

vm ∆P dx ,

which implies that∫Rd

vm

∣∣∣∣∣∆P−(∫

Rd v |∇P|2 dx)2∫

Rd vm dx

∣∣∣∣∣2

dx =

∫Rd

vm |∆P|2 dx−(∫

Rd v |∇P|2 dx)2∫

Rd vm dx.


Hence we get that F′′ = −σ (1−m)R, where

(2.3) R[v] := (σ − 1) (1−m)E[v]σ−1∫Rd

vm∣∣∣∣∆P−

∫Rd v |∇P|2 dx∫

Rd vm dx

∣∣∣∣2 dx+ 2E[v]σ−1

∫Rd

vm∥∥D2P− 1

d ∆P Id∥∥2 dx .

This proves that σ (1−m)G = F′ is nonincreasing (so that the function t 7→ F(t) is concave).

2.1.3. Large time asymptotics and consequences. The large time behavior of the solutionof (1.2) is governed by the source-type Barenblatt solutions

v?(t, x) :=1

κd(µ t)d/µB?(

x

κ (µ t)1/µ

)where µ := 2 + d (m− 1) = d (m−mc) , κ :=

( 2m

1−m

)1/µ,

where B? is the Barenblatt profile

B?(x) :=(1 + |x|2

)1/(m−1)with mass M? :=

∫Rd B? dx. We recall that (β, γ) = (0, 0) and, as a consequence, n = d and

µ = µ?: notations are consistent with those of (1.9). To obtain the expression of v?, it isstandard to rephrase the evolution equation (1.2) in self-similar variables as follows. If weconsider a solution v of (1.2) and make the change of variables

(2.4) v(t, x) =1

κdRdu(τ,

x

κR

)where

dR

dt= R1−µ , R(0) = R0 = κ−1 and τ(t) := 1

2 log

(R(t)

R0

),

then the function u solves

(2.5)∂u

∂τ+∇ ·

[u(∇um−1 − 2x

) ]= 0 , (τ, x) ∈ R+ × Rd .

It is straightforward to check that B? is a stationary solution of (2.5) and it is well knownthat B? attracts all nonnegative solutions with mass M? at least if m ∈ (mc, 1). Since

R(t) = (Rµ0 + µ t)1/µ

= (µ t)1/µ(1 + o(1)

)as t→ +∞ ,

this means that

v(t, x) ∼ v?(t, x) as t→ +∞ .

We refer to [5] for details and further references. As a consequence, we obtain that

limt→∞

G[v(t, ·)] = limt→∞

G[v?(t, ·)] = G[B?] ,

because

G[v] =

(∫Rd

vm dx

)σ−1 ∫Rd

v |∇P|2 dx

defined in Section 1.2 is scale invariant. The fact that G[v(t, ·)] is a nonincreasing functionof t means that

G[v0] ≥ G[v(t, ·)] ≥ G[B?]


for any t ≥ 0, which is exactly equivalent to (1.1) as noted in [27]. By keeping track of theremainder term, we get an improved inequality.

Proposition 1. Under the assumptions of Corollary 1, with R defined by (2.3), for all t ≥ 0,we have

G[v0] = G[v?] +

∫ ∞0

R[v(t, ·)] dt .

If we write vm−1/20 = M

m−1/2? w/ ‖w‖2q with q = 1

2m−1 , then this inequality amounts to

(2m−1)24m2

(G[v0]− G[B?]

)=‖∇w‖22 ‖w‖

2 (1−θ)/θq+1

‖w‖2/θ2q

− C2/θGN ≥ 0

with

CGN :=(

(2m−1)24m2 G[B?]

)θ/2.

In this way, we recover the Gagliardo-Nirenberg inequality that was established in [15], withan additional remainder term.

Corollary 2. If 1 < q ≤ dd−2 and d ≥ 3, or q > 1 and d = 1 or d = 2, then (1.1) holds with

optimal constant CGN as above. Moreover, equality holds in (1.1) if and only if w2q = B?,up to translations, multiplication by constants and scalings. With the above notations, onehas the improved inequality

‖∇w‖22 ‖w‖2 (1−θ)/θq+1 − C

2/θGN ‖w‖

2/θ2q = (2m−1)2

4m2 ‖w‖2/θ2q

∫ ∞0

R[v(t, ·)] dt ∀w ∈ Hp0,0(Rd)

if v solves (1.2) with initial datum v0 such that vm−1/20 = M

m−1/2? w/ ‖w‖2q and q =

1/(2m− 1).

Proof. The only point that deserves a discussion is the equality case. Solving simultaneously

∆P−∫Rd v |∇P|2 dx∫

Rd vm dx= 0 and D2P− 1

d ∆P Id = 0

shows that P(x) = a+b |x−x0|2 for some real constants a and b, and for some x0 ∈ Rd.

2.2. The entropy – entropy production method in rescaled variables. Here wefollow the computations of [26, Section 2] (also see [32, Proof of Theorem 2.4, pp. 33-36])and emphasize the role of the boundary terms when the problem is restricted to a ball.The major advantage of self-similar variables is that we control the sign of these boundaryterms. Such computations can be traced back to [12, 11, 13] and are directly inspired bythe carre du champ or Bakry-Emery method introduced in [2]. The algebra is slightly moreinvolved than the one of Section 2.1 because of the presence of a drift term. The mainadvantage of this framework is that boundary terms have a definite sign, which is importantin preparation of the computations of Section 3, in the weighted case.

For a while we will consider Eq. (2.5) written on ball BR instead of Rd, with no-fluxboundary condition. Let u = u(τ, x) be a solution of

(2.6)∂u

∂τ+∇ ·

[u(∇um−1 − 2x

) ]= 0 τ > 0 , x ∈ BR


where BR is a centered ball in Rd with radius R > 0, and assume that u satisfies no-fluxboundary conditions (

∇um−1 − 2x)· ω = 0 τ > 0 , x ∈ ∂BR .

On ∂BR, ω = x/|x| denotes the unit outgoing normal vector to ∂BR. We define

z(τ, x) := ∇um−1 − 2x

so that Eq. (2.6) can be rewritten with its boundary conditions as

∂u

∂τ+∇ · (u z) = 0 in BR , z · ω = 0 on ∂BR .

We recall that m ∈ [m1, 1) where m1 = 1− 1/d. It is straightforward to check that

∂z

∂τ= (1−m)∇

(um−2∇ · (u z)

).

With these definitions, the time-derivative of relative Fisher information

IR[u] :=

∫BR

u |z|2 dx =

∫BR

u∣∣∇um−1 − 2x

∣∣2 dxcan be computed as

d

dτ

∫BR

u |z|2 dx =

∫BR

∂u

∂τ|z|2 dx+ 2

∫BR

u z · ∂z∂τ

dx

=

∫BR

u z · ∇ |z|2 dx− 2

∫BR

u z · ∇(z · ∇um−1 + (m− 1)um−1∇ · z

)dx

using the above equations. By definition of z, we have

d

dτ

∫BR

u |z|2 dx

=

∫BR

u z · ∇ |z|2 dx− 2

∫BR

u z · ∇(|z|2 + 2 z · x+ (m− 1)um−1∇ · z

)dx

= −∫BR

u z · ∇ |z|2 dx− 2

∫BR

u z · ∇(2 z · x+ (m− 1)um−1∇ · z

)dx

= −∫BR

(m−1m ∇um − 2xu

)· ∇ |z|2 dx

− 2

∫BR

u z · ∇(2 z · x+ (m− 1)um−1∇ · z

)dx .


Let us denote by dσ the measure induced by Lebesgue’s measure on ∂BR. Taking intoaccount the boundary condition z · ω = 0 on ∂BR, we integrate by parts and get

d

dτ

∫BR

u |z|2 dx

=

∫BR

m−1m um ∆ |z|2 dx+ 2

∫BR

ux · ∇ |z|2 dx− 4

∫BR

u z · ∇(z · x) dx

+

∫∂BR

1−mm um

(ω · ∇ |z|2

)dσ

− 2 (m− 1)

∫BR

(u z · ∇um−1 (∇ · z) + um z · ∇ (∇ · z)

)dx

=

∫BR

m−1m um ∆ |z|2 dx+ 2

∫BR

ux · ∇ |z|2 dx− 4

∫BR

u z · ∇(z · x) dx

+

∫∂BR

1−mm um

(ω · ∇ |z|2

)dσ

− 2 (m− 1)

∫BR

(−m−1m um (∇ · z)2 + 1

m um z · ∇(∇ · z))dx .

Using the elementary identity

1

2∆ |∇q|2 =

∥∥D2q∥∥2 +∇q · ∇∆q ,

with q := um−1 − 1− |x|2 so that z = ∇q, we get that∫BR

um ∆ |z|2 dx = 2

∫BR

um∥∥D2q

∥∥2 dx+ 2

∫BR

um z · ∇(∇ · z) dx .

Moreover, since z · ω = 0 on ∂BR, we know from [29, Lemma 5.2], [34, Proposition 4.2]or [30] (also see [25] or [32, Lemma A.3]) that∫

∂BR

um(ω · ∇|z|2

)dσ ≤ 0 .

Therefore, we have shown that

d

dτ

∫BR

u |z|2 dx ≤ 2m− 1

m

∫BR

um(∥∥D2q

∥∥2 + (m− 1) (∆q)2)dx

+ 2

∫BR

ux · ∇ |z|2 dx− 4

∫BR

u z · ∇(z · x) dx

= − 21−mm

∫BR

um(∥∥D2q

∥∥2 − (1−m) (∆q)2)dx− 4

∫BR

u |z|2 dx

where, in the last step, we use the fact that∂zj∂xi

= ∂zi∂xj

to write that

2

∫BR

ux · ∇ |z|2 dx− 4

∫BR

u z · ∇(z · x) dx = − 4

∫BR

u |z|2 dx .

For any m ∈ [m1, 1), this establishes that

d

dτ

∫BR

u |z|2 dx+ 4

∫BR

u |z|2 dx ≤ − 21−mm

∫BR

um(∥∥D2q

∥∥2 − (1−m) (∆q)2)dx ≤ 0 .


This allows us to prove a result similar to the one of Corollary 2. The relative entropy

ER[u] := − 1

m

∫BR

(um − Bm? − mBm−1? (u− B?)

)dx

is such thatd

dτEr[u(τ, ·)] = −IR[u(τ, ·)]

according to [15, 5]. We deduce that

d

dτ

(IR[u(τ, ·)]− 4 ER[u(τ, ·)]

)≤ − 2

1−mm

∫BR

um(∥∥D2q

∥∥2 − (1−m) (∆q)2)dx .

It turns out that for all τ ≥ 0,

IR[u0]− 4 ER[u0] ≥ IR[u(τ, ·)]− 4 ER[u(τ, ·)] ≥ IR[B?]− 4 ER[B?] = 0 .

For functions on Rd, let us define the relative entropy

E [u] := − 1

m

∫Rd

(um − Bm? − mBm−1? (u− B?)

)dx ,

the relative Fisher information

I[u] :=

∫Rd

u |z|2 dx =

∫Rd

u∣∣∇um−1 − 2x

∣∣2 dxand

R[u] := 21−mm

∫Rd

um(∥∥D2q

∥∥2 − (1−m) (∆q)2)dx .

Proposition 2. If 1 < q ≤ dd−2 and d ≥ 3, or q > 1 and d = 1 or d = 2, then (1.1)

holds with optimal constant CGN as above. Moreover, equality holds in (1.1) if and onlyif w2q = B?, up to translations, multiplication by constants and scalings. With the abovenotations, one has the improved inequality

I[u0]− 4 E [u0] ≥∫ ∞0

R[u(τ, ·)] dτ ,

if u solves (2.5) with initial datum u0 ∈ L1+(Rd) such that um0 and u0 |x|2 are integrable.

Proof. To prove the result, one has to approximate a solution of (2.5) by the solution of (2.6)on the centered ball BR of radius R, and extend it to Rd \ BR by B?. By passing to thelimit as R→ +∞, the result follows.

To conclude this subsection, let us list a few comments.

(1) The method of entropy – entropy production method in rescaled variables is not asaccurate as the method of Renyi entropy powers. Boundary terms have a sign andcan be dropped, but at the end we get an inequality instead of an equality. On theother hand, the method is very robust and applicable not only to large balls butalso to any convex domain. This is the method that we shall extend to the case ofthe weighted evolution equation in Section 3.


(2) If we replace u by w2q, with 2q = m− 12 , the inequality I[u]− 4 E [u] ≥ 0 amounts

to

4 (m−1)2(2m−1)2 ‖∇w‖

22 +

4

m

(1− d (1−m)

)‖w‖q+1

q+1

≥ 4 (m−1)2(2m−1)2

∥∥∥∇Bm−1/2?

∥∥∥22

+4

m

(1− d (1−m)

) ∥∥∥Bm−1/2?

∥∥∥q+1

q+1.

This is a non scale-invariant, but optimal, form of the Gagliardo-Nirenberg inequal-

ity, as shown in [15]. The inequality written with w replaced by Mm−1/2? w/ ‖w‖2q

is, after optimization under scalings, equivalent to inequality (1.1). However R[u] isnot invariant under scaling. In order to replace the improved inequality of Propo-sition 2 by an improved inequality similar to the one of Corollary 2, one shoulduse delicate scaling properties involving the best matching Barenblatt instead of B?.See [26, 27] for further considerations in this direction.

(3) An interesting remark which is important in our computations and results is thatfor any function p,∥∥D2(p + |x|2

)∥∥2 − 1d

(∆(p + |x|2

) )2=∥∥D2p

∥∥2 − 1d (∆p)2 =

∥∥D2p− 1d ∆p Id

∥∥2 .As a consequence, the remainder terms in the entropy – entropy production methodin rescaled variables are very similar to the remainder terms in the Renyi entropypowers method, and the |x|2 term plays essentially no role.

2.3. The two methods are identical. The computations of Sections 2.1 and 2.2 looksimilar and are actually the same, if we do not take into consideration the boundary terms.Let us give some details.

2.3.1. A computation based on the time-dependent rescaling. If v is a solution of (1.2), thenthe function u defined by the time-dependent rescaling (2.4) solves (2.5). With the choiceR0 = 1/κ, the initial data are identical

u(τ = 0, ·) = u0 = v0 = v(t = 0, ·) .

As in Section 2.2, let us define z(x, τ) := ∇um−1 − 2x and consider the relative Fisherinformation

I[u] :=

∫Rd

u |z|2 dx =

∫Rd

u∣∣∇um−1 − 2x

∣∣2 dx=

∫Rd

u∣∣∇um−1∣∣2 dx+ 4

∫Rd

u |x|2 dx− 4 1−mm d

∫Rd

um dx .

• If m = m1 = 1− 1d , then 1−m

m d = 1m and, by undoing the time-dependent rescaling (2.4),

we obtain that ∫Rd

u∣∣∇um−1∣∣2 dx =

(1−mm

)2 ∫Rd

v |∇P|2 dx

with P = m1−m vm−1, because µ = 1 and so, since dt

dτ = 1−mm e2 τ , we get that

d

dτ

∫Rd

u∣∣∇um−1∣∣2 dx = 1−m

m e2 τd

dt

∫Rd

v |∇P|2 dx


is nonpositive by Corollary 1. By the computations of Section 2.1, we obtain that

d

dτI[u(τ, ·)] = 1−m

m e2 τd

dt

∫Rd

v |∇P|2 dx+ 4d

dτ

∫Rd

(u |x|2 − 1

mum)dx

≤ 4d

dτ

∫Rd

(u |x|2 − 1

mum)dx = − 4 I[u(τ, ·)] .

• If m ∈ [m1, 1), we observe that∫Rd

u∣∣∇um−1∣∣2 dx =

(1−mm

)2e4 (µ−1) τ

∫Rd

v |∇P|2 dx

and from R(t) = R0 e2 τ , we deduce that dt

dτ = 2Rµ = 1−mm e2µ τ . A computation similar to

the case m = m1 gives

d

dτ

(I[u(τ, ·)]− 4 E [u(τ, ·)]

)=(1−mm

)2e4 (µ−1) τ

(1−mm e2µ τ

d

dt

∫Rd

v |∇P|2 dx+ 4 (µ− 1)

∫Rd

v |∇P|2 dx)

+ 4 dm (m−m1)

d

dτ

∫Rd

um dx

with P = m1−m vm−1. Using the fact that

d

dτ

∫Rd

um dx = m

∫Rd

u∣∣∇um−1∣∣2 dx− 2 d (1−m)

∫Rd

um dx

on the one hand, and Corollary 1 on the other hand, we end up with

d

dτ

(I[u(τ, ·)]− 4 E [u(τ, ·)]

)= −R?[u(τ, ·)] ,

where

R?[u] := 2 e4 (µ−1) τ(

1−mm e2µ τ

∫Rd

vm(∥∥D2vm−1

∥∥2 − (1−m)(∆vm−1

)2 )dx

− 2 (µ− 1)

∫Rd

v∣∣∇vm−1∣∣2 dx)

− 4 dm (m−m1)

(m

∫Rd

u∣∣∇um−1∣∣2 dx− 2 d (1−m)

∫Rd

um dx

).

Notice that R?[u] does not depend on τ explicitly because, according to the time-dependentrescaling (2.4),

R?[u] = 2 1−mm

∫Rd

um∥∥D2um−1 − 1

d ∆um−1 Id∥∥2 dx

+ 2 (m−m1) 1−mm

∫Rd

um(∆um−1

)2dx− 8 d (m−m1)

∫Rd

u∣∣∇um−1∣∣2 dx

+ 8 d2

m (m−m1) (1−m)

∫Rd

um dx .


This can be rewritten as

R?[u] = 2 1−mm

∫Rd

um∥∥D2um−1 − 1

d ∆um−1 Id∥∥2 dx

+ 2 (m−m1) 1−mm

∫Rd

um∣∣∆um−1 − 2 d

∣∣2 dx .With these considerations, we obtain an improvement of Proposition 2, which goes as follows.

Proposition 3. If 1 < q ≤ dd−2 and d ≥ 3, or q > 1 and d = 1 or d = 2, then (1.1)

holds with optimal constant CGN as above. Moreover, equality holds in (1.1) if and onlyif w2q = B?, up to translations, multiplication by constants and scalings. With the abovenotations, one has the improved inequality

I[u0]− 4 E [u0] =

∫ ∞0

R?[u(τ, ·)] dτ ,

if u solves (2.5) with initial datum u0 ∈ L1+(Rd) such that um0 and u0 |x|2 are integrable.

2.3.2. A direct computation in rescaled variables. Although this is equivalent to the compu-tations of the previous subsection, it is instructive to redo the computation in the rescaledvariables. Let us define p := um−1 and observe that it solves

∂p

∂τ= (m− 1) p∆p− |∇p|2 + 2x · ∇p + 2 d (m− 1) p .

For simplicity, we consider only the case m = m1 and observe that

d

dτ

∫Rd

u |∇p|2 dx

=1−mm

∫Rd

∆(um) |∇p|2 dx+ 2

∫Rd

u∇p · ∇(

(m− 1) p∆p + |∇p|2)dx

+ 2

∫Rd

∇ · (xu) |∇p|2 dx+ 4

∫Rd

u∇p · ∇(x · ∇p + d (m− 1) p

)dx

=1−mm

∫Rd

∆(um) |∇p|2 dx+ 2

∫Rd

u∇p · ∇(

(m− 1) p∆p + |∇p|2)dx

− 4

∫Rd

u |∇p|2 dx

because d (m− 1) = −1 and 2∇ · (xu) |∇p|2 + 4u∇p · ∇ (x · ∇p) = 2∇ ·(xu |∇p|2

). If we

write that P := m1−m p, then the r.h.s. can be rewritten as

1−mm

∫Rd

∆(um) |∇p|2 dx+ 2

∫Rd

u∇p · ∇(

(m− 1) p∆p + |∇p|2)dx

=

(1−mm

)3 [∫Rd

∆(um) |∇P|2 dx+ 2

∫Rd

u∇P · ∇(

(m− 1)P∆P + |∇P|2)dx

]


and we are back to the computations of Section 2.1. Using (2.2) with v = u and P =m

1−m vm−1, we obtain that

d

dτI[u(τ, ·)] + 4 I[u(τ, ·)] =

d

dτ

∫Rd

u |∇p|2 dx+ 4

∫Rd

u |∇p|2 dx

= − 21−mm

∫Rd

um(∥∥D2p− 1

d ∆p Id∥∥2 dx

if u solves (2.5).This concludes the section on Gagliardo-Nirenberg inequalities (1.1) and fast diffusion

equations (1.2). So far proofs are rigorous. From now on, we shall work with weights, thatis, on Caffarelli-Kohn-Nirenberg inequalities (1.3) and weighted parabolic equations (1.5),and assume that integrations by parts can be carried out at x = 0 without any precaution.Corresponding results will henceforth be formal.

3. The case of the weighted diffusion equation

In order to study the weighted evolution equation (1.5), it is convenient to introduce asin [21, 22] a change of variables which amounts to rephrase our problem in a space of higher,artificial dimension n ≥ d (here n is a dimension at least from the point of view of thescaling properties), or to be precise to consider a weight |x|n−d which is the same in allnorms. With

α = 1 +β − γ

2and n = 2

d− γβ + 2− γ

,

we claim that Inequality (1.3) can be rewritten for a function W such that

w(x) = W(|x|α−1 x

)∀x ∈ Rd

as

‖W‖2p,δ ≤ Kα,n,p ‖DαW‖ϑ2,δ ‖W‖1−ϑp+1,δ , ∀W ∈ Hp

δ,δ(Rd) ,

with the notations

δ = d− n , r = |x| , ω =x

r, DαW =

(α∂rW, r

−1∇ωW),

where ∂r = ∂/∂r and ∇ω is the gradient in the angular derivatives, ω ∈ Sd−1. The optimalconstant Kα,n,p is explicitly computed in terms of Cβ,γ,p and the condition (1.4) is equivalentto

d ≥ 2 , α > 0 , n > d and p ∈ (1, p?] with p? =n

n− 2.

By our change of variables, w? is changed into

W?(x) :=(1 + |x|2

)−1/(p−1) ∀x ∈ Rd .

The symmetry condition (1.8) now reads

α ≤ αFS with αFS :=

√d− 1

n− 1.

For any α ≥ 1, note that the operator Dα can be rewritten as

Dα = ∇+ (α− 1)x

|x|2(x · ∇) = ∇+ (α− 1)ω ∂r .


If D∗α is the adjoint operator of Dα , with respect to the measure dµn := rn−1 dr dω, then

D∗αZ = − |x|δ∇ · (|x|−δ Z)− (α− 1) r1−n ω · ∂r(rn−1 Z)

for any vector-valued function Z and moreover we have the useful formula

D∗α (W Z) = −DαW · Z +W D∗αZ

if W and Z are respectively scalar- and vector-valued functions. Let us define the operatorLα by

Lα = −D∗αDα = α2

(∂2r +

n− 1

r∂r

)+

∆ω

r2,

where ∆ω denotes the Laplace-Beltrami operator on Sd−1.We introduce the weighted equation

∂g

∂t= Lαg

m ,

which is obtained from (1.5) by the change of variables

v(t, x) = g(t, |x|α−1 x

)∀ (t, x) ∈ R+ × Rd .

Next we use a self-similar change of variables similar to (2.4), but with a scaling which

corresponds to the artificial dimension n. With µ = 2 + n (m− 1) and κ =(

2m1−m

)1/µ, let

(3.1) g(t, x) =1

κnRnu(τ,

x

κR

)where

dRdt = R1−µ , R(0) = R0 = κ−1 ,

τ(t) = 12 log

(R(t)R0

).

We observe that µ? = αµ with the notations of (1.9) in Section 1.2.In self-similar variables the function u solves

(3.2)∂u

∂τ= D∗α (u z)

where

z(τ, x) := Dαum−1 − 2

αx = Dα

(um−1 − |x|

2

α2

)= Dαq , q := um−1 − Bm−1α

and

Bα(x) :=

(1 +|x|2

α2

) 1m−1

.

The exponent m is now in the range m1 ≤ m < 1 with m1 = 1 − 1/n. As in the casewithout weights, i.e. the case n = d, we also consider the problem restricted to a ball BRand assume no-flux boundary conditions, that is,

z · ω = 0 on ∂BR .

It is straightforward to check that

∂z

∂τ+ (1−m)Dα

(um−2 D∗α (u z)

)= 0


and, as a consequence,

d

dτ

∫BR

u |z|2 dµn =

∫BR

∂u

∂τ|z|2 dµn + 2

∫BR

u z · ∂z∂τ

dµn

=

∫BR

D∗α (u z) |z|2 dµn − 2 (1−m)

∫BR

u z · Dα(um−2 D∗α (u z)

)dµn .

Taking into account the boundary condition z · ω = 0 on ∂BR, a first integration by partsshows that ∫

BR

D∗α (u z) |z|2 dµn =

∫BR

u z · Dα |z|2 dµn .

Hence we get

d

dτ

∫BR

u |z|2 dµn

=

∫BR

u z · Dα |z|2 dµn − 2

∫BR

u z · Dα(z · Dαum−1 + (1−m)um−1 D∗α z

)dµn

=

∫BR

u z · Dα |z|2 dµn − 2

∫BR

u z · Dα(|z|2 +

2

αz · x+ (1−m)um−1 D∗α z

)dµn

= −∫BR

u z · Dα |z|2 dµn −4

α

∫BR

u z · Dα(z · x) dµn

− 2 (1−m)

∫BR

u z · Dα(um−1 D∗α z

)dµn .

Now, by expanding and integrating by parts we see that∫BR

u z · Dα(um−1 D∗α z

)dµn =

∫BR

(um z · Dα (D∗α z) +

m− 1

mDαu

m · z (D∗α z)

)dµn

=1

m

∫BR

um z · Dα (D∗α z) dµn −1−mm

∫BR

um (D∗α z)2dµn .

Integrating again by parts, we obtain∫BR

Dαum · Dα |z|2 dµn = −

∫BR

um Lα|z|2 dµn + αRn−d∫∂BR

um ω · Dα |z|2 dσ .

So, after observing that u z = m−1m Dαu

m − 2α ux, we finally get

d

dτ

∫BR

u |z|2 dµn =m− 1

m

∫BR

um(Lα|z|2 − 2 z · Dα (D∗α z) + 2 (m− 1) (D∗α z)

2)dµn

+2

α

∫BR

ux · Dα |z|2 dµn −4

α

∫BR


+1−mm

αRn−d∫∂BR

um(ω · Dα |z|2

)dσ .

Next, since z · ω = 0 on ∂BR, exactly for the same reasons as in Section 2.2, we know that∫∂BR

um(ω · Dα |z|2

)dσ ≤ 0 .


Let us define p := um−1 and

K[p] :=1

2Lα |Dαp|2 − Dαp · DαLαp− (1−m) (Lαp)2 .

By [22, Lemma 4.2], we know that

K[p] = α4(1− 1

n

) [p′′ − p′

r− ∆ω p

α2 (n− 1) r2

]2+

2α2

r2

∣∣∣∣∇ωp′ − ∇ωpr∣∣∣∣2 +

k[p]

r4

+(1n − (1−m)

)(Lαq)

2

where

k[p] := 12 ∆ω |∇ωp|2 −∇ωp · ∇ω∆ω p− 1

n−1 (∆ω p)2 − (n− 2)α2 |∇ωp|2 .

As a consequence, we can write that

K[p] = K[p− Bm−1α

]= 1

2

(Lα|z|2 − 2 z · Dα (D∗α z) + 2 (m− 1) (D∗α z)

2).

We also know that∫Sd−1

k[p]um dω =

∫Sd−1

Q[p]um dω + (n− 2)(α2FS − α2

) ∫Sd−1

|∇ωp|2 um dω ,

where, according to [22, Lemma 4.3] (see details in the proof), for some explicit constants aand b which depend only on α, n and d, Q[p] is such that

Q[p] = α2FS

n− 2

d− 2

∥∥∥∥(∇ω ⊗∇ω) p− 1

d− 1(∆ωp) + a

(∇ωp⊗∇ωp

p− 1

d− 1

|∇ωp|2

pg

)∥∥∥∥2+ b2

|∇ωp|4

|p|2if d ≥ 3 .

Here g denotes the standard metric on Sd−1. The case d = 2 has to be treated separately.According to [22, Lemma 4.3], there exists also an explicit constant, that we still denoteby b, such that ∫

Sd−1

Q[p]um dω ≥ b2∫Sd−1

|∇ωp|4

p2dω if d = 2 .

Collecting these observations, we have shown that

d

dτ

∫BR

u |z|2 dµn

≤ − 21−mm

∫BR

(α4(1− 1

n

) [p′′ − p′

r− ∆ω p

α2 (n− 1) r2

]2+

2α2

r2

∣∣∣∣∇ωp′ − ∇ωpr∣∣∣∣2)um dµn

− 21−mm

(1

n− (1−m)

)∫BR

(Lαq)2um dµn

− 21−mm

∫BR

Q[p]

r4um dµn − 2

1−mm

(n− 2)(α2FS − α2

) ∫BR

|∇ωp|2

r4um dµn

+2

α

∫BR


α

∫BR

u z · Dα(z · x) dµn .


Since x·Dα = α r ∂r, x·z = α r∂rq, x·∇ω = 0, and z·∂rz = z·∂r(Dαq) = z·Dα∂rq− 1r2 |∇ωq|

2,we have that

2

α

∫BR


α

∫BR


=

∫BR

(4 r u z · ∂rz − 4 r u z · Dα∂rq− 4αu∂rq (ω · z)) dµn

= − 4

∫BR

u

(α2 |∂rq|2 +

|∇ωq|2

r2

)dµn = − 4

∫BR

u |z|2 dµn .

After observing that(1n − (1−m)

)(Lαq)2 = (m−m1)

(Lαu

m−1 − 2n)2

, we conclude that

d

dτ

∫BR

u |z|2 dµn + 4

∫BR

u |z|2 dµn

≤ − 21−mm

∫BR

(α4(1− 1

n

) [p′′ − p′

r− ∆ω p

α2 (n− 1) r2

]2+

2α2

r2

∣∣∣∣∇ωp′ − ∇ωpr∣∣∣∣2)um dµn

− 21−mm

(m−m1)

∫BR

(Lαu

m−1 − 2n)2um dµn

− 21−mm

∫BR

Q[p]

r4um dµn − 2

1−mm

(n− 2)(α2FS − α2

) ∫BR

|∇ωp|2

r4um dµn .

We can extend the function u outside BR by the function Bα and pass to the limit as Rgoes to +∞. If now we consider a solution of (3.2) on Rd and if p := um−1, then we have

d

dτ

∫Rd

u |z|2 dµn + 4

∫Rd

u |z|2 dµn

≤ − 21−mm

∫Rd

(α4(1− 1

n

) [p′′ − p′

r− ∆ω p

α2 (n− 1) r2

]2+

2α2

r2

∣∣∣∣∇ωp′ − ∇ωpr∣∣∣∣2)um dµn

− 21−mm

(m−m1)

∫Rd

(Lαu

m−1 − 2n)2um dµn

− 21−mm

∫Rd

Q[p]

r4um dµn − 2

1−mm

(n− 2)(α2FS − α2

) ∫Rd

|∇ωp|2

r4um dµn .

This inequality implies (1.3) in a non scale-invariant form (as in Section 2.2 when there areno weights), but also provides an additional integral remainder term. With P = m

1−m gm−1

and v(t, x) = g(t, rα−1x

), r = |x|, let us define

R[v] :=

∫Rd

gm

(α4(1− 1

n

) ∣∣∣∣P′′ − P′

r− ∆ωP

α2 (n− 1) r2

∣∣∣∣2 +2α2

r2

∣∣∣∣∇ωP′ − ∇ωPr∣∣∣∣2)dµn

+

∫Rd

gm

((n− 2)

(α2FS − α2

) |∇ωP|2r4

+b2

r4|∇ωP|4

|P|2

)dµn .

Theorem 3. Let d ≥ 2. Under Condition (1.4), if

either β ≤ βFS(γ) ∀ γ ≤ 0 , or γ > 0 ,


then then there are two positive constants C1, C2, and a constant b, depending only on β, γand d such that the following property holds.

Assume that v0 satisfies ‖v0‖1,γ = M? and that there exist two positive constants C1

and C2 such that (1.11) holds. Let us consider a positive solution of (1.5) with initialdatum v0 such that v is smooth at x = 0 for any t ≥ 0. Then, with the above notations, wehave

(3.3) G[v(t, ·)]− G[v?] ≥ C1∫ ∞t

h(s)3µ−2 R[v(s, ·)] ds

+ C2∫ t

0

h(s)3µ−2∫Rd

∣∣LαP− 2n h(s)−µ∣∣2 dµn ds

for any t ≥ 0. Here µ and h are given by (1.10).

The expressions of the constants b, C1 and C2 are explicit. See [22] for details. Sinceα < αFS is equivalent to β < βFS and

R[g] ≥ (n− 2)(α2FS − α2

) ∫Rd

gm|∇ωP|2

r4dµn ,

Theorem 2 is a straightforward consequence of Theorem 3. In the opposite direction, bykeeping all terms in Q[p], it is possible to give a sharper estimate than (3.3), which hashowever no simple expression.

Under the above assumptions, Theorem 1 is a consequence of Theorem 3. Indeed, if wetake w2p = v0, then we know that d

dtG[v(t, ·)] = 0 at t = 0 because v0 is a critical point ofG under the mass constraint ‖v0‖1,γ = M?. Hence we know that

0 =d

dtG[v(t, ·)]|t=0 ≤ −R[v0] ≤ 0

by differentiating (3.3) at t = 0, so that R[v0] = 0, and this is enough to conclude. Wecan also notice that (3.3) implies (1.3) simply by dropping the right hand side and using adensity argument, if necessary.

Proof of Theorem 3. For any solution v of (1.5), we apply the above computations withv(t, x) = g

(t, |x|α−1x

)and u given by (3.1). Let us observe that

κR(t) = h(t)

if R and h are given by (1.10) and (3.1). It is then enough to undo the above changes ofvariables to obtain (3.3).

4. Linearization and optimality

4.1. The linearized fast diffusion flow and the spectral gap. Let us perform a formallinearization of (3.2) around the Barenblatt profile Bα by considering a solution uε withmass

∫Rd uε dx = M? such that

uε = Bα(1 + ε f B1−mα

),

and by taking formally the limit as ε→ 0. We obtain that f solves

∂f

∂t= Lα f where Lα f := (m− 1)Bm−2α D∗α (Bα Dαf) .


We define the scalar products

〈f1, f2〉 =

∫Rd

f1 f2 B2−mα dµn and 〈〈f1, f2〉〉 =

∫Rd

Dαf1 · Dαf2 Bα dµn

which correspond to weighted spaces, respectively of L2 and H1 type. It is straightforwardto check that the mass constraint results in the orthogonality condition

〈f, 1〉 = 0 ,

that constant functions span the kernel of Lα, and that Lα is self-adjoint on the space

X := L2(Rd, B2−mα dµn

)with norm given by ‖f‖2 = 〈f, f〉. Moreover,

1

2

d

dt〈f, f〉 = 〈f,Lα f〉 =

∫Rd

f (Lα f)B2−mα dµn = −∫Rd

|Dαf |2 Bα dµn = −〈〈f, f〉〉

if f belongs to the subspace

Y :=f ∈ L2

(Rd, B2−mα dµn

): 〈〈f, f〉〉 < +∞

,

and also1

2

d

dt〈〈f, f〉〉 =

∫Rd

Dαf · Dα(Lα f)Bα dµn = −〈〈f,Lα f〉〉

if f is smooth enough.Now let us consider the smallest positive eigenvalue λ1 of Lα on X . An eigenfunction

associated with λ1 solves the eigenvalue equation

−Lα f1 = λ1 f1 .

According to [7], we know that f1 ∈ Y ⊂ X , so that

〈〈f1, f1〉〉 = −〈f1,Lα f1〉 = λ1 〈f1, f1〉

and f1 yields the equality case in the Hardy-Poincare inequality

〈〈g, g〉〉 = −〈g,Lα g〉 ≥ λ1 ‖g − g‖2 ∀ g ∈ Y .

Here g := 〈g, 1〉/〈1, 1〉 denotes the average of g. It turns out that

〈〈f1, 1〉〉 = 0 and − 〈〈f1,Lα f1〉〉 = λ1 〈〈f1, f1〉〉

so that f1 is also optimal for the higher order inequality

−〈〈g,Lα g〉〉 ≥ λ1 〈〈g, g〉〉

written for any function g ∈ Y such that 〈〈g,Lα g〉〉 is finite. The proof of the inequalityitself is a simple consequence of the expansion of the square

−〈〈(g − g),Lα (g − g)〉〉 = 〈Lα (g − g),Lα (g − g)〉 = ‖Lα (g − g)‖2 ≥ 0 .

See [5, 6] and [16, 17] for more details on the results in X and Y respectively. It has beenobserved in [6] that the operator Lα on X and its restriction to Y are unitarily equivalentwhen (α, n) = (1, d) and the extension to the general case is straightforward. The kernel ofLα is generated by f0(x) = 1, and the eigenspaces corresponding to the next two eigenvaluesare generated by f1,k(x) = xk and f2(x) = |x|2 − c, for some explicit constant c. If (β, γ) =(0, 0), the eigenvalues λ1 and λ2 are strictly ordered if 1 − 1/d < m < 1 and coincide if


m = 1−1/d, but the spectrum is more complicated in the general case: see [7, Appendix B]for details. The key observation for our analysis is the fact that

(4.1) λ1 ≥ 4 ⇐⇒ α ≤ αFS :=

√d− 1

n− 1.

4.2. The optimality cases in the functional inequalities.

4.2.1. Symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities. It has been shown in[22] that the best constant in (1.3) is determined by the infimum of

J [w] := ϑ log(‖Dαw‖2,δ

)+ (1− ϑ) log

(‖w‖p+1,δ

)− log

(‖w‖2p,δ

)with δ = d − n and that symmetry holds if J [w] ≥ J [w?] for any w ∈ Hp

δ,δ(Rd), where

w?(x) = (1 + |x|2/α2)1

p−1 . As a result of [7, Section 2.4], we have that

J [w? + ε g] = ε2Q[g] + o(ε2) ,

where

2

ϑ‖Dαw?‖22,δ Q[g]

= ‖Dαg‖22,δ +p (2 + β − γ)

(p− 1)2[d− γ − p (d− 2− β)

] ∫Rd

|g|2 1

1 + α−2 |x|2dµn

− p (2 p− 1)(2 + β − γ)2

(p− 1)2

∫Rd

|g|2 1

(1 + α−2 |x|2)2 dµn

is a nonnegative quadratic form if and only if α ≤ αFS according to (4.1). Symmetrybreaking therefore holds if α > αFS.

4.2.2. An estimate on the information – production of information inequality. As shown inSection 3, we define I[u] :=

∫Rd u |z|2 dµn where z(τ, x) := Dαu

m−1 − 2α x and K[u] so that

d

dτI[u(τ, ·)] = −K[u(τ, ·)]

if u solves (3.2). If α ≤ αFS, then λ1 ≥ 4 and

u 7→ K[u]

I[u]− 4

is a nonnegative functional whose minimizer is achieved by u = Bα. With

uε = Bα(1 + ε f B1−mα

),

we observe that

4 ≤ C2 := infu

K[u]

I[u]≤ limε→0

inff

K[uε]

I[uε]= inf

f

〈〈f,Lα f〉〉〈〈f, f〉〉

=〈〈f1,Lα f1〉〉〈〈f1, f1〉〉

= λ1 .

Of course one has to take some precautions ensuring that the mass is normalized and thatdenominator in the above quotients is never zero. Summarizing, what we observe is thatthe infimum of KI is achieved in the asymptotic regime as u → Bα and determined by the

spectral gap of Lα when λ1 = 4, and that KI ≥ 4 if λ1 ≥ 4, that is, when α = αFS andα ≤ αFS respectively.


4.2.3. Symmetry in Caffarelli-Kohn-Nirenberg inequalities. If α ≤ αFS, the fact that KI ≥ 4has an important consequence. Indeed we know that

d

dτ(I[u(τ, ·)]− 4 E [u(τ, ·)]) ≤ 0

where

E [u] := − 1

m

∫Rd

(um − Bmα − mBm−1α (u− Bα)

)dµn ,

so thatI[u]− 4 E [u] ≥ I[Bα]− 4 E [Bα] = 0 .

This inequality is equivalent to J [w] ≥ J [w?], which establishes that optimality in (1.3) isachieved among symmetric functions. In other words, the computations of Section 3 showthat for α ≤ αFS, the function

τ 7→ I[u(τ, ·)]− 4 E [u(τ, ·)]is monotone decreasing. The condition α ≤ αFS is a sufficient condition, which is howevercomplementary of the symmetry breaking condition (because it depends only on the sign ofλ1 − 4), and this explains why the method based on nonlinear flows provides the optimalrange for symmetry.

4.2.4. Optimality of the information – production of information inequality. From Section4.2.2, we know that the infimum of KI is achieved in the asymptotic regime as u → Bαand determined by the spectral gap of Lα when λ1 = 4. This covers in particular thecase without weights of the Gagliardo-Nirenberg inequalities (1.1) and of the fast diffusionequation (1.2) studied in Section 2.1.

We also know that

C2 = infu

K[u]

I[u]≤ λ1 < 4

if α > αFS, and that

λ1 ≥ C2 = infu

K[u]

I[u]> 4

if α < αFS. The inequality is strict because, otherwise, if 4 was optimal, it would be achievedin the asymptotic regime and therefore would be equal to λ1 > 4, a contradiction.

4.2.5. Optimality of the entropy – production of entropy inequality. Arguing as in Sec-tion 4.2.3, we know that

I[u]− C2 E [u] ≥ I[Bα]− C2 E [Bα] = 0 .

As a consequence, we have that

C1 := infu

I[u]

E [u]≥ C2 = inf

u

K[u]

I[u].

With uε = Bα(1 + ε f B1−mα

), we observe that

4 ≤ C1 ≤ limε→0

inff

I[uε]

E [uε]= inf

f

〈f,Lα f〉〈f, f〉

=〈f1,Lα f1〉〈f1, f〉1

= λ1 = limε→0

inff

K[uε]

I[uε].

If α = αFS, then λ1 = 4 = C1 = C2. Again this covers in particular the case withoutweights of the Gagliardo-Nirenberg inequalities (1.1) and of the fast diffusion equation (1.2).If α < αFS, then C1 ≥ C2 > 4. Conversely, if α > αFS, then C1 ≤ λ1 < 4. We know from [7]


that C1 > 0, and also that the optimal constant is achieved, but the precise value of C1 is sofar unknown.

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J. Dolbeault: CEREMADE (CNRS UMR n 7534), PSL research university, Universite Paris-

Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, FranceE-mail address: [email protected]

URL: https://www.ceremade.dauphine.fr/~dolbeaul/

M.J. Esteban: CEREMADE (CNRS UMR n 7534), PSL research university, Universite Paris-Dauphine, Place de Lattre de Tassigny, 75775 Paris 16, France


URL: https://www.ceremade.dauphine.fr/~esteban/

M. Loss: School of Mathematics, Georgia Tech, 686 Cherry St., Atlanta, GA, 30332-0160, USA


URL: http://people.math.gatech.edu/~loss/

JEPE Vol 2, 2016, p. 297-321

SPREADING AND VANISHING FOR NONLINEAR STEFAN

PROBLEMS IN HIGH SPACE DIMENSIONS

YIHONG DU, BENDONG LOU AND MAOLIN ZHOU

Abstract. We classify the long-time behavior of solutions to nonlinear diffusive equa-

tions of the form ut − ∆u = f(u) for t > 0 and x over a variable domain Ω(t) ⊂ RN ,

with a Stefan condition for u over the free boundary Γ(t) = ∂Ω(t), and u(0, x) > 0 inΩ(0) = Ω0. For monostable type of f and bistable type of f , we obtain a rather complete

classification of the long-time dynamical behavior of the solution to this nonlinear Stefanproblem, and examine how the behavior changes when u(0, x) takes initial functions of

the form σφ(x) and σ > 0 is varied.

Dedicated to Professor David Kinderlehreron the occassion of his 75th birthday

1. Introduction

We are interested in the long-time behavior of nonlinear Stefan problems of the form

(1.1)

ut = ∆u+ f(u) for x ∈ Ω(t), t > 0,u = 0 and ut = µ|∇xu|2 for x ∈ Γ(t), t > 0,u(0, x) = u0(x) for x ∈ Ω0,

where Ω(t) ⊂ RN (N ≥ 2) is a varying domain with boundary Γ(t) (commonly called thefree boundary), µ > 0 is a constant. We assume that Ω(0) = Ω0 is a bounded domain whichagrees with the interior of its closure Ω0, ∂Ω0 satisfies the interior ball condition, and u0 istaken from the set

I(Ω0) := φ ∈ C(Ω0) ∩H1(Ω0) : φ(x) > 0 in Ω0, φ(x) = 0 on ∂Ω0.

The basic assumptions on f are:

(F )

(i) f(0) = 0 and f ∈ C1+α([0, δ0]) for some δ0 > 0, α ∈ (0, 1);

(ii) f(u) is locally Lipschitz in [0,∞), f(u) ≤ 0 in [M,∞) for some M > 0.

Under these assumptions, by [7], (1.1) has a unique weak solution and it is defined for allt > 0. Moreover, the following properties have been proved in [13]:

2010 Mathematics Subject Classification. Primary: 35B40, 35K55; Secondary: 35R35, 35K20.Key words and phrases. Nonlinear diffusion equation, free boundary problem, asymptotic behavior,

monostable nonlinearity, bistable nonlinearity, sharp threshold, spreading speed.Received 16/11/2016, accepted 22/11/2016.Research supported by the Australian Research Council and by the NSFC (No. 11671262).

297

298 YIHONG DU, BENDONG LOU AND MAOLIN ZHOU

Theorem A. (Theorem 1.1 in [13]) For each t > 0, Γ(t) := Γ(t)\co(Ω0) is a C2+α hy-

persurface in RN , and Γ := (t, x) : x ∈ Γ(t), t > 0 is a C2+α hypersurface in RN+1.

Here co(Ω0) denotes the closed convex hull of Ω0.

Theorem B. (Theorem 1.2 in [13]) Ω(t) is expanding in the sense that Ω0 ⊂ Ω(t) ⊂ Ω(s) if0 < t < s. Moreover, Ω∞ := ∪t>0Ω(t) is either the entire space RN , or it is a bounded set.Furthermore,(i) when Ω∞ = RN , for all large t, Γ(t) is a C2+α closed hypersurface in RN , and thereexists a continuous function M(t) such that

(1.2) Γ(t) ⊂ x : M(t)− πd0 ≤ |x| ≤M(t), where d0 is the diameter of Ω0;

(ii) when Ω∞ is bounded, limt→∞ ‖u(t, ·)‖L∞(Ω(t)) = 0.

Theorem C. (Theorem 1.3 in [13]) If f(u) = u(a − bu) with a, b positive constants, thenthere exists µ∗ ≥ 0 such that Ω∞ = RN if and only if µ > µ∗. Moreover, when Ω∞ = RN ,the following holds:

limt→∞

M(t)

t= k0(µ) > 0, lim

t→∞max|x|≤ct

∣∣∣u(t, x)− a

b

∣∣∣ = 0 ∀c ∈ (0, k0(µ)),

where k0(µ) is increasing in µ and limµ→∞ k0(µ) = 2√a.

The proof of the above results is built upon techniques in [2, 18, 19, 20] and a number ofnew ones. In particular, the following monotonicity lemma has played an important role.

Lemma D. (Lemma 4.2 in [13]) For any t > 0, x ∈ Ω(t)\co(Ω0) and ν ∈ SN−1 satisfyingν · (z − x) < 0 ∀z ∈ co(Ω0), we have ∂νu(t, x) < 0.

Problem (1.1) is often used to describe the spreading of an invasive species or a newspecies, whose population density at time t and space location x is given by u(t, x), and thepopulation range at time t is Ω(t), with Γ(t) = ∂Ω(t) representing the moving spreading frontof the species. In this context, when Ω∞ is a bounded set and limt→∞ ‖u(t, ·)‖L∞(Ω(t)) = 0,

one often says that the species is vanishing. On the other hand, the case for Ω∞ = RNdescribed in Theorem C indicates a situation that the species can invade the entire availablespace and establish itself, which is widely called the spreading case. Thus Theorem Cgives a spreading-vanishing dichotomy for the long-time dynamical behavior of u(t, x). Sucha dichotomy was first discovered in [8] for the one space dimension case, and the result hassubsequently been extended in several directions (see, for example, [6, 7, 9, 10, 13]).

The main purpose of this paper is to obtain sharp long-time behavior for (1.1) in the spiritof Theorem C above, but for much more general nonlinear functions f(u). Moreover, whileµ is treated as a parameter in Theorem C to distinguish the spreading-vanishing behaviors,here we will fix µ > 0 and take u0 = σφ, with φ ∈ I(Ω0) fixed and σ > 0 regarded as avarying parameter.

Such a problem in one space dimension has been treated in [9] and [10] (after the initialwork [8]), where a rather complete theory has been established for f belonging to threetypes of nonlinearities: Monostable type, bistable type and combustion type. However, inhigher space dimensions, the problem is much more difficult to treat, and the behavior of(1.1) may be more complicated and even different, especially in the combustion case (which

SPREADING AND VANISHING FOR NONLINEAR STEFAN PROBLEMS 299

is studied separately in [11]). Here we only consider the case that f is monostable and thecase that f is bistable, as precisely described below.

In this paper, the nonlinear function f is said to be of monostable type if

(fM )

(i) f(0) = 0 and f ∈ C1([0,∞));(ii) f(s) > 0 for s ∈ (0, 1) and f(s) < 0 for s ∈ (1,∞);(iii) f ′(0) > 0, f ′(1) < 0;(iv) f ∈ C1+α([0, δ1]) for some δ1 > 0 and α ∈ (0, 1).

A typical example of f(u) satisfying all the above conditions is u(1− u).The function f is said to be of bistable type if it satisfies

(fB)

(i) f(0) = 0 and f ∈ C1([0,∞));(ii) f(s) < 0 for s ∈ (0, θ) ∪ (1,∞) and f(s) > 0 for s ∈ (θ, 1);

(iii)∫ 1

0f(s)ds > 0;

(iv) f ′(0) < 0, f ′(1) < 0;(v) f(u)/(u− θ) is non-increasing in u ∈ (θ, 1), where θ ∈ (θ, 1)

is uniquely determined by∫ θ

0f(s)ds = 0;

(vi) limsθ f(s)/(s− θ)κ ∈ (0,∞],where κ = N

N−2 when N > 2, and κ ∈ (0,∞) when N = 2;

(vii) f ∈ C1+α([0, δ1]) for some δ1 > 0, α ∈ (0, 1).

A typical example of f(u) satisfying all the conditions in (fB) except for (v) is f(u) =u(u− θ)(1− u) with θ ∈ (0, 1

2 ), and (v) is also satisfied if θ ∈ ( 516 ,

12 ).

The definitions for monostable and bistable nonlinearities here are more restrictive thanthose in [9]. For monostable f , the extra requirement (iv) is due to the same restrictionused in [13]; for bistable f , the extra restrictions (v) and (vi) in (fB) are used to guaranteethe uniqueness of the “ground state solution” v for (1.3) below (cf. [21, 22]), and (vii) isdue to [13] as in the monostable case.

For monostable type of f our main results are contained in the following theorem.

Theorem 1. Assume that f satisfies (fM ). Let φ ∈ I(Ω0) and let uσ(t, x) be the solutionof (1.1) with initial function u0(x) = σφ(x), σ > 0. Then there exists σ∗ = σ∗(φ) ∈ [0,∞]such that

(i) for 0 < σ ≤ σ∗, vanishing happens, i.e., Ω∞ is bounded and

limt→∞

maxx∈Ω(t)

uσ(t, x) = 0;

(ii) for σ > σ∗, spreading happens, i.e., Ω∞ = RN and

limt→∞

uσ(t, x) = 1 locally uniformly in x ∈ RN ,

M(t) = c∗t− c∗(N − 1) log t+O(1) as t→∞,where c∗ and c∗ are positive constants independent of N and u0 (they depend on fand µ only), and M(t) is given in (1.2).

The constants c∗ and c∗ are given in [14]; see Theorems E and F in Section 2 below. Thecases σ∗ = 0 and σ∗ =∞ can actually happen; see Remark 3 for details.

The next theorem is about bistable type of f .


Theorem 2. Assume that f satisfies (fB). Let φ ∈ I(Ω0) and let uσ(t, x) be the solutionof (1.1) with initial function u0(x) = σφ(x), σ > 0. Then either vanishing happens for uσfor all σ > 0, or there exists σ∗ = σ∗(φ) ∈ (0,∞) such that

(i) for 0 < σ < σ∗, vanishing happens, i.e., Ω∞ is bounded and

limt→∞

maxx∈Ω(t)

uσ(t, x) = 0;

(ii) for σ > σ∗, spreading happens, i.e., Ω∞ = RN and

limt→∞

uσ(t, x) = 1 locally uniformly in x ∈ RN ,

M(t) = c∗t− c∗(N − 1) log t+O(1) as t→∞,where c∗ and c∗ are positive constants independent of N and u0 (they depend on fand µ only), and M(t) is given in (1.2);

(iii) for σ = σ∗, a transition case happens, where we have Ω∞ = RN and for anysequence tk increasing to ∞ as k → ∞, there is a subsequence tkj and a pointx0 ∈ co(Ω0) such that

limj→∞

uσ(tkj , x) = v(|x− x0|) locally uniformly for x ∈ RN ,

where v(r) is the unique positive solution to

(1.3) vrr +N − 1

rvr + f(v) = 0 for r > 0, vr(0) = 0, v(∞) = 0.

The rest of the paper is organized as follows. In Section 2, we study the radially symmetriccase of (1.1). In Section 3, we give the proofs for Theorems 1 and 2, where the results inSection 2 are used extensively.

2. The Radially Symmetric Case

In this section, we study the radially symmetric case of (1.1), which forms an importantfirst step towards the proof of Theorems 1 and 2. In such a case, u = u(t, r) and Ω(t) =r < h(t), with r = |x|. So the free boundary is given by r = h(t). To avoid confusion, wewill use w(t, r) to replace u(t, r) and rewrite (1.1) in the form

(2.1)

wt = ∆w + f(w), 0 < r < h(t), t > 0,wr(t, 0) = 0, w(t, h(t)) = 0, t > 0,h′(t) = −µwr(t, h(t)), t > 0,h(0) = h0, w(0, r) = w0(r), r ≤ h0,

where ∆w = wrr + N−1r wr and h0 is a positive constant. For simplicity, we will choose the

initial function w0 from

(2.2) K (h0) :=ψ ∈ C2([0, h0]) : ψ′(0) = ψ(h0) = 0, ψ(r) > 0 in [0, h0)

.

For (2.1), apart from results needed for the proof of Theorems 1 and 2, we also con-sider cases which are of independent interest. For that purpose, we will consider general fsatisfying

(2.3) f : [0,∞)→ R is C1 and f(0) = 0,


and

(2.4) f(w) ≤ Kw for all w ≥ 0 and some K > 0.

With f satisfying (2.3) and (2.4), by [6], for any given h0 > 0 and w0 ∈ K (h0), (2.1)has a classical solution (w(t, r), h(t)) belonging to C1,2(D) × C1([0,∞)), such that all theidentities in (2.1) are satisfied pointwisely, where

D :=

(t, r) : t ∈ (0,∞), r ∈ [0, h(t)].

In the rest of the paper, the solution of (2.1) may also be denoted by (w(t, r;w0), h(t;w0)),or simply (w, h), depending on the context.

By [6] we have h′(t) > 0 and w(t, r) > 0 for r ∈ [0, h(t)). So h∞ := limt→∞ h(t) ∈ (h0,∞]always exists.

2.1. Main Results. Our first result on (2.1) is a general convergence theorem, which is ananalogue of Theorem 1.1 in [12] and Theorem 1.1 in [9].

Theorem 3. Suppose that f satisfies (2.3) and (w, h) is a solution of (2.1) defined for allt > 0. If w(t, r) is bounded, namely

w(t, r) ≤ C for all t > 0, r ∈ [0, h(t)] and some C > 0,

then either h∞ <∞ and limt→∞ ‖w(t, r)‖L∞([0,h(t)]) = 0; or h∞ =∞ and limt→∞ w(t, r) =v(r) locally uniformly for r ∈ [0,∞), where v(r) satisfies

(2.5) v′′ +N − 1

rv′ + f(v) = 0 for r > 0, v′(0) = 0,

and either v ≡ constant or v′(r) < 0 for r ≥ h0; in the former case, the constant isnecessarily a nonnegative zero of f .

The proof of this theorem is given in subsection 2.4. Using this theorem we can prove thefollowing results, which give a rather complete description for the asymptotic behavior of(2.1) with monostable and bistable types of f . We remark that the condition (iv) in (fM )and (vii) in (fB) is not required for the radial case here.

Theorem 4. Assume that f satisfies (i)-(iii) in (fM ). If h0 > 0, ψ ∈ K (h0) and if wσ(t, r)is the solution of (2.1) with initial function w0 = σψ, then there exists σ∗ = σ∗(h0, ψ) ∈[0,∞] such that

(i) for 0 < σ ≤ σ∗, vanishing happens, i.e., h∞ <∞ and

limt→∞

max0≤r≤h(t)

w(t, r) = 0;

(ii) for σ > σ∗, spreading happens, i.e., h∞ =∞ and

limt→∞

w(t, r) = 1 locally uniformly in r ∈ [0,∞),

For bistable f we have a trichotomy result.

Theorem 5. Assume that f satisfies (i)-(vi) in (fB). If h0 > 0, ψ ∈ K (h0) and if wσ(t, r)is the solution of (2.1) with initial function w0 = σψ, then either wσ vanishes for everyσ > 0, or there exists σ∗ = σ∗(h0, ψ) ∈ (0,∞) such that

(i) for 0 < σ < σ∗, vanishing happens, i.e., h∞ <∞ and

limt→∞

max0≤r≤h(t)

w(t, r) = 0;


(ii) for σ > σ∗, spreading happens, i.e., h∞ =∞ and

limt→∞

w(t, r) = 1 locally uniformly in r ∈ [0,∞)

(iii) for σ = σ∗, h∞ =∞ and

limt→∞

|w(t, r)− v(r)| = 0 locally uniformly in r ∈ [0,∞),

where v is the unique positive solution to

(2.6) v′′ +N − 1

rv′ + f(v) = 0 for r > 0, v′(0) = 0, v(∞) = 0.

When spreading happens in Theorems 4 and 5, much better descriptions of the long-timebehavior of (w(t, r), h(t)) have been obtained in [14]. For convenience of reference and lateruse, we recall them below.

We need the following result for the semi-wave from [9].

Theorem E (Proposition 1.8 and Theorem 6.2 of [9]) Suppose that f satisfies (i)-(iii) in(fM ), or it satisfies (i)-(iv) in (fB). Then for any µ > 0 there exists a unique c∗ = c∗(µ) > 0and a unique solution qc∗ to

q′′ − cq′ + f(q) = 0, q > 0 in (0,∞), q(0) = 0, q(∞) = 1(2.7)

with c = c∗ such that q′c∗(0) = c∗

µ .

We remark that this function qc∗ is shown in [9] to satisfy q′c∗(z) > 0 for z ≥ 0, and it iscalled a semi-wave with speed c∗.

When spreading happens for (2.1), namely

h∞ =∞ and limt→∞

w(t, r) = 1 locally uniformly for r ∈ [0,∞),

it is shown in [14] that the following holds.

Theorem F (Theorem 4.1 of [14]) Let f be as in Theorem E, and suppose spreading happens

for (2.1). Then there exists c∗ > 0 independent of N , and h ∈ R1 such that

(2.8) limt→∞

[h(t)− c∗t+ (N − 1)c∗ log t

]= h,

limt→∞

supr∈[0, h(t)]

∣∣∣w(t, r)− qc∗(c∗t− (N − 1)c∗ log t+ h− r)∣∣∣ = 0.

Moreover, the constant c∗ is given by

c∗ =1

ζ c∗, ζ = 1 +

c∗

µ2∫∞

0q′c∗(z)

2e−c∗zdz.

2.2. Comparison results. We give some basic comparison results which will be used laterin the paper. In these results, we always assume that f satisfies (2.3).

Lemma 1. Suppose that T ∈ (0,∞), h ∈ C1([0, T ]), w ∈ C(DT ) ∩ C1,2(DT ) with DT =(t, r) ∈ R2 : 0 < t ≤ T, 0 < r < h(t), and

wt ≥ ∆w + f(w) for 0 < t ≤ T, 0 < r < h(t),

w(t, h(t)) = 0, h′(t) ≥ −µwr(t, h(t)) for 0 < t ≤ T,

wr(t, 0) ≤ 0 for 0 < t ≤ T.


If

h0 ≤ h(0) and w0(r) ≤ w(0, r) in [0, h0],

then the solution (w, h) of the free boundary problem (2.1) satisfies

h(t) ≤ h(t) in (0, T ], w(t, r) ≤ w(t, r) for t ∈ (0, T ] and r ∈ [0, h(t)).

Lemma 2. Suppose that T ∈ (0,∞), ξ, h ∈ C1([0, T ]), w ∈ C(D∗T ) ∩ C1,2(D∗T ) with

D∗T = (t, r) ∈ R2 : 0 < t ≤ T, ξ(t) < r < h(t), ξ(t) ≥ 0 in [0, T ] andwt ≥ ∆w + f(w) for 0 < t ≤ T, ξ(t) < r < h(t),w(t, ξ(t)) ≥ w(t, ξ(t)) for 0 < t ≤ T,w(t, h(t)) = 0, h

′(t) ≥ −µwr(t, h(t)) for 0 < t ≤ T,

with

ξ(0) ≤ h0 ≤ h(0), w0(r) ≤ w(0, r) in [ξ(0), h0],

where (w, h) solves (2.1). Then

h(t) ≤ h(t) in (0, T ], w(t, r) ≤ w(t, r) for t ∈ (0, T ] and ξ(t) ≤ r ≤ h(t).

The proof of Lemma 1 is identical to that of Lemma 5.7 in [8], and a minor modificationof this proof yields Lemma 2 (see also [9]).

Remark 1. The function w, or the pair (w, h) in Lemmas 1 and 2 is often called an uppersolution to (2.1). A lower solution can be defined analogously by reversing all the inequalities.We also have corresponding comparison results for lower solutions in each case.

The following result shows that an upper solution in one space dimension can often beused to construct an upper solution in high space dimensions.

Lemma 3. Suppose that t1 < t2 and ξ, h ∈ C1([t1, t2]), w(t, y) ∈ C(D∗) ∩ C1,2(D∗) withD∗ = (t, y) ∈ R2 : t1 < t < t2, 0 ≤ ξ(t) < y < h(t), and

wt ≥ wyy + f(w) for y ∈ [ξ(t), h(t)], t ∈ [t1, t2].

Assume that

wy(t, y) ≤ 0 for y ∈ [ξ(t), h(t)], t ∈ [t1, t2].

Then w(t, r) := w(t, r) satisfies

wt ≥ ∆w + f(w) for r ∈ [ξ(t), h(t)], t ∈ [t1, t2].

Proof. Since wy(t, y) ≤ 0, a direct calculation gives

wt −∆w − f(w) = wt − wrr −N − 1

rwr − f(w) ≥ wt − wrr − f(w) ≥ 0.


2.3. Local and global existence. The following local existence result can be proved inthe same way as in [6] (see Theorems 2.1 and 4.1 there).

Theorem 6. Suppose that (2.3) holds. For any given w0 ∈ K (h0) and any α ∈ (0, 1), thereis a T > 0 such that problem (2.1) admits a unique solution

(w, h) ∈ C(1+α)/2,1+α(GT )× C1+α/2([0, T ]);

moreover,

‖w‖C(1+α)/2,1+α(GT ) + ‖h‖C1+α/2([0,T ]) ≤ C,(2.9)

where GT = (t, r) ∈ R2 : t ∈ (0, T ], r ∈ [0, h(t)], C and T depend on h0, α and‖w0‖C2([0,h0]).

Remark 2. As in [6, 8], problem (2.1) can be converted to a problem in a fixed domain.Using the Schauder estimates to the latter problem, one can obtain additional regularity forthe solution. Therefore, we indeed have w ∈ C1+α/2,2+α(GT ).

Lemma 4. Suppose that (2.3) holds, (w, h) is a solution to (2.1) defined for t ∈ [0, T0) forsome T0 ∈ (0,∞), and there exists C1 > 0 such that

w(t, r) ≤ C1 for t ∈ [0, T0) and r ∈ [0, h(t)].

Then there exists C2 depending on C1 but independent of T0 such that

0 < h′(t) ≤ C2 for t ∈ (0, T0).

Moreover, the solution can be extended to some interval (0, T ) with T > T0.

The proof of this lemma is identical to that of [6, Lemma 4.2] (see also [8, Lemma 2.2]and [9, Lemma 2.6]). This lemma implies that the solution of (2.1) can be extended as longas w remains bounded. In particular, if f satisfies (2.3) and (2.4), then the solution of (2.1)exists for all t > 0.

2.4. Proof of Theorem 3. To prove Theorem 3, we need the following lemma.

Lemma 5. Suppose that (w(t, r), h(t)) is a solution of (2.1) as given in Theorem 3. Then

(2.10) wr(t0, r) < 0 for all t0 > 0, r ∈ (h0, h(t0)].

Proof. First wr(t0, h(t0)) < 0 follows from the Hopf lemma directly. For any r1 ∈ (h0, h(t0)),there exists a unique t1 ∈ (0, t0) such that h(t1) = r1 and h(t) > r1 for t > t1. By the Hopflemma again, we have wr(t1, r1) = wr(t1, h(t1)) < 0. To prove (2.10) we need to show thatwr(t0, r1) < 0.

We actually show that wr(t, r1) < 0 for t ∈ (t1, t0]. Note that u(t, x) := w(t, |x|) = w(t, r)is a radially symmetric solution of (1.1) for t ∈ [0,∞) and x ∈ Bh(t). For t > t1 denote

Gt := x = (x1, x′) ∈ Bh(t) : x1 > r1

and define

z(t, x) := u(t, x1, x′)− u(t, 2r1 − x1, x

′) for x ∈ Gt.Then we have

zt = ∆z + c(t, x)z for t > t1, x ∈ Gt and some c ∈ L∞,


and

z(t, x) = 0 for x ∈ ∂Gt ∩ x : x1 = r1, z(t, x) < 0 for x ∈ ∂Gt ∩ x : x1 > r1.

Hence we can apply the strong maximum principle and the Hopf lemma to deduce

z(t, x) < 0 in Gt, zx1(t, x) < 0 for x ∈ ∂Gt ∩ x : x1 = r1, t > t1.

We thus have 0 > zx1(t, r1, 0) = 2ux1

(t, r1, 0) = 2wr(t, r1) for t > t1.

Proof of Theorem 3: We will make use of Lemma 5 and follow the ideas of [9, 12] withconsiderable variations.

Let (w, h) be a solution of (2.1) as given in the theorem. Then h∞ is either a finitepositive number or h∞ = ∞. If h∞ < ∞, then the same reasoning as in [8, 9] shows thatlimt→∞ ‖w(t, ·)‖L∞([0,h(t)]) = 0.

Next we consider the case that h∞ =∞. Denote by ω(w) the ω-limit set of w(t, ·) in thetopology of L∞loc([0,∞)), namely, a function η ∈ ω(w) if and only if there exists a sequence0 < t1 < t2 < t3 < · · · → ∞ such that

(2.11) limn→∞

w(tn, r) = η(r) locally uniformly in [0,∞).

By standard parabolic estimates, we see that the convergence (2.11) implies convergence inthe C2

loc([0,∞)) topology. Thus the definition of ω(w) remains unchanged if the topology ofL∞loc((0,∞)) is replaced by that of C2

loc([0,∞)).It is well-known that ω(w) is compact and connected, and for any η ∈ ω(w) there exists

an entire orbit passing through it, namely a solution of ηt = ηrr + f(η) defined for all t ∈ Rand r ∈ [0,∞), that satisfies η(0, r) = η(r). We can find such an entire solution η(t, r) bychoosing a suitable sequence 0 < t1 < t2 < t3 < · · · → ∞ such that

(2.12) w(t+ tn, r)→ η(t, r) as n→∞.

Here the convergence is understood in the L∞loc sense in (t, r) ∈ R× [0,∞), but, by parabolic

regularity, it takes place in the C1,2loc (R× [0,∞)) sense as well.

For clarity we divide the arguments below into two parts, each proving a specific claim.

Claim 1: ω(w) consists of solutions of (2.5), namely functions v(r) satisfying

(2.13) vrr +N − 1

rvr + f(v) = 0 for r > 0, vr(0) = 0.

Let η(r) be an arbitrary element of ω(w) and η(t, r) be the entire orbit satisfying η(0, r) =η(r). Since η is a nonnegative solution of

ηt = ∆η + f(η), t ∈ R, r ∈ [0,∞),

and f(0) = 0, by the strong maximum principle we have either η(t, r) > 0 for all t ∈ R andr ∈ [0,∞), or η ≡ 0. In the latter case we have η ≡ 0, which is a solution to (2.13). In whatfollows we assume the former, namely η(r) > 0 for r ∈ [0,∞).

By Lemma 5, η′(r) ≤ 0 for r ∈ [h0,∞), and η′(0) = 0. Let v(r) be the solution of thefollowing initial value problem:

(2.14) vrr +N − 1

rvr + f(v) = 0 for r > 0, v(0) = η(0) > 0, vr(0) = 0.


This problem has a unique positive solution defined for all small r > 0 (see [22]). By uniquecontinuation, v(r) can be extended to a positive solution of (2.14) in [0,∞), or a solution of(2.14) with compact positive support, namely there exists R > 0 such that

v(r) > 0 in [0, R) and v(R) = 0 or v(R) =∞.

In case v has compact positive support, denote

h1(t) := minR, h(t) for t > 0.

Since h(t) is strictly increasing with h∞ = ∞, we can find some large t1 > 0 such thath(t) > h1(t) = R for t ≥ t1.

In case v > 0 in [0,∞), we simply take h1(t) := h(t) and t1 = 1. Then

z(t, r) := w(t, r)− v(r), r ∈ [0, h1(t)), t > t1

satisfies

(2.15)

zt = ∆z + c(t, r)z, r ∈ [0, h1(t)), t > t1,zr(t, 0) = 0, z(t, h1(t)) 6= 0, t > t1,

for some c ∈ L∞loc(D), with D := (t, r) : t ≥ t1, 0 ≤ r < h1(t). Denote by Z(z(t, ·)) thenumber of zeros of z(t, ·) on [0, h1(t)).

By Theorem 2.1 of [4]1, one sees that

(i) Z(z(t, ·)) <∞ for any t > t1;(ii) Z(z(t, ·)) is monotone nonincreasing in t;(iii) if z(t0, r0) = zr(t0, r0) = 0 for some t0 > t1 and 0 ≤ r0 < h1(t0), then Z(z(t, ·)) >

Z(z(s, ·)) for any t1 < t < t0 < s.

Consequently, for sufficiently large t, the function z has fixed number of simple zeros on[0, h1(t)). In view of this and the fact that

limn→∞

z(t+ tn, r) = η(t, r)− v(r) in C1,2loc (R× [0, h1(∞))),

we see (cf. [12, Lemma 2.6]) that for each t ∈ R, either η(t, r) ≡ v(r) on [0, h1(∞)), orη(t, r) − v(r) has only simple zeros on [0, h1(∞)). The latter, however, is impossible fort = 0 since r = 0 is a degenerate zero of η(0, r) − v(r) ≡ η(r) − v(r). Consequently,η(0, r) = η(r) ≡ v(r). This proves Claim 1.

Claim 2: ω(w) consists of a single function which is a solution of (2.5), and it is either anonnegative constant or a positive radially symmetric function strictly decreasing for r > h0.

By Claim 1, ω(w) consists of solutions of (2.5). First, we show that ω(w) is a singleton.Otherwise, ω(w) contains infinitely many elements since it is connected. Let v1, v2 andv3 be three distinct functions in ω(w). Then v1(0), v2(0) and v3(0) are different fromeach other since v1(r), v2(r) and v3(r) are distinct solutions of (2.14) with initial values(v(0), v′(0)) = (vi(0), 0) for i = 1, 2, 3, respectively. Without loss of generality we canassume v1(0) < v2(0) < v3(0). Using Lemma 6 below, we see that w(t, 0) − v2(0) neverchanges sign for sufficiently large t. This clearly contradicts the fact that v1, v3 ∈ ω(w).

1Theorem 2.1 of [4] treats the case where the linear parabolic equation is satisfied over (t, r) ∈ [T1, T2]×[0, 1]. However, the proof there easily carries over to the case of (2.15) where the boundary is r = h1(t)instead of r = 1.


So we have proved that ω(w) is a singleton. Suppose ω(w) = η. By Claim 1, η(r) is asolution of (2.5). Then ζ := ηr satisfies

ζ ′′ +N − 1

rζ ′ +

[f ′(η)− N − 1

r2

]ζ = 0 for r > 0.

It follows from Lemma 5 that ζ(r) ≤ 0 for r ≥ h0. Hence by Harnack’s inequality eitherζ(r) ≡ 0 for all r > h0, or ζ(r) < 0 for all r > h0. In the former case, by the uniquecontinuation property of the solutions to the above linear ODE, we deduce η(r) = 0 for allr > 0. This proves Claim 2.

The proof of Theorem 3 is now complete under the assumption that Lemma 6 bow isvalid. 2

Lemma 6. Let (w, h) be given in Theorem 3, with h∞ =∞. Suppose that v ∈ ω(w). Thenthere exists t∗ > 0 large such that w(t, 0)− v(0) does not change sign for t ≥ t∗.

Proof. The function z(t, r) := w(t, r) − v(r) satisfies an equation of the form (2.15), withh1(t) replaced by h(t), and t1 replaced by 0. Hence the number of zeros of z(t, ·) in [0, h(t)],denoted by Z(z(t, ·)), has the properties (i), (ii) and (iii) stated just below (2.15).

We now use the idea of [17]. Denote

A+ := (t, r) : t > 0, r ∈ [0, h(t)], z(t, r) > 0,

A− := (t, r) : t > 0, r ∈ [0, h(t)], z(t, r) < 0,

and for each s > 0 set

Σs := (t, r) : t ≥ s, r ∈ [0, h(t)], ls := s × [0, h(s)].

The proof of Lemma 2 in [17] yields the following conclusion: Fix t0 > 0 and let C beany connected component of A+ ∩Σt0 . Then C ∩ lt0 6= ∅. The same assertion holds for anyconnected component of A− ∩ Σt0 .

If there is a sequence t0 < t1 < t2 < ... < tk < ... satisfying

limk→∞

tk =∞, (−1)kz(tk, 0) > 0,

then let C+j be the connected component of A+ ∩Σt0 that contains (t2j , 0), and C−j be the

connected component of A− ∩ Σt0 that contains (t2j−1, 0), j = 1, 2, .... Clearly these aredisjoint connected sets, and the above stated conclusion indicates that

C+j ∩ lt0 6= ∅, C

−j ∩ lt0 6= ∅ for all j ≥ 1.

So we may choose x+j and x−j such that

(t0, x+j ) ∈ C+

j ∩ lt0 , (t0, x−j ) ∈ C−j ∩ lt0 , j = 1, 2, ...

Since tk is strictly increasing in k, it is easily seen that x−j < x+j < x−j+1 for j = 1, 2, .... This

implies that z(t0, r) changes sign infinitely many times for r ∈ [0, h(t0)], a contradiction toproperty (i) for Z(z(t, ·)).

Hence there exists some large t∗ > 0 such that z(t, 0) does not change sign for r ≥ t∗.


2.5. Stationary solutions. In this subsection we assume that f satisfies (i)-(iii) in (fM )or (i)-(vi) in (fB), and (w, h) solves (2.1) with h∞ = ∞. We want to better understandthe nonnegative solutions of (2.5) which belongs to ω(w). By Lemma 5, wr(t, r) ≤ 0 forr ∈ (h0, h(t)], so for any v ∈ ω(w), we easily see that

(2.16) 0 ≤ v(r) ≤ 1 for r > 0, v′(r) ≤ 0 for r > h0.

Therefore, we only focus on solutions of (2.5) which satisfy (2.16).

For monostable f , by Theorem 1.1 of [5], we immediately obtain

Lemma 7. Assume that f satisfies (i)-(iii) in (fM ), and v is a nonnegative solution of (2.5)satisfying 0 ≤ v ≤ 1. Then either v ≡ 0 or v ≡ 1.

For bistable f , we first prove

Lemma 8. Assume that f satisfies (i)-(vi) in (fB). Then (2.5) has a unique ground statesolution, namely a positive solution V0(r) satisfying V ′0(r) < 0 for r > 0 and V0(∞) = 0.

Proof. The following initial value problem

(2.17) v′′ +N − 1

rv′ + f(v) = 0 for r > 0, v(0) = β > 0, vr(0) = 0

has a unique solution defined on some maximal interval [0, r∗) with 0 < r∗ ≤ ∞, which wedenote by v(r;β).

Clearly v(r; θ) ≡ θ and v(r; 1) ≡ 1. For β ∈ I := (θ, 1), since f(v) > 0 for v ∈ (θ, 1), weeasily see from (2.17) that v′(r;β) < 0 for small r > 0. Set

I− := β ∈ I : ∃ r0 > 0 such that v(r0;β) = 0 and v′(r;β) < 0 for r ∈ (0, r0]

and

I+ := β ∈ I : ∃ r0 > 0 such that v′(r0;β) = 0 and v(r;β) > 0 for r ∈ [0, r0].

By [1, Lemma I.1] and its proof, I+ and I− are nonempty, disjoint, open and (θ, θ] ⊂ I+.Therefore, the set J := I\(I+ ∪ I−) ⊂ (θ, 1) is not empty, and for any β1 ∈ J , the uniquesolution v(r;β1) of (2.17) satisfies

v′(r;β1) < 0 for all r > 0, limr→∞

v(r;β1) = v∗ ≥ 0.

By [1, Lemma I.2], v∗ = θ or 0. However, due to (vi) in (fB), we can apply Theorem 1.1 of[5] to exclude the possibility of v∗ = θ. Therefore, v(r;β1) 0 as r →∞, and so v(r;β1) isa solution of (2.17) with β = β1 and

(2.18) v(r) > 0, v′(r) < 0 for r > 0, v(∞) = 0.

Such a solution is generally called a ground state solution.On the other hand, by [21, 22] the ground state solution is unique if (i), (ii) (iii) and (v)

in (fB) are satisfied. Hence, when f satisfies (i)-(vi) of (fB), the problem (2.5) has a uniqueground state solution V0(r) := v(r;β1).

The above proof indicates that

(2.19) J = V0(0) = β1, I+ = (θ, V0(0)) and I− = (V0(0), 1).


Lemma 9. Assume that f satisfies (i)-(vi) in (fB), and v is a solution of (2.5) satisfying(2.16). Then v ∈ 0, 1, θ, V0 or it satisfies

v(0) ∈ (0, V0(0)) \ θ, v(∞) = θ.

Proof. Let v(r) be a solution of (2.5) satisfying (2.16) and v 6∈ 0, 1, θ, V0. By the definitionof I− we see that v(0) 6∈ I− = (V0(0), 1). Clearly we also have v(0) 6∈ 0, 1, θ, V0(0). Hencenecessarily v(0) ∈ (0, θ) ∪ (θ, V0(0)).

By (2.16), v(r) γ as r → ∞ for some γ ≥ 0. Therefore, for some sequence rn → ∞satisfying v′(rn) → 0, by replacing r = rn in the equation of (2.17) and letting n → ∞ wededuce limn→∞ v′′(rn) = f(γ). This, together with v(∞) = γ, implies that f(γ) = 0, andso γ = θ or 0.

We claim that γ = 0 is impossible. In fact, when v(0) = β ∈ I+ = (θ, V0(0)), there existsr∗ > 0 such that v′(r) < 0 for r ∈ (0, r∗) and v′(r∗) = 0. Hence v′′(r∗) = −f(v(r∗)) ≥ 0.Clearly, f(v(r∗)) 6= 0, for otherwise, v ≡ v(r∗) is the unique solution of the equation in(2.17), a contradiction. Therefore, f(v(r∗)) < 0, which implies

(2.20) v(r∗) ∈ (0, θ), v′(r∗) = 0, v′′(r∗) > 0.

If γ = 0, that is, v(r) 0 as r → ∞, then there exists a sequence rn → ∞ such thatv(rn)→ 0, v′(rn)→ 0 as n→∞. Multiplying the equation in (2.17) by 2v′ and integratingit on [r∗, rn] we have

0 = [v′(rn)]2 +

∫ rn

r∗

2(N − 1)

r[v′(r)]2dr + 2

∫ v(rn)

v(r∗)

f(s)ds > [v′(rn)]2 + 2

∫ v(rn)

v(r∗)

f(s)ds.

Letting n→∞ we deduce ∫ v(r∗)

0

f(s)ds ≥ 0.

Since v(r∗) ∈ (0, θ) and f(s) < 0 for s ∈ (0, v(r∗)), the above inequality is impossible.If v(0) = β ∈ (0, θ) and γ = 0, we obtain a similar contradiction by taking r∗ = 0 in

the above argument. Hence γ = 0 is impossible, and v(r) satisfies v(0) ∈ (0, V0(0)) \ θ,v(∞) = θ.

Next we consider solutions of (2.5) which have compact positive support, namely, func-tions v(r) that satisfies (2.5) over some finite interval [0, R), with v(R) = 0, v(r) > 0 forr ∈ [0, R). These solutions will be used to find sufficient conditions for spreading of (2.1).

Lemma 10. Suppose that f satisfies (i)-(iii) in (fM ). Then there exists RM > 0 such thatfor any R ≥ RM , (2.5) has a solution vR(r) satisfying

(2.21) vR(0) ∈ (0, 1), v′R(r) < 0 for r ∈ (0, R], vR(R) = 0.

Proof. We use a standard upper and lower solution argument, the details are included herefor completeness. Let λR1 be the first eigenvalue of the problem

−∆u = λu in BR, u = 0 on ∂BR,

and φ > 0 the corresponding eigenfunction with ‖φ‖∞ = 1. It is well known that 0 < λR1 → 0as R→∞. Choose R0 > 0 large enough such that λR1 < f ′(0) for all R ≥ R0. Then for allsmall ε > 0, f(εφ(x)) ≥ λR1 εφ(x) in BR. It follows that

−∆(εφ) ≤ f(εφ) in BR, εφ = 0 on ∂BR.


Thus εφ is a lower solution to the problem

(2.22) −∆u = f(u) in BR, u = 0 on ∂BR.

Since f(1) = 0, clearly the constant function 1 is an upper solution of (2.22). Thereforeby the standard upper and lower solution argument we know that (2.22) has a solutionu satisfying 0 < u < 1 in BR, for every R ≥ R0. The well-known Gidas-Ni-Nirenbergsymmetry result infers that such a solution is radially symmetric, and u′(r) < 0 for r ∈(0, R].

Lemma 11. Assume that f satisfies (i)-(vi) in (fB). Then for every β ∈ (V0(0), 1), thereexists Rβ > 0 such that (2.5) has a solution vβ(r) satisfying

vβ(0) = β, v′β(r) < 0 for r ∈ (0, Rβ ], vβ(Rβ) = 0.

Proof. By (2.19), we have I− = (V0(0), 1), and the existence of vβ follows directly from thedefinition of I−.

2.6. Sufficient conditions for vanishing. For f satisfying (i)-(iii) in (fM ) or (i)-(vi) in(fB), we give some sufficient conditions ensuring that vanishing happens for (2.1). Thefollowing upper bound for w is an easy consequence of the standard comparison principle.

Lemma 12. Assume that f satisfies (2.3) and (2.4). Then, for any h0 > 0 and anyw0 ∈ K (h0), the solution w(t, r) of (2.1) satisfies

(2.23) w(t, r) ≤ eKt

(4πt)N/2

∫Bh0

w0(|ξ|)dξ for 0 ≤ r ≤ h(t), t > 0.

Proof. Consider the Cauchy problem

(2.24)

wt = ∆w +Kw, x ∈ RN , t > 0,w(0, x) = Φ(|x|), x ∈ RN ,

where

Φ(r) =

w0(r), r ∈ [0, h0],0, r ∈ [h0,∞).

Using the fundamental solution we have

w(t, |x|) =eKt

(4πt)N/2

∫RN

e−|x−ξ|2

4t Φ(|ξ|)dξ ≤ eKt

(4πt)N/2

∫Bh0

w0(|ξ|)dξ.

By the standard comparison theorem, we have w(t, r) ≤ w(t, r) for t > 0 and r ∈ [0, h(t)],and the required inequality follows.

Proposition 1. Let h0 > 0 and φ ∈ K (h0). Then vanishing happens to (2.1) with u0 = φ,namely

h∞ <∞ and limt→∞ ‖w(t, r;φ)‖L∞([0,h(t)]) = 0,

if one of the following conditions holds:

(i) f satisfies (i)-(iii) in (fM ), h0 <√λ1/f ′(0) and ‖φ‖L∞ is sufficiently small, where

λ1 > 0 is the first eigenvalue of

(2.25) −∆ϕ = λϕ in B1(0) ⊂ RN , ϕ = 0 on ∂B1(0);

(ii) f satisfies (i)-(vii) in (fB), and ‖φ‖L∞ ≤ θ;


(iii) f satisfies (i)-(vii) in (fB), and

(2.26)

∫Bh0

φ(x)dx ≤ θ ·(2πN

eK

)N/2.

Proof. (i) Since h20f′(0) < λ1, there exists a small δ > 0 such that

(2.27)λ1

(1 + δ)2h20

− f ′(0) ≥ 2δ.

Moreover, there exists an s > 0 small such that

−2µϕ′1(1)s ≤ δ2h20, and f(u) ≤ (f ′(0) + δ)u for u ∈ [0, s],

where ϕ1 = ϕ1(|x|) is the first eigenfunction of (2.25) corresponding to λ1 which satisfies

ϕ1(r) > 0 for r ∈ [0, 1), ϕ′1(0) = 0 and ϕ′1(r) < 0 for r ∈ (0, 1].

Set

k(t) := h0

(1 + δ − δ

2e−δt

)and w(t, r) := se−δtϕ1

( r

k(t)

).

Clearly w(t, k(t)) = wr(t, 0) = 0. A direct calculation shows that, for t > 0 and r ∈ [0, k(t)],

wt −∆w − f(w) ≥(

λ1

h20(1 + δ)2

− f ′(0)− 2δ

)w ≥ 0.

On the other hand, by the choice of s we have

µwr(t, k(t)) = µse−δtϕ′1(1)

k(t)≥ −δ

2h0

2e−δt = −k′(t).

Choose ε1 := sϕ1( 22+δ ). When ‖φ‖L∞ ≤ ε1, we have

φ(r) ≤ ε1 ≤ sϕ1

( 2r

h0(2 + δ)

)= w(0, r) for r ∈ [0, h0].

Therefore, (w(t, r), k(t)) is an upper solution to (2.1). By Lemma 1 we have

h(t) ≤ k(t) ≤ h0(1 + δ), h∞ <∞.

By Theorem 3, limt→∞ ‖w(t, ·)‖L∞([0,h(t)]) = 0. This proves (i).By Lemma 3 we can use the same upper solutions as in the proof of Theorem 3.2 of [9]

to prove (ii). Finally, (iii) follows from (ii) by making use of (2.23) with t = N2K , and regard

this t value as the initial time.

From this proposition, we immediately obtain

Corollary 1. If f satisfies (i)-(vi) in (fB) , then limt→∞ ‖w(t, ·)‖L∞([0,h(t)]) = 0 impliesthat h∞ <∞.


2.7. Sufficient conditions for spreading. We will use Lemmas 10, 11 and the notationsvR, RM , vβ and Rβ there.

Proposition 2. Let (w, h) be the solution of (2.1). Then spreading happens, i.e.,

h∞ =∞ and limt→∞

w(t, ·) = 1 locally uniformly in [0,∞),

provided that one of the following conditions is satisfied:(i) f satisfies (i)-(iii) in (fM ) and h0 ≥ R, w0 ≥ vR on [0, R] for some R > RM ;(ii) f satisfies (i)-(vi) in (fB) and h0 ≥ Rβ, w0 ≥ vβ on [0, Rβ ] for some β ∈ (V0(0), 1).

Proof. (i) In this case, by the comparison principle we have w(t, r) ≥ vR(r) for all t > 0 andr ∈ [0, R]. The convergence result (Theorem 3) implies that w converges to a solution v of∆v + f(v) = 0 on [0,∞). We thus obtain v(r) ≥ vR(r) for r ∈ [0, R]. In view of Lemma 7,we must have v ≡ 1.

(ii) Similarly by the comparison principle we have w(t, r) ≥ vβ(r) for all t > 0 andr ∈ [0, Rβ ]. The convergence result (Theorem 3) indicates that w converges to a solution v of∆v+f(v) = 0 on [0,∞). Hence v(r) ≥ vβ(r) for r ∈ [0, Rβ ]. It follows that v(0) ≥ β > V0(0).By Lemma 9 we necessarily have v ≡ 1.

2.8. Dichotomy and sharp threshold for monostable f . In this subsection, based onthe results in the previous subsections, we give a complete description of the long-timedynamical behavior of the solutions of (2.1) for monostable f .

Theorem 7 (Dichotomy). Suppose f satisfies (i)-(iii) in (fM ), and h0 > 0, u0 ∈ K (h0)and (w, h) is the solution of (2.1). Then either spreading happens, namely, h∞ =∞ and

limt→∞

w(t, r) = 1 locally uniformly in [0,∞),

or vanishing happens, namely, h∞ <∞ and

limt→∞

maxr∈[0,h(t)]

w(t, r) = 0.

Moreover, when vanishing happens, h∞ ≤√

λ1

f ′(0) , where λ1 is the first eigenvalue of (2.25).

Proof. This is a simple variation of that of [9, Theorem 5.1]; the details are omitted.

Theorem 8 (Sharp threshold). Let f satisfy (i)-(iii) in (fM ). Suppose that h0 > 0, ψ ∈K (h0), and (wσ, hσ) is the solution of (2.1) with w0 = σψ, σ > 0. Then we have thefollowing conclusions:

(i) If h0 ≥√

λ1

f ′(0) then spreading happens for wσ for every σ > 0.

(ii) If h0 <√

λ1

f ′(0) , then there exists σ∗ = σ∗(h0, φ) ∈ (0,∞] such that spreading happens

when σ > σ∗, and vanishing happens when 0 < σ ≤ σ∗.

Proof. The proof is a simple variation of that of [9, Theorem 5.2]. We again omit thedetails.


2.9. Trichotomy and sharp thresholds for bistable f .

Theorem 9 (Trichotomy). Suppose that f satisfies (i)-(vi) in (fB), h0 > 0, w0 ∈ K (h0)and (w, h) is the solution of (2.1). Then exactly one of the following three cases happens:

(i) Spreading: h∞ =∞ and

limt→∞

w(t, r) = 1 locally uniformly in [0,∞),

(ii) Vanishing: h∞ <∞ and

limt→∞

max0≤x≤h(t)

w(t, r) = 0,

(iii) Transition: h∞ =∞ and

limt→∞

w(t, r) = V0(r) locally uniformly in [0,∞),

where V0 is the unique ground state solution of (2.5).

Proof. By Theorem 3, we have either h∞ < ∞ or h∞ = ∞, and in the former case,limt→∞maxr∈[0,h(t)] w(t, r) = 0.

Suppose now h∞ = ∞. By Lemma 5, Theorem 3 and Lemma 9, v(r) := limt→∞ w(t, r)is a function in 0, θ, 1, V0, or it satisfies v(0) ∈ (0, V0(0)) \ θ, v(∞) = θ.

We now show that v ∈ 1, V0. Since h∞ = ∞, by Corollary 1, v ≡ 0 is impossible. Itremains to exclude the case v ≡ θ, and the case v(0) ∈ (0, V0(0)) \ θ, v(∞) = θ. Clearlyin both cases we have v(∞) = θ.

For the solution w of (2.1), we set

(2.28) E(t) :=

∫Bh(t)(0)

[1

2|∇w|2 − F (w)

]dx =

∫ h(t)

0

ωN

[1

2w2r − F (w)

]rN−1dr,

with ωN denoting the surface area of the unit sphere in RN , and

F (w) =

∫ w

0

f(s)ds.

It is easily checked that E(t) is well-defined and E′(t) ≤ 0.Next we make use of E(t) to show that v(∞) = θ leads to a contradiction. Choose R0 > 0

large such that v(r) ≤ 12 (θ + θ) for r ≥ R0. For any fixed R ≥ R0, we can find T = TR > 0

large so that

h(t) > R, w(t, R) ≤ 1

3(θ + 2θ) for t ≥ T.

Due to the monotonicity of w(t, r) in r for r ≥ h0, it follows that w(t, r) ≤ w(t, R) < θ fort ≥ T and r ≥ R. Hence F (w(t, r)) < 0 for such (t, r), and we have, for t ≥ T ,

E(0) ≥ E(t) >

∫ R

0

ωN

[1

2w2r − F (w)

]rN−1dr ≥ −

∫ R

0

ωNF (w)rN−1dr.

Letting t→∞ we obtain

−∫ R

0

ωNF (v(r))rN−1dr ≤ E(0) for all R ≥ R0.

Since F (v(r))→ F (θ) < 0 as r →∞, the above inequality cannot hold for all large R. Thiscontradiction shows that v(∞) = θ is impossible. Hence v ∈ 1, V0.


Theorem 10 (Sharp threshold). Suppose that f satisfies (i)-(vi) in (fB), and h0 > 0,φ ∈ K (h0), and (w, h) is a solution of (2.1) with w0 = σφ, σ > 0. Then there existsσ∗ = σ∗(h0, φ) ∈ (0,∞] such that spreading happens when σ > σ∗, vanishing happens when0 < σ < σ∗, and transition happens when σ = σ∗.

Proof. By Proposition 1 (ii) we find that vanishing happens if σ < θ/‖φ‖∞. Hence

σ∗ = σ∗(h0, φ) := supσ0 : vanishing happens for σ ∈ (0, σ0]

∈ (0,+∞].

If σ∗ = +∞, then there is nothing left to prove. So we assume that σ∗ is a finite positivenumber.

By definition, vanishing happens for all σ ∈ (0, σ∗). We now consider the case σ = σ∗. Inthis case, we cannot have vanishing, for otherwise we have, for some large t0 > 0, w(t0, r) < θin [0, h(t0)], and due to the continuous dependence of the solution on the initial values, wecan find ε > 0 sufficiently small such that the solution (wε, hε) of (2.1) with w0 = (σ∗ + ε)φsatisfies

wε(t0, r) < θ in [0, hε(t0)].

Hence we can apply Proposition 1 (ii) to conclude that vanishing happens to wε, a contra-diction to the definition of σ∗. Thus at σ = σ∗ either spreading or transition happens.

We show next that spreading cannot happen at σ = σ∗. Suppose this happens. Let vβand Rβ with β ∈ (V0(0), 1) be given by Lemma 11. Then we can find t0 > 0 large (dependingon β) such that

(2.29) h(t0) > Rβ , w(t0, r) > vβ(r) in [0, Rβ ].

By the continuous dependence of the solution on initial values, we can find a small ε > 0 suchthat the solution (wε, hε) of (2.1) with w0 = (σ∗ − ε)φ satisfies (2.29), and by Proposition2, spreading happens for (wε, hε). But this is a contradiction to the definition of σ∗.

Hence transition must happen when σ = σ∗. We show next that spreading happens whenσ > σ∗. Let (w, h) be the solution of (2.1) with w0 = σφ for some σ > σ∗, and denote thesolution of (2.1) with w0 = σ∗φ by (w∗, h∗). By the comparison results we know that

h∗(1) < h(1), w∗(1, r) < w(1, r) on [0, h∗(1)].

Hence we can find ε0 > 0 small such that for any given e ∈ SN−1 and any ε ∈ [0, ε0] we have

Bh∗(1)(εe) ⊂ Bh(1)(0)

and

w∗(1, |x− εe|) < w(1, |x|) for x ∈ Bh∗(1)(εe).

Now define

wε(t, x) = w∗(t+ 1, |x− εe|), hε(t) = h∗(t+ 1).

Clearly (wε, Bhε(t)(εe)) is a solution of (1.1) with initial function w0(x) = w∗(1, |x − εe|).By the comparison principle for the solutions of (1.1) (see [7, 13]) we have, for all t > 0 andε ∈ (0, ε0],

Bhε(t)(εe) ⊂ Bh(t+1)(0), wε(t, x) ≤ w(t+ 1, |x|) in Bhε(t)(εe).

If ω(w) 6= 1, then by Theorem 9 necessarily w(t, x) → V0(|x|) as t → ∞, and by lettingt→∞ in the above inequality we obtain

V0(|x− εe|) ≤ V0(|x|) for all x ∈ RN .


Taking x = εe we obtain V0(0) ≤ V0(ε), a contradiction to the fact that V ′(r) < 0 for r > 0.Thus we must have ω(w) = 1. This proves that spreading happens for σ > σ∗.

3. The General Case

In this section we prove Theorems 1 and 2 for the general problem (1.1).

3.1. Proof of Theorem 1. We first prove that Ω∞ = RN implies spreading, i.e., limt→∞u(t, x) = 1 locally uniformly in x ∈ RN .

In fact, when Ω∞ = RN , by Theorem B (i), there exists T > 0 such that Ω(T ) ⊃ BR0(0),

where R0 :=√

λ1

f ′(0) and λ1 > 0 is the first eigenvalue of (2.25). So there exists R1 > R0

such that Ω(T ) ⊃ BR1(0). We can then select a radially symmetric function φ0(r) such that

(3.1) φ0 ∈ K (R1), 0 < φ0(|x|) < u(T, x) for x ∈ BR1(0).

Then Theorem 8 and the comparison principle imply that, as t→∞,

(3.2) u(t+ T, x) ≥ w(t, |x|;φ0)→ 1 locally uniformly in |x| ∈ [0,∞).

On the other hand, limt→∞maxx∈Ω(t) u(t, x) ≤ 1 follows easily from the assumption (fM ).

Therefore, limt→∞ u(t, x) = 1 locally uniformly in x ∈ RN . So we have proved that spreadinghappens when Ω∞ = RN .

Denote

Σ := σ > 0 : limt→∞

uσ(t, ·) = 1 locally uniformly in RN

and define

σ∗ := inf Σ if Σ 6= ∅, σ∗ =∞ if Σ = ∅.Clearly there are only three possible cases:

(i) σ∗ = 0; (ii) σ∗ ∈ (0,∞); (iii) σ∗ =∞.

In cases (i), by the comparison principle we have Σ = (0,∞) and nothing further is left toprove. In case (iii), by Theorem B and what we have proved above, Ω∞ must be a boundedset and so vanishing happens for every σ > 0.

It remains to consider case (ii). By the comparison principle we easily see that Σ ⊃(σ∗,∞). We now show that σ∗ 6∈ Σ. If σ∗ ∈ Σ then as above we can find φ0(r) satisfying(3.1) with u = uσ∗ . By continuous dependence on initial data, for sufficiently small ε > 0,(3.1) also holds for u = uσ∗−ε. Hence (3.2) holds for u = uσ∗−ε and so spreading happensfor uσ∗−ε, contradicting the definition of σ∗. Thus Σ = (σ∗,∞), and for σ ≤ σ∗, Ω∞ mustbe a bounded set, and so vanishing happens.

Finally we prove the estimate for M(t) when spreading happens. Let φ0 and w(t, r;φ0)satisfy (3.1) and (3.2). Clearly

M(t+ T ) ≥ h(t;φ0) for t > 0.

Fix R2 > R1 such that Ω(T ) ⊂ BR2(0), and then choose a radially symmetric function

φ0(r) such that

φ0 ∈ K (R2), φ0(|x|) > u(T, x) for x ∈ Ω(T ).

Then the comparison principle implies that

w(t, |x|;φ0) ≥ u(t+ T, x) for t > 0, x ∈ Ω(t).


It follows that

M(t+ T ) ≤ h(t;φ0) for t > 0.

By Theorem 7 and Theorem F, we have

h(t;φ0)− [c∗t− c∗(N − 1) log t]→ h0 ∈ R1 as t→∞,

h(t;φ0)− [c∗t− c∗(N − 1) log t]→ h0 ∈ R1 as t→∞.

It thus follows from h(t;φ0) ≤M(t+ T ) ≤ h(t;φ0) that

M(t) = c∗t− c∗(N − 1) log t+O(1) as t→∞.

The proof of Theorem 1.1 is now complete. 2

Remark 3. (i) The above proof shows that σ∗ = 0 if there exists x0 ∈ RN such that

Ω0 ⊃ BR0(x0), R0 :=

√λ1

f ′(0) .

(ii) The case σ∗ = ∞ is also possible. This happens, for example, when Ω0 is containedin a small ball and f(u) converges to −∞ sufficiently fast as u→∞. In [9] we proved that,when

(3.3) lim infu→∞

−f(u)

u8> 0,

there exists h∗ > 0 such that, for any initial function u0 with support in [−h∗, h∗], thesolution u of (1.1) (with N = 1) vanishes. By Lemma 3, we can use such a solution for theone dimensional problem to construct an upper solution for the problem (1.1). Therefore,when Ω0 ⊂ Bh∗(0) and f(u) satisfies (3.3), any solution uσ of (1.1) with initial functionσφ ∈ I(Ω0) vanishes no matter how large σ is.

3.2. Proof of Theorem 2. By Proposition 1 and by the comparison principle, it is easilyseen that

Σ0 := σ > 0 : limt→∞

‖uσ(t, ·)‖L∞(Ω(t)) = 0

is a non-empty open interval of the form (0, σ∗). In case Σ0 = (0,∞), there is nothing leftto prove. So, in what follows, we suppose Σ0 = (0, σ∗) with σ∗ <∞.

Define

Σ1 := σ > 0 : limt→∞

uσ(t, ·) = 1 in L∞loc(RN ).

By Proposition 2 and by the comparison principle, one sees that the set Σ1 is an openinterval of the form (σ∗,∞) if it is not empty. Set

Σ∗ := (0,∞)\(Σ0 ∪ Σ1).

To complete the proof of Theorem 2 we only need to show that

σ∗ = σ∗ (and so Σ∗ = σ∗)

and

ω(uσ∗) ⊂ V0(x0 + ·) : x0 ∈ co(Ω0).

We prove these conclusions by several lemmas.


Lemma 13. Suppose that u is a solution of (1.1) with Ω∞ = RN , and there exist a constantc > 0 and a sequence (tn, xn) satisfying

u(tn, xn) ≥ c, limn→∞

|xn| =∞.

Then for any given R0, R > 0, there exist n and some yn ∈ RN , such that

u(tn, x) ≥ c for x ∈ BR(yn), BR(yn) ∩BR0(0) = ∅.

Proof. Since |xn| → ∞ as n → ∞, necessarily tn → ∞ as n → ∞. By Theorem B, for alllarge n, Γ(tn) is a C2+α closed hypersurface in RN , and

Γ(tn) ⊂ BM(tn)(0) \BM(tn)−πd0(0).

Let r0 > 0 be chosen such that B0 := Br0(0) ⊃ Ω0. For any x ∈ RN \B0, denote

S0x := ν ∈ SN−1 : ν · (z − x) < 0 ∀z ∈ B0.

Then

(3.4) ν ∈ SN−1 : ν · (z − x) < 0 ∀z ∈ co(Ω0) ⊃ S0x,

and it follows from Lemma D and (3.4) that

(3.5) ∂νu(t, x) < 0 for x ∈ Ω(t) \B0 and ν ∈ S0x.

It is easily seen that the cone

Kx := y ∈ RN : y − x = sν, ν ∈ S0x, s > 0

is circular, with vertex at x, and axis passing through 0 and x. Let us denote its openningangle by θx. Clearly θx depends only on |x|, and is a strictly increasing function of |x|, with

lim|x|→r0

θx = 0, lim|x|→∞

θx = π.

For any given R > 0, r > r0 + R and x ∈ RN with |x| > r + R, we consider anothercircular cone

Kxr := y ∈ RN : y − x = s(z − x), z ∈ BR(rx/|x|), s > 0.

This cone has vertex at x, axis passing through 0 and x, and contains the ball BR(rx/|x|).If we denote its opening angle by θx, then it depends only on |x| − r, is a strictly decreasingfunction of |x| − r, and

θx → 0 as |x| − r →∞, θx → π as |x| − r → R.

We now consider the set

Kxr := x ∈ Kx

r : |x| > r ∪BR(rx/|x|).By the regularity of Γ(tn) for large n, and the above properties of θx, it is easily seen that

Kxnr ⊂ Ω(tn) for all large n.

Moreover, in view of the properties of θx, a simple geometrical consideration shows that,if r is chosen large enough, then for all large |x| and y ∈ Kx

r , we have x−y|x−y| ∈ S0

y , and

Kxr ∩ BR0

(0) = ∅. Hence, by (3.5), for such r and all large n, u(tn, ·) is decreasing from

y ∈ Kxnr to xn along the line segment yxn. We thus obtain

u(tn, y) ≥ u(tn, xn) ≥ c for y ∈ Kxnr .


In particular, u(tn, x) ≥ c for x ∈ BR(yn), yn = rxn/|xn|, and BR(yn) ∩BR0(0) = ∅.

Lemma 14. Assume σ ∈ Σ∗ and that uσ(t, x) is extended to [0,∞)×RN by letting uσ(t, x) =0 for x ∈ RN \ Ω(t), t ≥ 0. Then there exists some large R0 > 0 such that uσ(t, x) ≤V0(|x| −R0) for t > 0 and |x| ≥ R0, where V0(s) is the unique solution of the problem

v′′ + f(v) = 0 in R1, v′(0) = 0, v(±∞) = 0.

Proof. It is well known that V0(0) = θ ∈ (θ, 1). Fix θ ∈ (θ, V0(0)). First we prove that, forany given large R0,

(3.6) uσ(t, x) ≤ θ for all t > 0 and |x| ≥ R0.

Otherwise, for any positive integer n, there exists tn > 0 and xn ∈ Ω(tn) such that

uσ(tn, xn) > θ and |xn| ≥ n. By Lemma 13, for any R > 0, we can find some large n and aball BR(yn) such that

(3.7) uσ(tn, x) > θ for x ∈ BR(yn).

We show below that this would imply spreading happens for uσ, contradicting our assump-tion that σ ∈ Σ∗.

We start with an argument similar to the proof of [3, Lemma 3.2] (see also [15]). Let η(t)be the solution of

η′ = f(η), η(0) = θ.

Clearly η(t) is an increasing function with η(∞) = 1.Let w(t, x) be the unique solution to

wt −∆w = f(w) for x ∈ BR(0), t > 0,w = 0 for x ∈ ∂BR(0), t > 0,

w(0, x) = θ for x ∈ BR(0).

By the comparison principle clearly w(t, x) ≤ η(t) for t > 0, x ∈ BR(0).Denote

ρ(x) = (1 + |x|2)−1, ζ(t, x) = ρ(x)[w(t, x)− η(t)].

A simple calculation gives

ζt −∆ζ − 4x

1 + |x|2· ∇ζ =

[2N

1 + |x|2+ f ′(θ(t, x)

]ζ

with θ(t, x) lying between w(t, x) and η(t), and hence θ(t, x) ∈ [0, 1). It follows that

2N

1 + |x|2+ f ′(θ(t, x)) ≤M := 2N + max

u∈[0,1]f ′(u) for t > 0, x ∈ BR(0).

This implies that, for any T > 0, the constant function v := −η(T )/(1 + R2) is a lowersolution of the equation satisfied by e−Mtζ(t, x) over [0, T ]×BR(0). It follows that

ζ(t, x) ≥ −eMtη(T )/(1 +R2) for t ∈ [0, T ], x ∈ BR(0).

In particularζ(T, x) ≥ −eMT η(T )/(1 +R2) for x ∈ BR(0),

which gives

w(T, x) ≥ η(T )

[1− (1 + |x|2)eMT

1 +R2

]for x ∈ BR(0).


For β ∈ (V0(0), 1), let vβ and Rβ be given by Lemma 11. Then we can find T > 0 largesuch that η(T ) > β. From the above estimate for w(T, x), we can subsequently find R > 0large so that

w(T, x) > β for x ∈ BRβ (0).

For R > 0 chosen this way, we now have a ball BR(yn) and tn > 0 such that (3.7) holds.The comparison principle then yields

uσ(tn + t, x) ≥ w(t, x− yn) for x ∈ BR(yn).

It follows thatuσ(tn + T, x) > β ≥ vβ(|x− yn|) for x ∈ BRβ (yn).

Then by the spreading condition in Proposition 2 and by the comparison principle we seethat spreading happens for uσ, contradicting the assumption that σ ∈ Σ∗. This proves (3.6).

Next, we define v1(x) := V0(|x| −R0) for x ∈ RN \BR0. Then, it is easily seen that

−∆v1 ≥ f(v1) for x ∈ RN \BR0(0).

We may now use (3.6) and a simple comparison argument to show that

uσ(t, x) ≤ v1(x) = V0(|x| −R0) for t > 0, |x| > R0.

This proves the lemma.

Lemma 15. Assume σ ∈ Σ∗. Then ω(σφ) is non-empty, and for each U ∈ ω(σφ),

supx∈co(Ω0)

U(x) = supx∈RN

U(x) ∈ [θ, 1].

Proof. By Lemma D, for every t > 0,

supx∈co(Ω0)

uσ(t, x) = supx∈RN

uσ(t, x).

By Proposition 1 and by the comparison principle we see that vanishing happens ifsupx∈RN uσ(t, x) ≤ θ for some t ≥ 0. Since σ ∈ Σ∗, we must have

supx∈co(Ω0)

uσ(t, x) = supx∈RN

uσ(t, x) > θ for every t ≥ 0.

It follows thatsup

x∈co(Ω0)

U(x) = supx∈RN

U(x) ≥ θ.

Since lim supt→∞ uσ(t, x) ≤ 1 always holds, we must also have U ≤ 1.

Lemma 16. Assume σ ∈ Σ∗ and U ∈ ω(σφ). Then U(x) = V (|x − x0|) for some x0 ∈co(Ω0).

Proof. For all large t > 0, say t ≥ T , such that Γ(t) is a smooth closed hypersurface in RN ,define the energy of uσ(t, x) by

E[uσ](t) :=

∫Ω(t)

[1

2|∇uσ|2 − F (uσ)

]dx

where F (u) =∫ u

0f(s)ds. A direct calculation gives

d

dtE[uσ](t) = −

∫Ω(t)

(uσ)2tdx−

µ

2

∫Γ(t)

|∇uσ|3dS ≤ 0.


By Lemma 14, for all large |x|, say |x| ≥ R1, uσ(t, x) < θ. Hence F (uσ(t, x)) ≤ 0 for x ≥ R1.It follows that E[uσ](t) is bounded from below.

So, by a simple variant of the standard argument involving a Lyapunov functional (see,e.g. [16, Theorem 4.3.4]) we see that each element U ∈ ω(σφ) is a stationary solution, thatis, ∆U + f(U) = 0 for x ∈ RN . By Lemma 15 we see that supU ≥ θ. By Lemma 14 wededuce U(x)→ 0 as |x| → ∞. By the moving plane method, one further concludes that U isradially symmestric about some x0 ∈ RN , and is strictly decreasing in the radial directionsaway from x0. We may now apply Lemma 9 to conclude that U(x) ≡ V (|x−x0|). By Lemma15, the maximum of U(x) is achieved at some point in co(Ω0). Hence x0 ∈ co(Ω0).

Lemma 17. Σ∗ consists of exactly one element, that is, σ∗ = σ∗.

Proof. We argue by contradiction. Assume σ1, σ2 ∈ Σ∗ and σ1 < σ2. Then there exista time sequence tn tending to infinity such that uσi(tn, x) → V0(|x − xi|) (i = 1, 2) asn→∞ in the topology of C2

loc(RN ), where x1, x2 ∈ co(Ω0).By comparison we have

uσ1(T, x) < uσ2(T, x) for x ∈ Ω1(T ) ⊂ Ω2(T ),

where Ωi(t) (i = 1, 2) denote the supporting domain of uσi(t, ·), and T > 0 is chosen largeenough such that ∂Ωi(T ) (i = 1, 2) are closed smooth hypersurfaces in RN . Hence thereexists ε0 > 0 such that

uσ1(T, x) < uσ2

(T, x+ εe) for all x ∈ Ω1(T ), e ∈ SN−1, ε ∈ (0, ε0).

By the comparison principle again we have

uσ1(t, x) < uσ2(t, x+ εe) for all t > T, x ∈ Ω1(t), e ∈ SN−1, ε ∈ (0, ε0).

Taking t = tn and letting n→∞ we thus deduce

V0(|x− x1|) ≤ V0(|x+ εe− x2|) for all x ∈ RN , e ∈ SN−1, ε ∈ (0, ε0).

In particular,

V0(0) ≤ V0(|εe+ x1 − x2|) for all e ∈ SN−1, ε ∈ (0, ε0).

This contradicts the fact that V0(0) > V0(r) for r > 0.This completes the proof of the lemma and hence the proof of Theorem 1.2.

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Y. Du: School of Science and Technology, University of New England, Armidale, NSW 2351,Australia


B. Lou: Mathematics & Science College, Shanghai Normal University, Shanghai 200234, China


M. Zhou: School of Science and Technology, University of New England, Armidale, NSW2351, Australia


JEPE Vol 2, 2016, p. 323-339

NONLOCAL NEUMANN PROBLEM WITH CRITICAL EXPONENT

FROM THE POINT OF VIEW OF THE TRACE

FRANCISCO JULIO S.A. CORREA AND AUGUSTO CESAR DOS REIS COSTA

Abstract. In this work, we are concerned with questions of existence and multiplicity

of solutions for a nonlocal and non-homogeneous Neumann boundary value problemsinvolving the p(x)-Laplace operator and critical growth, from the point of view of the

trace, via a truncation argument on generalized Lebesgue-Sobolev spaces.

Dedicated to Prof. David Kinderlehrer for his relevant mathematical achievements.

1. Introduction

In this work, we are going to study questions of existence and multiplicity of solutions forthe nonlocal and non-homogeneous equation, under Neumann boundary condition involvingcritical growth from the point of view of the trace, given by

(1.1)

M (ψ(u)) (−div(|∇u|p(x)−2∇u) + |u|p(x)−2u) = λf(x, u)

[∫Ω

F (x, u)dx

]r− |u|h(x)−2u

in Ω,

M (ψ(u)) |∇u|p(x)−2 ∂u

∂ν= γ|u|q(x)−2u on ∂Ω,

where Ω ⊂ RN is a bounded smooth domain of RN , ψ(u) =

∫Ω

1

p(x)(|∇u|p(x) + |u|p(x))dx,

p, q, h ∈ C(Ω), f : Ω × R → R, and M : R+ → R+ are continuous functions enjoying some

conditions which will be stated later, F (x, u) =

∫ u

0

f(x, ξ)dξ,∂u

∂νis the outer unit normal

derivative, λ, r, γ are real parameters, and ∆p(x) is the p(x)-Laplace operator, that is,

∆p(x)u =

N∑i=1

∂

∂xi

(|∇u|p(x)−2 ∂u

∂xi

), 1 < p(x) < N.

Problems like (1.1), involving the p(x)-Laplace operator, nonlinearities with non stan-dard growth and generalized Lebesgue-Sobolev spaces have been studied by several authors,mainly under homogeneous Dirichlet boundary condition.

2010 Mathematics Subject Classification. Primary: 35J60; Secondary: 58E05, 35J70.Key words and phrases. nonlocal problem; Neumann boundary conditions; trace; Sobolev spaces with

variable exponent; critical exponent; genus.Received 13/10/2016, accepted 07/12/2016.Partially supported by CNPq-Brazil under Grant 301807/2013-2.

323

324 FRANCISCO JULIO S.A. CORREA AND AUGUSTO CESAR DOS REIS COSTA

Problems of this class have several interesting motivations from both physical and mathe-matical point of view. They arise, for instance, in the so-called model of motion of electrorhe-ological fluids, characterized by their capability to change in drastic way the mechanicalproperties when influenced by an exterior electromagnetic field. See, for example, [23].

As we have said in [9], the first major discovery on electrorheological fluids is due to WillisWinslow in 1949.

These fluids have the interesting property that their viscosity depends on the electric fieldin the fluid. He noticed that in such fluids (for instance lithium polymetachrylate) viscosityis inversely proportional to the strength of the field. The fields induces string-like formationsin the fluids, which are parallel to the field. They can raise the viscosity by as much as fiveorders of magnitude. This phenomenon is known as the Winslow effect. Electrorheologicalfluids have been used in robotic and space technology. The experimental research has beendone mainly in the USA, for instance in NASA laboratories.

For more information on this subject, the reader may consult [23] and the referencestherein.

Another application of such a kind of equation is related to image processing. See [24]and the references therein.

In this article, we discuss existence and multiplicity of solutions for the nonlocal Neumannproblem (1.1) with critical growth on its boundary. We study the nonlocal condition forthe two following important classes of functions: M(τ) = a + bτη with a ≥ 0, b > 0, η ≥ 1and m0 ≤ M(τ) ≤ m1, where m0 and m1 are positive constants. Note that the originalKirchhoff term, M(τ) = a+ bτ with a ≥ 0, b > 0, is included in our analysis.

We will study the problem with the following critical Sobolev exponent

(1.2) p∗(x) =(N − 1)p(x)

N − p(x),

where p∗ is a critical exponent from the point of view of the trace.Problems in the form (1.1) are associated with the energy functional

(1.3)Jλ,γ(u) = M (ψ(u))− λ

r + 1

[∫Ω

F (x, u)dx

]r+1

+

∫Ω

1

h(x)|u|h(x)dx

−γ∫∂Ω

1

q(x)|u|q(x)dS

for all u ∈ W 1,p(x)(Ω), where M(t) =

∫ t

0

M(τ)dτ , dS denotes the boundary measure, and

W 1,p(x)(Ω) is the generalized Lebesgue-Sobolev space whose precise definition and propertieswill be established in section 2.

Depending on the behavior of the functions p, q and h, the above functional is differen-tiable and its Frechet-derivative is given by

(1.4)

J ′λ,γ(u)v = M(ψ(u))

∫Ω

(|∇u|p(x)−2∇u∇v + |u|p(x)−2uv

)dx+

∫Ω

|u|h(x)−2uvdx

−λ[∫

Ω

F (x, u)dx

]r ∫Ω

f(x, u)vdx− γ∫∂Ω

|u|q(x)−2uvdS

NONLOCAL NEUMANN PROBLEM 325

for all u, v ∈W 1,p(x)(Ω). So, u ∈W 1,p(x)(Ω) is a weak solution of problem (1.1) if, and onlyif, u is a critical point of Jλ,γ .

We use a truncation argument, the concentration-compactness principle of Lions [22], tothe variable exponent spaces from the point of view of the trace, extended by Bonder andSilva [5], and an appropriate mini-max class of critical points via the classical concept andproperties of the genus, to prove our main result as follows:

Theorem 1.(i) Assume q : ∂Ω→ [1,∞) and A := x ∈ ∂Ω : q(x) = p∗(x) 6= ∅ and M(τ) = a+bτη, witha ≥ 0, b > 0 and η ≥ 1. Moreover, assume the existence of functions p(x), q(x), h(x), β(x) ∈C+(Ω), see section 2, positive constants A1, A2 such that A1t

β(x)−1 ≤ f(x, t) ≤ A2tβ(x)−1

for all t ≥ 0 and for all x ∈ Ω, with f(x, t) = −f(x,−t) for all t ∈ R and for all x ∈ Ω.

Furthermore, β+(r+1) < p− and(η + 1)(p+)η+1

(p−)η< h− ≤ h+ < q−. Then there exists λ > 0

such that for all 0 < λ < λ there exists infinitely many solutions to (1.1) in W 1,p(x)(Ω).(ii) Assume q : ∂Ω → [1,∞) and A := x ∈ ∂Ω : q(x) = p∗(x) 6= ∅. Moreover, assumethe existence of functions p(x), q(x), h(x), β(x) ∈ C+(Ω), see section 2, positive constantsA1, A2 such that A1t

β(x)−1 ≤ f(x, t) ≤ A2tβ(x)−1 for all t ≥ 0 and for all x ∈ Ω, with

f(x, t) = −f(x,−t) for all t ∈ R and for all x ∈ Ω. Furthermore, assume there exists 0 < m0

and m1 such that m0 ≤ M(τ) ≤ m1, withp+m1

m0< h− ≤ h+ < q− and β−(r + 1) < p−.

Then there exists λ > 0 such that for all 0 < λ < λ there exists infinitely many solutions to(1.1) in W 1,p(x)(Ω).

We should point out that the novelty in the present paper is the appearance of the integral

terms,

[∫Ω

F (x, u)dx

]r, the critical growth on Neumann boundary conditions, the use of

the p(x)-Laplacian and the generalized Lebesgue-Sobolev spaces.We should point out that some ideas contained in this paper were inspired by the articles

Ambrosetti and Rabinowitz [1], Azorero and Alonso [3], Bonder and Silva [4], Bonder,Saintier and Silva [[5], [6]], Correa and Costa [10], Correa and Figueiredo [[11], [12]], Fan[[14], [15]], Liang and Zhang [21], Guo and Zhao [19] and Yao [26].

This paper is organized as follows: In section 2 we present some preliminaries on thevariable exponent spaces. In section 3, we give some basic notions on the Krasnoselskii’sgenus. In section 4, we proof of our main result.

2. Preliminaries on variable exponent spaces

First of all, we set

C+(Ω) :=z; z ∈ C(Ω), z(x) > 1 for all x ∈ Ω

and for each z ∈ C+(Ω) we define

z+ := maxΩ

z(x) and z− := minΩz(x).

We denote by M(Ω) the set of real measurable functions defined on Ω.


For each p ∈ C+(Ω), we define the generalized Lebesgue space by

Lp(x)(Ω) :=

u ∈M(Ω);

∫Ω

|u(x)|p(x)dx <∞.

We consider Lp(x)(Ω) endowed with the Luxemburg norm

|u|p(x) := inf

µ > 0;

∫Ω

∣∣∣∣u(x)

µ

∣∣∣∣p(x)

dx ≤ 1

.

We denote by Lp′(x)(Ω) the conjugate space of Lp(x)(Ω), where

1

p(x)+

1

p ′(x)= 1, for all x ∈ Ω.

The proof of the following propositions may be found in Bonder and Silva [4], Bonder,Saintier and Silva [[5], [6]], Fan, Shen and Zhao [16], Fan and Zhang [17] and Fan and Zhao[18].

Proposition 1. (Holder Inequality) If u ∈ Lp(x)(Ω) and v ∈ Lp ′(x)(Ω), then∣∣∣∣∫Ω

uvdx

∣∣∣∣ ≤ ( 1

p−+

1

p ′−

)|u|p(x)|v|p ′(x).

Proposition 2. Set ρ(u) :=

∫Ω

|u(x)|p(x)dx. For all u ∈ Lp(x)(Ω), we have:

(i) For u 6= 0, |u|p(x) = λ ⇔ ρ(uλ ) = 1;(ii) |u|p(x) < 1 (= 1;> 1) ⇔ ρ(u) < 1 (= 1;> 1);

(iii) If |u|p(x) > 1, then |u|p−

p(x) ≤ ρ(u) ≤ |u|p+

p(x);

(iv) If |u|p(x) < 1, then |u|p+

p(x) ≤ ρ(u) ≤ |u|p−

p(x);

(v) limk→+∞

|uk|p(x) = 0 ⇔ limk→+∞

ρ(uk) = 0;

(vi) limk→+∞

|uk|p(x) = +∞ ⇔ limk→+∞

ρ(uk) = +∞.

The generalized Lebesgue - Sobolev space W 1,p(x)(Ω) is defined by

W 1,p(x)(Ω) :=u ∈ Lp(x)(Ω); |∇u| ∈ Lp(x)(Ω)

with the norm

‖u‖ := |u|p(x) + |∇u|p(x).

Denoting ρ1,p(x) :=

∫Ω

(|u|p(x) + |∇u|p(x))dx ∀u ∈ W 1,p(x)(Ω), we have the following

proposition:

Proposition 3. For all u, uj ∈W 1,p(x)(Ω), we have:(i) ‖u‖ < 1 (= 1;> 1) ⇔ ρ1,p(x)(u) < 1 (= 1;> 1);

(ii) If ‖u‖ > 1, then ‖u‖p− ≤ ρ1,p(x)(u) ≤ ‖u‖p+ ;(iii) If ‖u‖ < 1, then ‖u‖p+ ≤ ρ1,p(x)(u) ≤ ‖u‖p− ;(iv) lim

j→+∞‖uj‖ = 0 ⇔ lim

j→+∞ρ1,p(x)(uj) = 0;

(v) limj→+∞

‖uj‖ = +∞ ⇔ limj→+∞

ρ1,p(x)(uj) = +∞.


Proposition 4. Suppose that Ω is a bounded smooth domain in IRN and p ∈ C(Ω) withp(x) < N for all x ∈ Ω. If p1 ∈ C(Ω) and 1 ≤ p1(x) ≤ p∗(x) (1 ≤ p1(x) < p∗(x)) forx ∈ Ω, then there is a continuous (compact) embedding W 1,p(x)(Ω) → Lp1(x)(Ω), where

p∗(x) =Np(x)

N − p(x).

The Lebesgue spaces on ∂Ω are defined as

Lq(x)(∂Ω) := u| u : ∂Ω→ R is measurable and

∫∂Ω

|u|q(x)dS <∞,

and the corresponding (Luxemburg) norm is given by

‖u‖Lq(x)(∂Ω) := ‖u‖q(x),∂Ω := inf

λ > 0 :

∫∂Ω

∣∣∣∣u(x)

λ

∣∣∣∣q(x)

dS ≤ 1

.

Proposition 5. Suppose that Ω is a bounded smooth domain in IRN and p, q ∈ C(Ω) withp(x) < N for all x ∈ Ω. Then there is a continuous and compact embedding W 1,p(x)(Ω) →

Lq(x)(∂Ω), where q(x) < p∗(x) =(N − 1)p(x)

N − p(x).

Proposition 6. (Fan and Zhang [17]) Let Lp(x) : W 1,p(x)(Ω) → (W 1,p(x)(Ω))′ be suchthat

Lp(x)(u)(v) =

∫Ω

|∇u|p(x)−2∇u∇vdx, ∀ u, v ∈W 1,p(x)(Ω),

then(i) Lp(x) : W 1,p(x)(Ω) → (W 1,p(x)(Ω))′ is a continuous, bounded and strictly monotoneoperator;(ii) Lp(x) is a mapping of type S+, i.e., if un u in W 1,p(x)(Ω) and lim sup(Lp(x)(un) −Lp(x)(u), un − u) ≤ 0, then un → u in W 1,p(x)(Ω);

(iii) Lp(x) : W 1,p(x)(Ω)→ (W 1,p(x)(Ω))′ is a homeomorphism.

Proposition 7. (Bonder, Saintier and Silva [5]) Let (uj)j∈N ⊂W 1,p(x)(Ω) be a sequence

such that uj u weakly inW 1,p(x)(Ω). Then there exists countable set I of positive numbers(µi)i∈I and (νi)i∈I and points xi ⊂ A := x ∈ ∂Ω : q(x) = p∗(x) such that

(2.1) |uj |q(x)dS ν = |u|q(x)dS +∑i∈I

νiδxi , weakly − ∗ in the sense of measures.

(2.2) |∇uj |p(x)dx µ ≥ |∇u|p(x)dx+∑i∈I

µiδxi ,weakly − ∗ in the sense of measures.

(2.3) T xiν1

q(xi)

i ≤ µ1

p(xi)

i ,

where T xi = supε>0

T (p(.), q(.),Ωε,i,Γε,i) is the localized Sobolev trace constant where

Ωε,i = Ω ∩Bε(xi) and Γε,i = ∂Bε(xi) ∩ Ω.


Definition 1. We say that a sequence (uj) ⊂W 1,p(x)(Ω) is a Palais-Smale sequence for thefunctional Jλ,γ if

(2.4) Jλ,γ(uj)→ c∗ and J ′λ,γ(uj)→ 0 in (W 1,p(x)(Ω))′,

where

(2.5) c∗ = infh∈C

supt∈[0,1]

Jλ,γ(h(t)) > 0

andC = h : [0, 1]→W 1,p(x)(Ω) : h continuous and h(0) = 0, Jλ,γ(h(1)) < 0.

The number c∗ is called the energy level c∗.When (2.4) implies the existence of a subsequence of (uj), still denoted by (uj), which

converges in W 1,p(x)(Ω), we say that Jλ,γ satisfies the Palais-Smale condition.

3. Preliminaries on Krasnoselskii’s genus

In this section, we present some basic notions on the Krasnoselskii’s genus, whose detailsmay be found in Clark [8] and Krasnoselskii [20], that we will use in the proof of our mainresult. Let X be a real Banach space. Let us denote by U the class of all closed subsetsA ⊂ X \ 0 which are symmetric with respect to the origin, that is, u ∈ A implies −u ∈ A.

Definition 2. Let A ∈ U . The genus γ(A) of A is defined as being the least positive integerk such that there is an odd mapping φ ∈ C(A,Rk) such that φ(x) 6= 0 for all x ∈ A. If sucha k does not exist we set γ(A) =∞. Furthermore, by definition γ(∅) = 0.

In the sequel, we will establish only the properties of genus that will be used throughthis work. More information on this subject may be found in the references Ambrosetti [2],Castro [7], Costa [13] and Krasnoselskii [20].

Theorem 2. Let X = RN and ∂Ω be the boundary of an open, symmetric and boundedsubset Ω ⊂ RN with 0 ∈ Ω. Then γ(∂Ω) = N .

Corollary 1. γ(SN−1) = N.

As a consequence of this, if X is a separable infinite dimensional vector space and S isthe unit sphere in X, then γ(S) =∞.

We now establish a result due to Clark [8].

Theorem 3. Let J ∈ C1(X,R) be a functional satisfying the Palais-Smale condition. Fur-thermore, let us suppose that:

(i) J is bounded from below and even;(ii) There is a compact set K ∈ U such that γ(K) = k and supx∈K J(x) < J(0).Then J possesses at least k pairs of distinct critical points and their corresponding critical

values are less than J(0).


Proof. (i) The proof follows from several lemmas.

Lemma 1. If (uj) ⊂ W 1,p(x)(Ω) is a Palais-Smale sequence, with energy level c, then (uj)

is bounded in W 1,p(x)(Ω).


Proof. Since (uj) is a Palais-Smale sequence with energy level c, we have Jλ,γ(uj)→ c andJ ′λ,γ(uj)→ 0. Then, taking θ such that

(4.1) h+ < θ < q−,

we obtain

C + ‖uj‖ ≥(Jλ,γ(uj)−

1

θJ ′λ,γ(uj)uj

)= aψ(uj) +

b

η + 1ψη+1(uj)− γ

∫∂Ω

1

q(x)|uj |q(x)dS

− λ

r + 1

[∫Ω

F (x, uj)dx

]r+1

− a

θ

∫Ω

|∇uj |p(x)−2∇uj∇ujdx−a

θ

∫Ω

|uj |p(x)−2ujujdx

+λ

θ

[∫Ω

F (x, uj)dx

]r ∫Ω

f(x, uj)ujdx+γ

θ

∫∂Ω

|uj |q(x)−2ujujdS −b

θψη(uj)ρ1,p(x)(uj)

+

∫Ω

1

h(x)|uj |h(x)dx− 1

θ

∫Ω

|uj |h(x)−2ujujdx.

Thus,

C + ‖uj‖ ≥(a

p+− a

θ

)ρ1,p(x)(uj) +

(b

(η + 1)(p+)η+1− b

θ(p−)η

)ρη+1

1,p(x)(uj)

+

(λAr+1

1

θ(β+)r− λAr+1

2

(r + 1)(β−)r+1

)[∫Ω

|u|β(x)dx

]r+1

+

(γ

θ− γ

q−

)∫∂Ω

|uj |q(x)dS

+

(1

h+− 1

θ

)∫Ω

|uj |h(x)dx.

Now, let us suppose that (uj) is unbounded in W 1,p(x)(Ω). Passing to a subsequence ifnecessary, we get ‖uj‖ > 1 and we obtain

C + ‖uj‖ ≥(a

p+− a

θ

)‖u‖p

−+

(b

(η + 1)(p+)η+1− b

θ(p−)η

)‖u‖(η+1)p−

+ λ

(Ar+1

1

θ

1

(β+)r− Ar+1

2

r + 1

1

(β−)r+1

)‖u‖β

±(r+1),

which is a contradiction because p− > β±(r+ 1) > 1. Hence (uj) is bounded in W 1,p(x)(Ω).

Lemma 2. Let (uj) ⊂ W 1,p(x)(Ω) be a Palais-Smale sequence with energy level c andt0 = lim

j→∞ψ(uj). If

c <

(1

θ− 1

q−A

)infi∈I

(γ1−1/p(xi)a1/p(xi)T xi

) p(xi)p∗(xi)p∗(xi)−p(xi)

+K minλ

(h/β)−

(h/β)− − (r + 1) , λ

(h/β)+

(h/β)+ − (r + 1) ,where a = t1 with 0 < t1 < btη0 and K independent on λ. Then, there exists λ0 > 0 suchthat, for all 0 < λ < λ0 the index set I given in the Proposition 7 is empty and uj → u

strongly in Lq(x)(∂Ω) for some u ∈W 1,p(x)(Ω).

Proof. Assume that uj u weakly in W 1,p(x)(Ω). By Proposition 7 and Lemma 1, we have


|uj |q(x)dS ν = |u|q(x)dS +∑i∈I

νiδxi , weakly − ∗ in the sense of measures.

|∇uj |p(x)dx µ ≥ |∇u|p(x)dx+∑i∈I

µiδxi ,weakly − ∗ in the sense of measures.

T xiν1

q(xi)

i ≤ µ1

p(xi)

i , ∀i ∈ I.If I = ∅ then uj → u strongly in Lq(x)(∂Ω). Suppose that I 6= ∅. Let xi be a singular

point of the measures µ and ν. We consider φ ∈ C∞0 (RN ), such that 0 ≤ φ(x) ≤ 1, φ(0) = 0

and support in the unit ball of RN . Consider the functions φi,ε(x) = φ

(x− xiε

)for all

x ∈ RN and ε > 0.As J ′λ,γ(uj)→ 0 in

(W 1,p(x)Ω

)′we obtain

lim J ′λ,γ(uj)(φi,εuj) = 0.

J ′λ,γ(u)φi,εuj = (a+ bψη(uj))

∫Ω

(|∇uj |p(x)−2∇uj∇φi,εuj + |uj |p(x)−2ujφi,εuj

)dx

−γ∫∂Ω

|uj |q(x)−2ujφi,εujdS − λ[∫

Ω

F (x, uj)dx

]r ∫Ω

f(x, uj)(φi,εuj)dx

+

∫Ω

|uj |h(x)−2ujφi,εujdx→ 0,

i.e.,

J ′λ,γ(u)φi,εuj = (a+ bψη(uj))

∫Ω

|∇uj |p(x)−2∇uj∇φi,εujdx− γ∫∂Ω

|uj |q(x)φi,εdS

+(a+ bψη(uj))

∫Ω

|∇uj |p(x)∇φi,εdx+ (a+ bψη(uj))

∫Ω

|uj |p(x)φi,εdx

−λ[∫

Ω

F (x, uj)dx

]r ∫Ω

f(x, uj)(φi,εuj)dx+

∫Ω

|uj |h(x)φi,εdx→ 0.

When j →∞ we get

0 = lim

[(a+ bψη(uj))

∫Ω

(|∇uj |p(x)−2∇uj∇φi,εuj)dx+ (a+ bψη(uj))

∫Ω

|u|p(x)φi,εdx

](a+ btη0)

∫Ω

φi,εdµ− γ∫∂Ω

φi,εdν − λ[∫

Ω

F (x, u)dx

]r ∫Ω

f(x, u)(φi,εu)dx.

We may show that,

limε→0

(a+ btη0)

∫Ω

(|∇u|p(x)−2∇u∇φi,εu)dx→ 0, see Shang and Wang [25].

On the other hand,

limε→0

(a+ btη0)

∫Ω

φi,εdµ = (a+ btη0)µφ(0), γ limε→0

∫∂Ω

φi,εdν = γνφ(0)

and

λ

[∫Ω

F (x, u)dx

]r ∫Ω

f(x, u)(φi,εu)dx→ 0, (a+ bt0)

∫Ω

|u|p(x)φi,εdx→ 0,∫Ω

|uj |h(x)φi,εdx→ 0 as ε→ 0.

Then,


(a + btη0)µiφ(0) = γνiφ(0) implies that γ−1aµi ≤ νi. By T xiν1/p∗(xi)i ≤ µ

1/p(xi)i we

obtain γ−1a(T xi)p(xi)

νp(xi)/p∗(xi)i ≤ γ−1aµi ≤ νi. Thus γ−1a

(T xi)p(xi) ≤ ν1−p(xi)/p∗(xi)

i =

ν(p∗(xi)−p(xi))/p∗(xi)i and γ−1/p(xi)a1/p(xi)T xi ≤ ν

(p∗(xi)−p(xi))/p(xi)p∗(xi)i . Therefore

νi ≥(γ−1/p(xi)a1/p(xi)T xi

) p(xi)p∗(xi)p∗(xi)−p(xi) .

On the other hand, using θ satisfying (4.1)

c = lim Jλ,γ(uj) = lim

(Jλ,γ(uj)−

1

θJ ′λ,γ(uj)uj

).

Thus,

c ≥ lim

(λ

(Ar+1

1

θ

1

(β+)r− Ar+1

2

r + 1

1

(β−)r+1

)[∫Ω

|uj |β(x)

]r+1

+

+

(1

h+− 1

θ

)∫Ω

|uj |h(x)dx+ γ

∫∂Ω

(1

θ− 1

q(x)

)|uj |q(x)

).

Now, setting Aδ =⋃x∈A

(Bδ(x) ∩ Ω) = x ∈ Ω : dist(x,A) < δ, we obtain

c ≥ λ

(A1

θ

1

(β+)r− Ar+1

2

r + 1

1

(β−)r+1

)[∫Ω

|u|β(x)

]r+1

+

(1

h+− 1

θ

)∫Ω

|u|h(x)dx

+γ

(1

θ− 1

q−

)∫∂Ω

|u|q(x)dS + γ

(1

θ− 1

q−Aδ

)∑i∈I

νi.

Since δ > 0 is arbitrary and q is continuous, we get

c ≥ λ

(Ar+1

1

θ

1

(β+)r− Ar+1

2

r + 1

1

(β−)r+1

)[∫Ω

|u|β(x)

]r+1

+

(1

h+− 1

θ

)∫Ω

|u|h(x)dx

+

(1

θ− 1

q−A

)infi∈I



Applying now Holder inequality, we find

c ≥ λ

(Ar+1

1

θ

1

(β+)r− Ar+1

2

r + 1

1

(β−)r+1

)[||u|β(x)|h(x)/β(x)|Ω|

h+−β−

h−

]r+1

(1

h+− 1

θ

)∫Ω

|u|h(x)dx+

(1

θ− 1

q−A

)infi∈I



If ||u|β(x)|h(x)/β(x) ≥ 1, we have

c ≥ c1∣∣∣|u|β(x)

∣∣∣(h/β)−

h(x)/β(x)− λc2

∣∣∣|u|β(x)∣∣∣r+1

h(x)/β(x)+ c3,

where 0 < c2 =

(Ar+1

2

r + 1

1

(β−)r+1− Ar+1

1

θ

1

(β+)r

)|Ω|

(h+−β−)(r+1)

h− and

c3 =

(1

θ− 1

q−A

)infi∈I



So, if g1(t) = c1t(h/β)− − λc2tr+1, this function attains its absolute minimum, for t > 0, at

the point

t =

((r + 1)λc2(h/β)−c1

) 1

(h/β)− − (r + 1).


Note that,

g1(t) = c1

((r + 1)λc2(h/β)−c1

) (h/β)−

(h/β)− − (r + 1) − λc2

((r + 1)λc2(h/β)−c1

) r + 1

(h/β)− − (r + 1), which

implies

g1(t) = c1

((r + 1)λc2c1(h/β)−

) (h/β)−

(h/β)− − (r + 1)

− (r + 1)c1(h/β)−

(r + 1)c1(h/β)−λc2

((r + 1)λc2c1(h/β)−

) r + 1

(h/β)− − (r + 1).

Thus,

g1(t) = c1

((r + 1)λc2c1(h/β)−

) (h/β)−

(h/β)− − (r + 1)

−c1(h/β)−

r + 1

((r + 1)λc2c1(h/β)−

) r + 1

(h/β)− − (r + 1)+ 1

.

So,

g1(t) = c1

((r + 1)λc2c1(h/β)−

) (h/β)−

(h/β)− − (r + 1)(

1− (h/β)−

r + 1

).

Using the fact that β+(r + 1) < h−, we can write

g1(t) = λ

(h/β)−

(h/β)− − (r + 1)K, where K is a negative constant depending only on A1, A2,r, h, β and Ω.

If ||u|β(x)|h(x)/β(x) < 1, we have

c ≥ c1∣∣∣|u|β(x)

∣∣∣(h/β)+

h(x)/β(x)− λc2

∣∣∣|u|β(x)∣∣∣r+1

h(x)/β(x)+ c3,

where 0 < c2 =

(Ar+1

2

r + 1

1

(β−)r+1− Ar+1

1

θ

1

(β+)r

)|Ω|

(h+−β−)(r+1)

h− and

c3 =

(1

θ− 1

q−A

)infi∈I



So, if g2(t) = c1t(h/β)+ − λc2tr+1, this function attains its absolute minimum, for t > 0, at

the point

t =

((r + 1)λc2(h/β)+c1

) 1

(h/β)+ − (r + 1).

Thus, we obtain


g2(t) = λ

(h/β)+

(h/β)+ − (r + 1)K, whereK is a negative constant depending only onA1, A2, r, h, βand Ω. Then

c ≥(

1

θ− 1

q−A

)infi∈I



+K minλ

(h/β)−

(h/β)− − (r + 1) , λ

(h/β)+

(h/β)+ − (r + 1) .Therefore I = ∅.

Lemma 3. Let (uj) ⊂W 1,p(x)(Ω) be a Palais-Smale sequence with energy level c. If

c <

(1

θ− 1

q−A

)infi∈I



+K minλ

(h/β)−

(h/β)− − (r + 1) , λ

(h/β)+

(h/β)+ − (r + 1) ,there exist u ∈ W 1,p(x)(Ω) and a subsequence, still denoted by (uj), such that uj → u in

W 1,p(x)(Ω).

Proof. From

J ′λ,γ(uj)→ 0,

we have

J ′λ,γ(uj)(uj − u) = (a+ bψη(uj))

∫Ω

(|∇uj |p(x)−2∇uj∇(uj − u)dx

+(a+ bψη(uj))

∫Ω

|uj |p(x)−2uj(uj − u)

)dx+

∫Ω

|uj |h(x)−2uj(uj − u)dx

−γ∫∂Ω

|uj |q(x)−2uj(uj − u)dS − λ[∫

Ω

F (x, uj)dx

]r ∫Ω

f(x, uj)(uj − u)dx→ 0,

Note that there exists nonnegative constants c1, c2, c3 and c4 such that

c1 ≤ a+ bψη(uj) ≤ c2and

c3 ≤[∫

Ω

F (x, uj)dx

]r≤ c4.

But using Holder inequality, we have∣∣∣∣∫Ω

|uj |p(x)−2uj(uj − u)dx

∣∣∣∣ ≤ ∫Ω

|uj |p(x)−1|uj − u|dx ≤ C1|u|p(x)/p(x)−1|uj − u|p(x),∣∣∣∣∫Ω


∣∣∣∣ ≤ ∫Ω

|uj |h(x)−1|uj − u|dx ≤ C2|u|h(x)/h(x)−1|uj − u|h(x),

and∣∣∣∣∫Ω

f(x, uj)(uj − u)dx

∣∣∣∣ ≤ A2

∫Ω

|uj |β(x)−1|uj − u|dx ≤ C3|u|β(x)/β(x)−1|uj − u|β(x),

where C1, C2 and C2 are positive constants. Thus


∣∣∣∣∫Ω

|uj |p(x)−2uj(uj − u)dx

∣∣∣∣→ 0,

∣∣∣∣∫Ω


∣∣∣∣→ 0

and ∣∣∣∣∫Ω

f(x, uj)(uj − u)dx

∣∣∣∣→ 0.

By Lemma 2, uj → u in Lq(x)(∂Ω) and using Holder inequality we obtain∣∣∣∣∫∂Ω

|uj |q(x)−2uj(uj − u)

∣∣∣∣ dx→ 0.

Taking

Lp(x)(uj)(uj − u) =

∫Ω

|∇uj |p(x)−2∇uj∇(uj − u)dx,

we obtain Lp(x)(uj)(uj − u)→ 0. We also have Lp(x)(u)(uj − u)→ 0. So

(Lp(x)(uj)− Lp(x)(u), uj − u)→ 0.

From Proposition 6 we have uj → u in W 1,p(x)(Ω).

Lemma 4. The energy functional Jλ,γ associated to (1.1) is unbounded below.

Proof. Recall that

Jλ,γ(u) = aψ(u) +b

η + 1ψη+1(u)− λ

r + 1

[∫Ω

F (x, u)dx

]r+1

+

∫Ω

1

h(x)|u|h(x)dx

−γ∫∂Ω

1

q(x)|u|q(x)dS.

Take 0 < w ∈W 1,p(x)(Ω). For t > 1,

Jλ,γ(tw) ≤ a

p−tp

+

ρ1,p(x)(w) +btp

+(η+1)

(η + 1)(p−)η+1ρη+1

1,p(x)(w) +1

h−th

+

∫Ω

|w|h(x)dx

+λ

r + 1

(A2

β−

)r+1

tβ−(r+1)

(∫Ω

|w|β(x)dx

)r+1

− γ

q+tq−∫∂Ω

|w|q(x)dS.

Then we have

limt→∞

Jλ,γ(tw) = −∞,

and the proof is over.

In what follows we will use a truncation, like in Azorero and Alonso [3], on the functionalJλ,γ , to obtain a special bounded from below functional, as follows.

Assuming ‖u‖ ≤ 1 sufficiently small, by Proposition 2 and (4) we have


Jλ,γ(u) ≥ a

p+‖u‖p

+

+b

(η + 1)(p+)η+1‖u‖(η+1)p+ +

1

h+

1

Sh+

h

‖u‖h+

− λ

r + 1

(A2

β−

)r+11

Sβ−(r+1)β

‖u‖β−(r+1) − γ

q−1

Sq−q

‖u‖q−≥ J1,λ,γ(‖u‖),

where

J1,λ,γ(t) =a

p+th

+

+b

(η + 1)(p+)η+1th

+

+1

h+

1

Sh+

h

th+

− λ

r + 1

(A2

β−

)r+11

Sβ−(r+1)β

tβ−(r+1) − γ

q−1

Sq−q

tq−.

First note that J1,λ,γ(t) < 0 for t ≈ 0, because β−(r + 1) < h+.Furthermore,(

a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh+

h

)th+ − γ

q−1

Sq−q

tq−

=

th+

(a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh+

h

− γ

q−1

Sq−q

tq−−h+

).

In view of q− > h+, we can take R1 small enough, such that(a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh+

h

)Rh

+

1 − γ

q−1

Sq−q

Rq−

1 > 0.

Let us define

λ1 =(r + 1)

2

(β−

A2

)r+1 Sβ−(r+1)β

Rβ−(r+1)1

((a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh+

h

)Rh

+

1 − γ

q−1

Sq−q

Rq−

1

)and R0 = max0 < t ≤ R1; J1,λ1 ≤ 0. Thus, there exist 0 < λ1, R0 and R1 with R0 < R1,such that

Jλ,γ(u) ≥ J1,λ,γ(‖u‖) ≥ J1,λ1(‖u‖) =

(a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh+

h

)‖u‖h

+

− λ1

r + 1

(A2

β−

)r+11

Sβ−(r+1)β

‖u‖β−(r+1) − γ

q−1

Sq−q

‖u‖q−.

for all ‖u‖ < R1 and 0 < λ < λ1, with J1,λ1(R1) > 0 and J1,λ1(R0) = 0.We can choose a nonincreasing function τ1 : [0,∞)→ [0, 1], τ1 ∈ C∞([0,∞)) such that

τ1(x) = 1, se x ≤ R0

and

τ1(x) = 0, se x ≥ R1.

If ‖u‖ > 1, we obtain

Jλ,γ(u) ≥ a

p+‖u‖p

−+

b

(η + 1)(p+)η+1‖u‖(η+1)p− +

1

h+

1

Sh±h

‖u‖h±

− λ

r + 1

(A2

β−

)r+11

Sβ±(r+1)β

‖u‖β±(r+1) − γ

q−1

Sq±q

‖u‖q±.

So,


Jλ,γ(u) ≥(a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh±h

)‖u‖p

−

− λ

r + 1

(A2

β−

)r+11

Sβ±(r+1)β

‖u‖β+(r+1) − γ

q−1

Sq±q

‖u‖q+

= J2,λ,γ(‖u‖),

where

J2,λ,γ(t) =

(a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh±h

)tp−− λ

r + 1

(A2

β−

)r+11

Sβ±(r+1)β

tβ+(r+1)−

γ

q−1

Sq±q

tq+

.

First note that J2,λ,γ(t) < 0 for t ≈ 0+, because β−(r + 1) < p−.Furthermore,(

a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh±h

)tp−− γ

q−1

Sq±q

tq+

=

tp−

((a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh±h

)− γ

q−1

Sq±q

tq+−p−

).

In view of the q+ > p−, we can take R1, sufficiently small, such that(a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh±h

)Rp−

1 −γ

q−1

Sq±q

Rq+

1 > 0.

Let us define

λ2 =(r + 1)

2

(β−

A2

)r+1 Sβ±(r+1)β

Rβ+(r+1)1

((a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh±h

)Rp−1 − γ

q−1

Sq−q

Rq+

1

)and R0 = max0 < t ≤ R1; J2,λ1 ≤ 0. Thus, there are 0 < λ2, R0 and R1 with R0 < R1,such that

Jλ,γ(u) ≥ J2,λ,γ(‖u‖) ≥ J2,λ2(‖u‖) =

(a

p++

b

(η + 1)(p+)η+1+

1

h+

1

Sh±h

)‖u‖p

−

− λ2

r + 1

(A2

β−

)r+11

Sβ±(r+1)β

‖u‖β+(r+1) − γ

q−1

Sq±q

‖u‖q+

.

for all ‖u‖ < R1 and 0 < λ < λ2, with J2,λ2(R1) > 0 and J2,λ2(R0) = 0.We can choose a nonincreasing function τ2 : [0,∞)→ [0, 1], τ2 ∈ C∞([0,∞)), such that

τ2(x) = 1, se x ≤ R0

and

τ2(x) = 0, se x ≥ R1.

Finally, we define

τ(t) =

τ1(t) , t ≤ 1τ2(t) , t > 1

.

Now, we consider the truncated functional, with 0 < λ < λ′ = minλ1, λ2,


Iλ(u) = aψ(u) +b

η + 1ψη+1(u)− λ

r + 1

[∫Ω

F (x, u)dx

]r+1

+

∫Ω

1

h(x)|u|h(x)dx

−γ∫∂Ω

1

q(x)|u|q(x)τ(‖u‖)dS.

Note that, if ‖u‖ ≤ R0 then Jλ,γ(u) = Iλ(u) and if ‖u‖ ≥ R1, then

Iλ(u) = aψ(u) +b

η + 1ψη+1(u) +

∫Ω

1

h(x)|u|h(x)dx− λ

r + 1

[∫Ω

F (x, u)dx

]r+1

.

We can see that Iλ is coercive, and, hence Iλ is bounded from below.

Lemma 5. Iλ is C1(W 1,p(x)(Ω),R), if Iλ(u) ≤ 0 then ‖u‖ < R0 and Iλ(v) = Jλ(v) for all vin a small enough neighborhood of u. Moreover, Iλ satisfies a local Palais-Smale conditionfor c ≤ 0.

Proof. It is immediate that Iλ ∈ C1(W 1,p(x)(Ω),R). If Iλ(u) ≤ 0 then ‖u‖ < R0 byconstruction of truncated functional. Now for all u ∈ BR0

(0) there exists ε > 0 such thatBε(u) ⊂ BR0

(0) and Iλ(v) = Jλ(v) for all v ∈ Bε(u), since ‖v‖ < R0. To prove a localPalais-Smale condition for c ≤ 0, observe que that all Palais-Smale sequences for Iλ withc ≤ 0 must be bounded because the functional is coercive. By Lemma 2, there exists λ0

such that for 0 < λ < λ0

c ≤ 0 <

(1

θ− 1

q−A

)infi∈I



+K minλ

(h/β)−

(h/β)− − (r + 1) , λ

(h/β)+

(h/β)+ − (r + 1) .Hence by Lemma 3, we conclude that Iλ enjoys a local Palais-Smale condition.

Lemma 6. For every n ∈ N there exists ε > 0 such that

γ(I−ελ ) ≥ n,

where I−ελ = u ∈W 1,p(x)0 (Ω); I−ελ (u) ≤ −ε and γ is the Krasnoselskii’s genus.

Proof. Let En ⊂W 1,p(x)(Ω) be an n-dimensional subspace. Hence we have for u ∈ En suchthat ‖u‖ = 1 and 0 < t < R0, we get

Iλ(tu) = aψ(tu) +b

η + 1ψη+1(tu)− λ

r + 1

[∫Ω

F (x, tu)dx

]r+1

+

∫Ω

1

h(x)|tu|h(x)dx

−γ∫∂Ω

1

q(x)|tu|q(x)τ(‖u‖)dS.

Iλ(tu) ≤ atp−

p−ρ1,p(x)(u) +

bt(η+1)p−

(η + 1)(p−)η+1ρη+1

1,p(x)(u) +th−

h−

∫Ω

|u|h(x)dx

− λAr+11 tβ

+(r+1)

(r + 1)(β+)r+1

(∫Ω

|u|β(x)dx

)r+1

− γtq+

q+

∫∂Ω

|u|q(x)dS.

Iλ(tu) ≤ atp−

p−+

bt(η+1)p−

(η + 1)(p−)η+1+th−

h−an −

λAr+11 tβ

+(r+1)

(r + 1)(β+)r+1bn −

γtq+

q+cn,

where

an = sup

(∫Ω

|u|h(x)

);u ∈ En, ‖u‖ = 1

,


,

bn = inf

(∫Ω

|u|β(x)

)r+1

;u ∈ En, ‖u‖ = 1

,

and

cn = inf

∫∂Ω

|u|q(x)dS;u ∈ En, ‖u‖ = 1

.

Then,

Iλ(tu) ≤ atp−

p−+

bt(η+1)p−

(η + 1)(p−)η+1+th−

h−an −

λAr+11 tβ

+(r+1)

(r + 1)(β+)r+1bn

Note that an > 0, bn > 0 and cn > 0, because En is finite dimensional and the normW 1,p(x)(Ω) and Lβ(x)(Ω) are equivalent on En. As β+(r + 1) < p− and 0 < t < R0 weobtain that there exists positive constants ρ and ε such that

Iλ(ρu) < −ε for u ∈ En, ‖u‖ = 1.

Therefore, if we set Sρ,n = u ∈ En : ‖u‖ = ρ, we have that Sρ,n ⊂ I−ελ . Hence, bymonotonicity of genus,

γ(I−ελ ) ≥ γ(Sρ,n) = n

as we wanted to show.

Lemma 7. Let Σ = A ⊂W 1,p(x)(Ω)−0 : A is closed, A = −A, Σk = A ⊂ Σ : γ(A) ≥ kwhere γ stands for the Krasnoselskii’s genus. Then

ck = infA∈Σk

supu∈A

Jλ,γ(u)

is a negative critical value of Jλ,γ and moreover, if c = ck = · · · = ck+r, then γ(Kc) ≥ r + 1

where Kc = u ∈W 1,p(x)(Ω) : Jλ,γ(u) = c, J ′λ,γ(u) = 0.

Proof. The proof follows exactly the steps in Azorero and Alonso [3], using Lemma 6.

Remark 1. The item (ii) of Theorem 1, is shown following the same steps as item (i).

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Mat. Iberoamericana, 1 (1985), 145–201.[23] M. Mihailescu & V. Radulescu, A multiplicity result for a nonlinear degenerate problem arising in the

theory of electrorheological fluids, Proceedings of the Royal Society A, 462 (2006), 2625–2641.[24] M. Ruzicka, Electrorheological Fluids: Modelling and Mathematical Theory, Springer - Verlag, Berlin,

2000.

[25] X. Shang & Z. Wang, Existence of solutions for discontinuous p(x)-Laplacian problems with criticalexponents, Electron. J. Differential Equations, Vol. 2012, no. 25, 1–12.

[26] J. Yao, Solutions for Neumann boundary value problems involving p(x)-Laplace operators, Nonlinear

Anal. 68 (5) (2008), 1271–1283.

Universidade Federal de Campina Grande, Centro de Ciencias e Tecnologia, Unidade Academicade Matematica, CEP:58.109-970, Campina Grande - PB - Brazil


Universidade Federal do Para, Instituto de Ciencias Exatas e Naturais, Faculdade de Matematica,CEP:66.075-110, Belem - PA - Brazil


JEPE Vol 2, 2016, p. 341-355

ON THE SOLVABILITY IN SOBOLEV SPACES AND RELATEDREGULARITY RESULTS FOR A VARIANT OF THE TV-IMAGE

RECOVERY MODEL: THE VECTOR-VALUED CASE

M. BILDHAUER, M. FUCHS, J. MÜLLER AND C. TIETZ

Abstract. We study classes of variational problems with energy densities of lineargrowth acting on vector-valued functions. Our energies are strictly convex variants ofthe TV-regularization model introduced by Rudin, Osher and Fatemi [15] as a powerfultool in the field of image recovery. In contrast to our previous work we here try to figureout conditions under which we can solve these variational problems in classical spaces,e.g. in the Sobolev class W 1,1.

1. Introduction

In their fundamental paper [15] Rudin, Osher and Fatemi proposed to study the variationalproblem (“TV-regularization“)

I[u] :=

∫Ω

|∇u|dx+λ

2

∫Ω

|u− u0|2 dx→ min(1.1)

as a suitable model for the denoising of a grey-scale image u0 : Ω → [0, 1]. Here Ω is abounded domain in R2 and λ > 0 denotes a given parameter. As a matter of fact problem(1.1) in general admits no solution in the Sobolev classW 1,1(Ω) (see, e.g., [1] for a definitionof the spaces W k,p

loc (Ω,RN )), and therefore one has to pass to the space BV (Ω) consistingof functions u ∈ L1(Ω) with finite total variation (compare [13] or [2]). Further unpleasantfeatures of problem (1.1) are that the energy density |∇u| is neither differentiable nor strictlyconvex. So, from the analytical point of view, it seems reasonable to replace (1.1) by moreregular problems being still of linear growth in ∇u including even the case of vector-valuedfunctions u : Rn ⊃ Ω→ RN in more than two variables and combine the denoising procedurewith simultaneous inpainting. We wish to note that such a modification of (1.1) is not onlyof theoretical interest, the practical importance is indicated in the paper [9].

2010 Mathematics Subject Classification. Primary: 49N60; Secondary: 62H35.Key words and phrases. variational problems of linear growth, TV-regularization, denoising and inpaint-

ing of multicolor images, existence of solutions in Sobolev spaces.Received 25/10/2016, accepted 07/12/2016.

341

Next we fix our precise assumptions and state the main results: let Ω denote a boundedLipschitz region in Rn, n ≥ 2 (the case n = 1 is discussed in [11]), and consider a Ln-measurable subset D of Ω such that

0 ≤ Ln(D) < Ln(Ω).(1.2)

The set D represents the inpainting region, on which the data are missing, i.e. in contrastto problem (1.1) our noisy data u0 : Ω−D → RN can just be observed on the region Ω−D,and we require

u0 ∈ L∞(Ω−D,RN ).(1.3)

For a fixed positive parameter λ > 0 we then look at the variational problemJ [u] :=

∫Ω

F (∇u) dx+λ

2

∫Ω−D

|u− u0|2 dx→ min

among functions u : Ω→ RN ,

(1.4)

where the choice D = ∅ corresponds to pure denoising. In the case Ln(D) > 0 the idea of(1.4) is to denoise the incomplete data u0 through the solution u : Ω → RN , which at thesame time fills in the observed image on the missing region D. Concerning the density Four assumptions are as follows: there are constants νi > 0 such that

F ∈ C2(RnN ), F ≥ 0 and (w.l.o.g.) F (0) = 0,(1.5)|DF (Z)| ≤ ν1,(1.6)

F (Z) ≥ ν2|Z| − ν3,(1.7)ν4(1 + |Z|)−µ|X|2 ≤ D2F (Z)(X,X) ≤ ν5(1 + |Z|)−1|X|2(1.8)

hold for all X, Z ∈ RnN . Condition (1.8) is known as µ-ellipticity, and we always require

µ ∈ (1,∞).(1.9)

We note that clearly the second inequality in (1.8) implies (1.6), and the reader, who isinterested in minimal requirements concerning F in specific situations, should consult thereferences given below. However, it turned out, that the hypotheses (1.2)-(1.9) are sufficientfor proving the following results.

I. Existence and uniqueness.

i) The relaxed variant of problem (1.4) defined on the space BV (Ω,RN ) admits at leastone solution u being unique on Ω−D.

ii) The absolutely continuous part∇au (with respect to Lebesgue’s measure) of the matrix-valued measure ∇u is unique.

iii) Any minimizer of the relaxed problem occurs as a (L1-) limit of a J-minimizing sequencefrom the space W 1,1(Ω,RN ).

We refer to the papers [12] and [14], earlier contributions in more specific settings can befound for instance in [5] and [7].

342

II. Duality.

The problem being in duality to (1.4) has a unique solution σ ∈ L∞(Ω,RnN ), and it holdsσ = DF (∇au) a.e. on Ω.

The details can be found in [12] and [7].

III. Regularity.

i) Suppose that N = 1 or assume

F (Z) = Φ(|Z|),(1.10)

if the case N > 1 is considered. In addition we require (compare (1.9))

µ ∈ (1, 2).(1.11)

Then problem (1.4) admits a unique solution u in the spaceW 1,1(Ω,RN ). This solutionsatisfies the "maximum principle"

supΩ|u| ≤ sup

Ω−D|u0|.(1.12)

ii) Under the assumptions of i) u is of class C1,α(Ω,RN ) for any α ∈ (0, 1).iii) If condition (1.11) is dropped, then - keeping the hypothesis (1.10) - we have partial

C1-regularity for any solution u ∈ BV (Ω,RN ) of the relaxed variant of (1.4), i.e. thereexists an open subset Ω0 of Ω such that Ln(Ω−Ω0) = 0 and u ∈ C1,α(Ω0,RN ) for anyα ∈ (0, 1).

A discussion of i) can be found in [5] and [6], for a general proof of i) we refer to Section 3.4of [17].

In case n = 2 ii) was established in [8], Section 3.5 of [17] is devoted to the general case.

Finally, statement iii) can be found in Section 3.3 of [17]. Originally it was proved in [14],and the approach heavily benefits from the work [16]. We wish to emphasize that in thevector-case N > 1 the proof of the regularity results III depend on the structure condition(1.10) in an essential way, since (1.10) "always" implies inequality (1.12), which in turn givesthe boundedness of the (relaxed) minimizer u on account of (1.3).

So the natural question occurs what kind of regularity results can be expected in the vector-case without imposing (1.10). As a matter of fact, we can not hope for everywhere regularityin the sense of ii), but i) and iii) seem to be in reach. To begin with, we look at the casen = 2.

Theorem 1. Let n = 2 and suppose that (1.2), (1.3) as well as (1.5)-(1.9) are valid. Then,if

either: µ < 2 together with D = ∅or: µ < 3

2 in case of general D,

the following statements hold:

a) Problem (1.4) admits a unique solution u in the space W 1,1(Ω,RN ).343

b) The function u is in any space W 1,ploc (Ω,RN ), p ∈ [1,∞), in addition it holds

u ∈W 2,sloc (Ω,RN ), s < 2.

c) There is an open subset Ω0 of Ω such that H-dim(Ω− Ω0) = 0, i.e. Hε(Ω− Ω0) = 0 forall ε > 0, and u ∈ C1,α(Ω0,RN ) for any choice of α ∈ (0, 1).

Remark 1. Theorem 1 follows from Theorem 1.4 in [10] through simplification: we justlet m = 1 in this reference and observe that for m = 1 the density result [10], Theorem 1.1holds automatically, if n = 2 (compare Lemma 2.1, Lemma 2.2 and Remark 2.1, Remark2.2 in [12]), which means that our hypotheses on D are sufficient for proving Theorem 1.1.

Remark 2. Of course partial regularity for BV -minimizers u of the relaxed problem in thesense that u ∈ C1,α(Ω0,RN ) for an open set Ω0 with L2(Ω − Ω0) = 0 should hold for anyvalue µ > 1. However, a proof of this statement would require an inspection of the argumentsoutlined in [3], which means that the data term λ

2

∫Ω−D |u−u0|2 dx has to be incorporated.

Since we are interested in the Sobolev space solvability of problem (1.4), we have to imposethe upper bounds µ < 2 and µ < 3/2, respectively on the parameter µ. Hence these boundsnaturally occur in Theorem 1 c), and at the same time guarantee better estimates for thesize of the singular set (compare the discussion of the size of Ω− Ω0 in [10]).

In the higher-dimensional case of pure denoising we have the following version of Theorem1 a), b):

Theorem 2. Let n ≥ 3 and assume the validity of (1.3),(1.5)-(1.9) together with D = ∅. Ifwe assume

µ < 2,(1.13)

then problem (1.4) is uniquely solvable in the space W 1,1(Ω,RN ). Moreover, the solution is

of class W 1,2loc (Ω,RN ) ∩W 2, 4

2+µ

loc (Ω,RN ).

The proof of Theorem 2 can be traced e.g. in [5], however we will sketch its main ideas inSection 2. If we consider the case Ln(D) > 0 together with n ≥ 3, then we only succeededin proving the existence of a solution u in the space W 1,1(Ω,RN ), if the growth order r ofthe data term is not too large and at the same time a bound of the form µ < µ(n, r) holdsfor the value of µ.

To be precise we replace the functional J from (1.4) through the expression

K[u] :=

∫Ω

F (∇u) dx+

∫Ω−D

ω(|u− u0|

)dx(1.14)

with ω : [0,∞)→ [0,∞), ω(0) = 0, being strictly increasing and strictly convex.

In order not to overload our survey we restrict ourselves to the linear growth case and fixthe example

ω(t) :=√β2 + t2 − β, t ≥ 0,(1.15)

344

with a fixed parameter β > 0. We leave it to the reader to discuss the case of growth ratesr > 1 and to figure out what kind of bounds µ < µ(n, r) have to replace the condition (1.16)below.

Theorem 3. Assume that Ln(D) > 0, let n ≥ 3, and suppose that (1.3),(1.5)-(1.9) arevalid. Moreover, we define K and ω as in (1.14) and (1.15), respectively. Suppose furtherthat

µ <3n

3n− 2(1.16)

is satisfied. Then it holds:

a) The problem K → min in W 1,1(Ω,RN ) admits a unique solution u for which we have

u ∈W 1,ploc (Ω,RN ) ∩W 2,s

loc (Ω,RN ),

p =(

1− µ

2

) 2n

n− 2, s =

(2− µ)n

n− µ.

b) There is an open set Ω0 ⊂ Ω such that Ln(Ω− Ω0) = 0 and u ∈ C1(Ω0,RN ).

The proof of Theorem 3 is presented in Section 3.

Remark 3. As a matter of fact Theorem 3 remains valid in the case D = ∅.

The results from Theorem 3 suffer from the fact that the admissible range for the parameterµ stated in (1.16) decreases, if the dimension n of the domain Ω increases. At the same time,the dimensionless bound stated in (1.13) can only be established for the case of pure denoising(D = ∅) together with the particular (quadratic) data term occuring in the functional J from(1.4).

However, following ideas outlined in Chapter 4.2 of [4], we can state a result on “Sobolevspace solvability” of the problem K → min with general data term ω covering even the caseof a non-empty inpainting region D.

Theorem 4. Under the conditions (1.2) and (1.3) for D and u0, respectively, assume thatthe density F satisfies (1.5)-(1.8) with parameter

(1.17) µ ∈ (1, 2).

For δ > 0 let uδ denote the unique solution of

(1.18) Kδ[w] :=δ

2

∫Ω

|∇w|2 dx+K[w]→ min in W 1,2(Ω,RN )

with K defined in (1.14), the function ω: [0,∞) → [0,∞) being strictly convex and strictlyincreasing with ω(0) = 0. Then, if for any subdomain Ω∗ b Ω it holds

(1.19) supδ>0‖uδ‖L∞(Ω∗,RN ) ≤ c(Ω∗) <∞,

the problem

(1.20) K → min in W 1,1(Ω,RN )

admits a unique solution u, which satisfies u ∈W 1,ploc (Ω,RN ) for any p < 4− µ.

345

Remark 4. The (quadratic) regularization introduced in (1.18) is a well - established toolin connection with linear growth problems for the reason that first the functions uδ aresufficiently regular in order to carry out certain calculations (in particular to establish Cac-cioppoli’s inequality) and second, the minimizers uδ converge (in an appropriate sense)towards the solution of e.g. (1.20) as δ → 0. For an overview of the properties of the uδ werefer to e.g. [4] and [17], some more specific details are presented in the opening lines of thesubsequent sections.

Remark 5. i) With respect to the Sobolev space solvability of the problems (1.4) and(1.20) the upper bound µ < 2 for the ellipticity exponent µ seems to be optimal and werefer the interested reader to Remark 1.4 in [11] for a more detailed discussion of thecritical value µ = 2 even in the case n = 1.

ii) Let us look at variational problems of minimal surface type∫Ω

F (∇u) dx→ min in Φ+W

1,1(Ω,RN )

(compare problem (P) on p. 97 in [4]) and its relaxed variant (see problem (P’) on p. 99in [4]) with F satisfying (1.5)-(1.8) and a sufficiently regular boundary datum Φ.

As it is outlined in Theorems 4.14 and 4.16 of [4], now the choice µ = 3 seems tobe critical and an inspection of the proof of Theorem 4 (see Remark 7) will show: ifµ < 3 and if (1.19) holds, then we can find a solution u, which is of class W 1,1(Ω,RN )∩W 1,p

loc (Ω,RN ) for any p < 4− µ.This solution is unique up to a constant. Note that this result is a slight improvement

of Theorem 4.16 in [4], since it does not require the structure condition for N > 1.

2. The proof of the theorem 2

Following Remark 4 we consider the δ-regularization of problem (1.4), i.e. for δ > 0 we letuδ ∈W 1,2(Ω,RN ) denote the unique solution of

Jδ[u] :=δ

2

∫Ω

|∇u|2 dx+ J [u]→ min in W 1,2(Ω,RN ),(2.1)

whose properties are summarized e.g. in Lemma 3.2 of [10], where for the first order caseat hand some obvious modifications in the statements have to be carried out. We claim thevalidity of

ϕδ :=(1 + |∇u|

)1−µ2 ∈W 1,2loc (Ω)(2.2)

uniformly with respect to the parameter δ. In order to justify (2.2), we observe that theJδ-minimality of uδ implies (recall (2.1) and set Fδ := F + δ

2 | · |2)∫

Ω

D2Fδ(∇uδ)(∂α∇uδ,∇v) dx = λ

∫Ω−D

(u− u0) · ∂αv dx,(2.3)

α = 1, ..., n, for any v ∈ W 1,2(Ω,RN ) with compact support in Ω. Letting v := η2∂αuδ forsome function η ∈ C1

0 (Ω), 0 ≤ η ≤ 1, we obtain (dropping the index δ and using summation346

w.r.t. α) by inserting v into equation (2.3)∫Ω

η2D2F (∇u)(∂α∇u, ∂α∇u) dx+

∫Ω

2D2F (∇u)(η∂α∇u,∇η ⊗ ∂αu) dx

= λ

∫Ω−D

(u− u0) · ∂α(η2∂αu) dx.

(2.4)

If we apply the Cauchy-Schwarz inequality (for the bilinear form D2F (∇u) = D2Fδ(∇uδ))and Young’s inequality to the second term on the left-hand side of (2.4), it follows∫

Ω

η2D2F (∇u)(∂α∇u, ∂α∇u) dx ≤2

∫Ω

D2F (∇u)(∇η ⊗ ∂αu,∇η ⊗ ∂αu) dx

+ 2λ

∫Ω−D

(u− u0) · ∂α(η2∂αu) dx

︸︷︷︸=: T

,

and the second part of (1.8) yields on account of

supδ>0

∫Ω

|∇uδ|dx <∞

the bound ∫Ω

η2D2F (∇u)(∂α∇u, ∂α∇u) dx ≤ c(η) + 2λT.(2.5)

We discuss the quantity T . Performing an integration by parts we get

T = −∫Ω

η|∇u|2 dx−∫Ω

u0∂α(η2∂αu) dx

︸︷︷︸=: T1

and the boundedness of u0 implies

|T1| ≤ c∫Ω

|∇η||∇u|+ η2|∇2u|dx

≤ c(η) + c

∫Ω

η2|∇2u|dx

≤ c(η) + c

∫Ω

η2(1 + |∇u|)−µ2 |∇2u|(1 + |∇u|)

µ2 dx

≤(1.8)

c(η) + ε

∫Ω

D2F (∇u)(∂α∇u, ∂α∇u)η2 dx+ c(ε)

∫Ω

η2|∇u|µ dx.

347

Going back to (2.5) and choosing ε in an appropriate way, we find∫Ω

η2D2F (∇u)(∂α∇u, ∂α∇u) dx+

∫Ω

η2|∇u|2 dx

≤ c(η) + c

∫Ω

η2|∇u|µ dx.

Since we assume µ < 2 (recall (1.13)), we end up with (introducing the parameter δ again)∫Ω∗

D2Fδ(∇uδ)(∂α∇uδ∂α∇uδ) dx+

∫Ω∗

|∇uδ|2 dx ≤ c(Ω∗)(2.6)

for any subdomain Ω∗ b Ω with a finite constant c(Ω∗) independent of δ. Clearly (2.6)implies (2.2), and since according to (2.6) we have a local uniform bound for ‖uδ‖W 1,2

loc (Ω,RN ),the first claim of Theorem 2 follows along the lines of the proof of Theorem 1.3 in [5],moreover, we can choose p = 2. If s ∈

(1, 4

2+µ

]is given, we obtain from Hölder’s inequality∫

Ω∗

|∇2uδ|s dx =

∫Ω∗

(1 + |∇uδ|

)−µ s2 |∇2uδ|s(1 + |∇uδ|

)µ s2 dx

≤

∫Ω∗

(1 + |∇uδ|

)−µ|∇2uδ|2 dx

s2∫

Ω∗

(1 + |∇uδ|

) µs2−s dx

1− s2

,

and since µs/2−s ≤ 2 by our choice of s, we deduce from (2.6) that we can select s = 4/2+µ,which completes the proof of Theorem 2.

Remark 6. Sobolev’s inequality combined with (2.2) yields(1 + |∇uδ|

)(2−µ) nn−2 ∈ L1

loc(Ω)

with exponent (2−µ) nn−2 > 1 iff µ satisfies µ < n+2

n . Moreover, (2−µ) nn−2 ≥ 2 is equivalent

to the unnatural requirement µ ≤ 4n even contradicting (1.9) in case n ≥ 4. Thus we can

not improve the above bound on the integrability exponent s for ∇2uδ by merely exploiting(2.2).


For the reader’s convenience we summarize some properties of the solutions uδ of problem(1.18) with K and ω as defined in (1.14) and (1.15), respectively.

Lemma 1. i) supδ ‖uδ‖W 1,1(Ω,RN ) <∞.ii) (uδ)δ is a K-minimizing sequence.iii) uδ ∈W 2,2

loc (Ω,RN ).

Proof of Lemma 1. The claim i) is obvious while the corresponding reference for a proofof part ii) is quoted in Remark 4, i). Finally, the last statement iii) follows by using thewell-known difference quotient technique.

348

Let us now come to the proof of Theorem 3.

Ad a). In what follows, we again suppress the index δ for notational simplicity and empha-size, that u = uδ as well as F = Fδ. For the proof of part a), we go back to inequality (2.5),in which the quantity T is replaced by

T :=

∫Ω−D

ω′(|u− u0|

) u− u0

|u− u0|︸︷︷︸=: ξ

· ∂α(η2∂αu) dx

with ξ ∈ L∞(Ω) due to (1.15). Note, that by Young’s inequality we have

|T | ≤ c∫Ω

|∂α(η2∂αu)|dx(1.8)≤ ε

∫Ω

D2F (∇u)(∂α∇u, ∂α∇u)η2 dx

+c(ε)

∫Ω

η2(1 + |∇u|

)µdx+ c

∫Ω

|∇η||∇u|dx

︸︷︷︸≤ c(η)

.

The first term in the above estimate can be absorbed in the left-hand side of (2.5), and weobtain ∫

Ω

D2F (∇u)(∂α∇u, ∂α∇u)η2 dx ≤ c∫Ω

η2(1 + |∇u|

)µdx+ c(η).(3.1)

We let

ϕ :=(1 + |∇u|

)1−µ2 , ψ :=(1 + |∇u|

)µ(1− 1n )

and recall

|∇ϕ|2 ≤ cD2F (∇u)(∂α∇u, ∂α∇u).(3.2)

Moreover, we observe the identities (recall 0 ≤ η ≤ 1)

ψnn−1 =

(1 + |∇u|

)µ,

η2(1 + |∇u|

)µ= η2ψ

nn−1 ≤ η

nn−1ψ

nn−1 .

Now from Sobolev’s and Poincaré’s inequality it follows∫Ω

η2(1 + |∇u|

)µdx ≤ ‖ηψ‖

nn−1

Ln/n−1(Ω)≤ c‖∇(ηψ)‖

nn−1

L1(Ω)

≤(∫

Ω

|∇ηψ|dx

︸︷︷︸=: S1

) nn−1

+

(∫Ω

η|∇ψ|dx

︸︷︷︸=: S2

) nn−1

.

In order to proceed further, we observe that (1.16) implies

µ

(1− 1

n

)≤ 1 ⇔ µ ≤ n

n− 1.(3.3)

349

From (3.3) it follows ψ ≤ (1 + |∇u|), henceS1 ≤ c(η).(3.4)

For handling the quantity S2, we write

ψ = ϕ2

2−µµn−1n

(with exponent 22−µµ

n−1n > 1 according to µ > 1). Hölder’s and Young’s inequality then

yield

Snn−1

2 =

∫Ω

η|∇ψ|dx

nn−1

≤ c

∫Ω

ηϕ2µ

2−µn−1n −1|∇ϕ|dx

nn−1

≤ c

∫Ω

η2|∇ϕ|2 dx

12

nn−1

∫Ω

ϕ4µ

2−µn−1n −2 dx

12

nn−1

≤ ε∫Ω

η2|∇ϕ|2 dx+ c(ε)

∫spt(η)

ϕ4µ

2−µn−1n −2 dx

nn−2

.

Finally we observe

ϕ4µ

2−µn−1n −2 ≤ c(1 + |∇u|)

which is a consequence of the inequality(4µ

2− µn− 1

n− 2

)(1− µ

2

)≤ 1

being equivalent to (1.16). Now, by (3.4) along with the above discussion of the quantityS2 (and reintroducing the parameter δ) we deduce from (3.1):∫

Ω∗

D2Fδ(∇uδ)(∂α∇uδ, ∂α∇uδ) dx ≤ c(Ω∗)(3.5)

uniform in δ for all compact subsets Ω∗ b Ω, i.e. (compare (3.2))

ϕδ ∈W 1,2loc (Ω) and therefore |∇u| ∈ Lploc(Ω)

with p = (1− µ/2) (2n/(n−2)). Combining this information with the arguments used at theend of the proof of Theorem 2 we obtain

|∇2uδ| ∈ Lsloc(Ω),

where s = (2−µ)nn−µ , and part a) of Theorem 3 is proved.

Ad b). Here we benefit from Corollary 3.3 in [16] choosing p = 2 in this reference. In orderto justify the application of this corollary in our setting we first formulate a propositionwhich shows that we are actually in the situation of [16].

350

Proposition 1. Under the hypotheses of Theorem 3 we have:

a) for all P ∈ RnN the density F satisfies the hypotheses (H1)-(H4) of [16] (see Section 2in this reference);

b) setting g : Ω × RN → R, g(x, y) := χΩ−Dω(|y − u0(x)|) the following statements holdtrue:i) g is a Borel function;ii) there is a constant C > 0 such that we have

|g(x, y1)− g(x, y2)| ≤ C|y2 − y1|(3.6)

for all x ∈ Ω, y1, y2 ∈ RN .

Proof of Proposition 1. In accordance with the hypotheses of Theorem 3 we can state thatthe density F satisfies (1.5)-(1.8) for our fixed µ < 3n

3n−2 . For proving assertion a) we notethat on account of (1.8) we deal with the non-degenerate case. Quoting Remark 2.6 in [16]we then choose p = 2 in this reference and as a consequence (H2), (H3) as well as (H4)in [16] correspond to the requirement that F is of class C2(RnN ) with D2F (P ) > 0 for allP ∈ RnN . Thus, F satisfies (H2)-(H4) by recalling (1.8). Furthermore, F fulfills (H1) sinceF is (strictly) convex on RnN (see (1.8) again) and of linear growth. In order to verify thestatements of part b) we first remark that assertion b), i) is immediate whereas a calculationof ∇yg(x, y) directly gives (3.6) (recall that the function ω is defined as in (1.15)).

Using Corollary 3.3 in [16] we may immediately conclude that there exists an open subsetΩ0 of Ω such that u ∈ C1(Ω0,RN ) together with Ln(Ω−Ω0) = 0. This completes the proofof Theorem 3.


W.l.o.g. we replace (1.19) by the global bound

(4.1) supδ>0‖uδ‖L∞(Ω,RN ) <∞

and as usual drop the index δ. From (1.18) we infer

(4.2)∫

Ω

DF (∇u) : ∇(η2uΓ

α2

)dx =

∫Ω

Θ · uη2Γα2 dx

for a suitable function Θ ∈ L∞(Ω,RN ) (uniform in δ).

Here we have abbreviated Γ := 1+ |∇u|2, α denotes a positive parameter and η is a functionin C∞(B2R(x0)), B2R(x0) b Ω, such that η = 1 on BR(x0), 0 ≤ η ≤ 1 and |∇η| ≤ c/R.From the convexity of F (see (1.8)) together with (1.7) and F (0) = 0 it follows

F (∇u) : ∇u ≥ c|∇u|,351

hence (4.2) yields on account of (4.1)

∫Ω

η2Γα+12 dx ≤ c

[∫Ω

η2Γα2 dx+

∫Ω

|DF (∇u)||∇η2||u|Γα2 dx

+

∫Ω

|DF (∇u)|η2|u|∣∣∇Γ

α2

∣∣ dx]

≤ c

[∫Ω

η2Γα2 dx+

∫Ω

η|∇η|Γα2 dx+

∫Ω

η2|∇2u|Γα−12 dx

].

Writing Γα/2 = Γ(α−1)/4Γ(α+1)/4 and applying Young’s inequality, we find∫Ω

η2Γα+12 dx ≤ c

[∫Ω

η2Γα−12 dx+

∫Ω

|∇η|2Γα−12 dx

+

∫Ω

η2|∇2u|Γα−12 dx︸︷︷︸

=: T

].(4.3)

For discussing the integral T we observe (compare (4.2))∫Ω

D2F (∇u)(∇∂iu,∇(η2∂iu)) dx = −∫

Ω

Θ∂i(η2∂iu) dx

(summation w.r.t. i = 1, . . . , n), hence∫Ω

D2F (∇u)(∇∂iu,∇∂iu)η2 dx

= −2

∫Ω

D2F (∇u)(∇∂iuη,∇η ⊗ ∂iu) dx−∫

Ω

Θ∂i(η2∂iu) dx.

Proceeding as done after (2.4) (quoting (1.8)) it follows as usual

∫Ω

Γ−µ2 |∇2u|2η2 dx ≤ c

[∫Ω

|∇η|2|∇u|dx+

∫Ω

|∇η||∇u|dx

+

∫Ω

η2|∇2u|dx

]

≤ c(η) + c

∫Ω

η2|∇2u|dx,

and a proper application of Young’s inequality to the last term on the r.h.s. yields

(4.4)∫

Ω

η2Γ−µ2 |∇2u|2 dx ≤ c(η) + c

∫Ω

η2Γµ2 dx,

352

hence (using (4.4))

T =

∫Ω

|∇2u|Γ−µ4 ηηΓ

µ4 +α−1

2 dx

≤ c

[∫Ω

η2|∇2u|2Γ−µ2 dx+

∫Ω

η2Γα−1+µ2 dx

]

≤ c(η) + c

[∫Ω

η2Γµ2 dx+

∫Ω

η2Γα−1+µ2 dx

].

Inserting this inequality into (4.3) we find∫Ω

η2Γα+12 dx ≤ c(η)

∫Ω

Γα−12 dx

+c

[∫Ω

η2Γµ2 dx+

∫Ω

η2Γα−1+µ2 dx

].(4.5)

Let us assume

(4.6) α ≤ 2.

Then

(4.7)∫

Ω

Γα−12 dx ≤ c

[1 +

∫Ω

|∇u|dx

]≤ c <∞

for a constant c independent of δ, and if we further require

α+ 1 > µ,(4.8)α+ 1 > 2α− 2 + µ⇔ 3 > α+ µ(4.9)

then Young’s inequality applied to the last two terms on the r.h.s. of (4.5) yields for subdo-mains Ω∗ b Ω

(4.10)∫

Ω∗|∇u|α+1 dx ≤ c(Ω∗) <∞

uniform in δ. Recalling our assumption (1.17), i.e. µ ∈ (1, 2), we see that α < 3 − µ canbe chosen arbitrarily close to the number 3− µ > 1 satisfying the requirements (4.6), (4.8),(4.9), and (4.10) reads (after introducing the parameter δ again) as

(4.11) supδ>0‖uδ‖W 1,p(Ω∗,Rn) ≤ c(Ω∗, p) <∞

for any Ω∗ b Ω and all p < 4− µ.

Let us fix such a p ∈ (1, 4− µ). From

supδ>0‖uδ‖W 1,1(Ω,RN ) <∞

we deduce the existence of u ∈ BV (Ω,RN ) such that, e.g., uδ → u in L1(Ω,RN ) and a.e. for asubsequence. Moreover, by (4.11), it holds u ∈W 1,p

loc (Ω,RN ) and therefore u ∈W 1,1(Ω,RN ).

The lower semicontinuity of the functional K (more precisely of its relaxed variant) implies

K[u] ≤ lim infδ→0

K[uδ]

353

(compare, e.g., the proof of Theorem 1.1 in [12] and use the fact that u ∈ W 1,1(Ω,RN )),moreover, the minimality of uδ shows (as δ → 0)

K[uδ] ≤ Kδ[uδ] ≤ Kδ[v]→ K[v]

for any v ∈W 1,2(Ω,RN ), thusK[u] ≤ K[v]

for v as above.

Quoting Lemma 2.1 from [12] we end up with

K[u] ≤ K[w] for allw ∈W 1,1(Ω,RN ),

hence u is the (unique) solution of problem (1.20) in the Sobolev space W 1,1(Ω,RN ) satis-fying in addition ∇u ∈ Lploc(Ω,RnN ) for p < 4− µ.

Remark 7. Let us look at the minimal surface case described in Remark 5 ii), where wecan choose Θ = 0 in (4.2). In place of (4.5) we obtain∫

Ω

η2Γα+12 dx ≤ c(η)

∫Ω

|∇η|2Γα−12 dx+ c

∫Ω

η2Γα−1+µ2 dx,

which makes the condition (4.8) superfluous, and the requirement (4.9) can be satisfied atleast for some α > 0, provided we impose the bound µ < 3 on the ellipticity parameter µ.

References

[1] R. A. Adams. Sobolev spaces, volume 65 of Pure and Applied Mathematics. Academic Press, New-York-London, 1975.

[2] L. Ambrosio, N. Fusco, and D. Pallara. Functions of bounded variation and free discontinuity problems.Clarendon Press, Oxford, 2000.

[3] G. Anzelotti and M. Giaquinta. Convex functionals and partial regularity. Archive for Rational Me-chanics and Analysis, 102(3):243 – 272, 1988.

[4] M. Bildhauer. Convex Variational Problems: Linear, nearly Linear and Anisotropic Growth Conditions.Lecture Notes in Mathematics. Springer Berlin Heidelberg, 2003.

[5] M. Bildhauer and M. Fuchs. A variational approach to the denoising of images based on different variantsof the TV-regularization. Appl. Math. Optim., 66(3):331 – 361, 2012.

[6] M. Bildhauer and M. Fuchs. On some perturbations of the total variation image inpainting method.part I: regularity theory. J. Math. Sciences, 202(2):154 – 169, 2014.

[7] M. Bildhauer and M. Fuchs. On some perturbations of the total variation image inpainting method.part II: relaxation and dual variational formulation. J. Math. Sciences, 205(2):121 – 140, 2015.

[8] M. Bildhauer, M. Fuchs, and C. Tietz. C1,α-interior regularity for minimizers of a class of variationalproblems with linear growth related to image inpainting. Algebra i Analiz, 27(3):51–65, 2015.

[9] M. Bildhauer, M. Fuchs, and J. Weickert. Denoising and inpainting of images using TV-type energies.to appear in J. Math. Sciences.

[10] M. Fuchs and J. Müller. A higher order TV-type variational problem related to the denoising andinpainting of images. Nonlinear Analysis: Theory, Methods & Applications, 2016.

[11] M. Fuchs, J. Müller, and C. Tietz. Signal recovery via TV-type energies, 2016. Technical Report No.381, Department of Mathematics, Saarland University.

[12] M. Fuchs and C. Tietz. Existence of generalized minimizers and of dual solutions for a class of variationalproblems with linear growth related to image recovery. J. Math. Sciences, 210(4):458 – 475, 2015.

[13] E. Giusti. Minimal surfaces and functions of bounded variation, volume 80 of Monographs in Mathe-matics. Birkhäuser, Basel, 1984.

354

[14] J. Müller and C. Tietz. Existence and almost everywhere regularity of generalized minimizers for a classof variational problems with linear growth related to image inpainting, 2015. Technical Report No. 363,Department of Mathematics, Saarland University.

[15] L. I. Rudin, S. Osher, and E. Fatemi. Nonlinear total variation based noise removal algorithms. PhysicaD, 60:259 – 268, 1992.

[16] Thomas Schmidt. Partial regularity for degenerate variational problems and image restoration modelsin BV. Indiana Univ. Math. J., 63(1):213 – 279, 2014.

[17] C. Tietz. Existence and regularity theorems for variants of the TV-image inpainting method in higherdimensions and with vector-valued data. PhD thesis, Saarland University, 2016.

Department of Mathematics, Saarland University, P.O. Box 151150, 66041 Saarbrücken, Ger-many,

E-mail address: [email protected], [email protected], [email protected],[email protected]

URL: http://www.math.uni-sb.de/ag/bildhauer/index.html,http://www.math.uni-sb.de/ag/fuchs/

355

JEPE Vol 2, 2016, p. 357-370

SINGULAR PERTURBATIONS OF FORWARD-BACKWARD

p-PARABOLIC EQUATIONS

S.N. ANTONTSEV AND I.V. KUZNETSOV

Abstract. In this paper we have proved the existence of entropy measure-valued solu-

tions to forward-backward p-parabolic equations. We have obtained these solutions as

singular limits of weak solutions to (p, q)-elliptic regularized boundary-value problems asε → 0+. When q > 1 and q 6= 2 we have not defined yet admissible initial and final

conditions even in the form of integral inequalities.

Dedicated to Professor David Kinderlehrer on the occasion of his 75th birthday.

1. Introduction

In this paper we deal with entropy measure-valued solutions to forward-backward p-parabolic equations which are obtained as singular limits of weak solutions to elliptic bound-ary value problems. Singular limits of weak solutions to anisotropic elliptic boundary valueproblems were studied in [1], [2], [3]. The main difference with the present case is that firstorder terms were not involved there.

It is worth to mention that the presence of p-Laplacian can make the problem verycomplicated. For example, gradient Young measures occur in representation of a solution top(x, t)-wave equation [4], p(x, t) > 1, nonlinear evolution equation [5] and forward backward-parabolic equations [6], [7].

In this paper we deal with singular limits of weak solutions to elliptic boundary valueproblem Πp,q,ε with anisotropic (p, q)-Laplacian and quasi-linear first order terms wherep, q > 1. In the limit case as ε → +0 we formally obtain forward-backward p-parabolicequations. In two recent papers [8], [9] two cases (p = 2, q = 2) and, correspondingly,(p > 1, q = 2) were analyzed. In the first case the existence and the uniqueness of entropysolutions were proved. In the second case we have only constructed entropy measure-valuedsolutions with the help of methods invented in [10]–[16].

It is known that in the hyperbolic case there is the equivalence of entropy solutions fordifferent types of approximation [17], [18] only for the Cauchy problem, but nothing is knownhow q-Laplacian influences on vanishing viscosity method applied to the initial boundaryvalue problem.

2010 Mathematics Subject Classification. Primary: 35D99, 35K55, 35K65, 35K92; Secondary: 28A33,35B50, 35R25.

Key words and phrases. entropy solution, forward-backward parabolic equation, gradient Young measure,

maximum principle.Received 11/10/2016, accepted 18/11/2016.Research supported by project III.22.4.2 Boundary value problem in dynamics of heterogeneous media.

357

358 S.N. ANTONTSEV AND I.V. KUZNETSOV

In comparison with [9], when q > 1 and q 6= 2 we have not formulated entropy boundaryconditions in the form of integral inequalities. Therefore, it is still an open question.

This paper is organized as follows. Section 2 is devoted to auxiliary functions and defi-nitions. In section 3 we have formulated problem Πp,q,ε. Here we have defined an entropymeasure-valued solution obtained as a singular limit of weak solutions up,q,ε as ε→ 0+. Insections 4 and 5 we have proved results announced in section 3.

2. Auxiliary definitions and functions

2.1. Boundary entropy-entropy flux triples.

Definition 1. A triple of functions (H,Q,Φ) is called a boundary entropy-entropy fluxtriple if H, Q ∈ C2(R2), Φ ∈ C2(R2;Rd) and for any (z, k) ∈ R2:∂1Q(z, k) = a′(z)∂1H(z, k), ∂1Φj(z, k) = ϕ′j(z)∂1H(z, k), ∂2

1H(z, k) ≥ 0, Q(z, z) =∂1Q(z, z) = Φj(z, z) = ∂1Φj(z, z) = H(z, z) = ∂1H(z, z) = 0, j = 1, . . . , d, Φ(z, k) =(Φ1(z, k), . . . ,Φd(z, k)),where ∂1 means differentiation with respect to the first variable.

Note that function a and vector function ϕ = (ϕ1, . . . , ϕd) are defined by Condition 1formulated in subsection 3.1.Example 1. We consider the following class of boundary entropy-entropy flux triples(Hδ, Qδ,Φδ):

Hδ(z, k) =√

(z − k)2 + δ2 − δ,

Qδ(z, k) =

∫ z

k

a′(λ)∂1Hδ(λ, k) dλ,

Φδ(z, k) =

∫ z

k

ϕ′(λ)∂1Hδ(λ, k) dλ.

Passing to the limit as δ → 0+, we obtain

H0(z, k) = |z − k| , Q0(z, k) = sign(z − k)(a(z)− a(k)),

Φ0(z, k) = sign(z − k)(ϕ(z)−ϕ(k)).

Example 2. We introduce boundary entropy-entropy flux triple (Hδ,w, Qδ,w,Φδ,w) by therule:

Hδ,w(z, k) =((H0(z, k) +H0(z, w)−H0(k,w))2 + δ2

)1/2 − δ,Qδ,w(z, k) =

∫ z

k

a′(λ)∂λHδ,w(λ, k) dλ,

Φδ,w(z, k) =

∫ z

k

ϕ′(λ)∂λHδ,w(λ, k) dλ, ∀ (k, z, w) ∈ R3,

where H0(z, k) + H0(z, w) − H0(k,w) = 2dist(z, I[w, k]), where I[w, k] is an interval withits endpoints w and k. In the limit as δ → 0+ we get

H0,w(z, k) = H0(z, k) +H0(z, w)−H0(k,w),

Q0,w(z, k) = Q0(z, k) +Q0(z, w)−Q0(k,w),

Φ0,w(z, k) = Φ0(z, k) + Φ0(z, w)−Φ0(k,w).

SINGULAR PERTURBATIONS OF FORWARD-BACKWARD p-PARABOLIC EQUATIONS 359

It is important to note that

Q0,w(z, k) =

2(a(k)− a(z)) if z ≤ k ≤ w,2(a(w)− a(z)) if z ≤ w ≤ k,0 if w ≤ z ≤ k,0 if k ≤ z ≤ w,2(a(z)− a(w)) if k ≤ w ≤ z,2(a(z)− a(k)) if w ≤ k ≤ z.

2.2. Essential limits of bounded variation functions.Note that the variation V (f, (0, T )) of arbitrary functions f ∈ BV ((0, T )) has the form

(2.1) V (f, (0, T )) = sup∫ T

0

α′(t)f(t) dt : α ∈ C10 ((0, T )), ‖α‖L∞((0,T )) ≤ 1

.

Without limiting the generality of the foregoing, in formula (2.1) we can replace thecondition ‖α‖L∞((0,T )) ≤ 1 by the condition 0 ≤ α(t) ≤ 1 for a.e. t ∈ (0, T ). Let f(t) ∈L1((0, T )) and

essliminft→0+

f(t) := supε>0

essinf(0,ε)

f(t), esslimsupt→0+

f(t) := infε>0

esssup(0,ε)

f(t),

essliminft→T−

f(t) := supε>0

essinf(T−ε,T )

f(t), esslimsupt→T−

f(t) := infε>0

esssup(T−ε,T )

f(t).

Also we assume that f(t) ∈ BV ((0, T )). This implies the existence of essential limits (see[19, Theorem 9.89],[20, Theorem 3.28])

(2.2) esslimt→0+

f(t) := essliminft→0+

f(t) = esslimsupt→0+

f(t) = limρ→0+

1

ρ

∫ ρ

0

f(t) dt,

(2.3) esslimt→T−

f(t) := essliminft→T−

f(t) = esslimsupt→T−

f(t) = limρ→0+

1

ρ

∫ T

T−ρf(t) dt.

Definition 1, Examples 1, 2 and essential limits (2.2), (2.3) are used in Definition 3 of anentropy measure-valued solution to equation (3.6).

3. Singular limit limε→0+

up,q,ε

3.1. Existence of solutions to problem Πp,q,ε.In this subsection we deal with the elliptic regularization (3.1) of forward-backward p-

parabolic equation (3.6) where an anisotropic p, q-Laplacian is used. Moreover, we are goingto construct a singular limit of weak solutions up,q,ε for non-homogeneous Dirichlet problemΠp,q,ε as ε→ 0+.

Problem Πp,q,ε. For given initial and final conditions u0, uT ∈ L∞(Ω) ∩ W 1,p0 (Ω) the

problem is to find unknown function up,q,ε : GT → R which satisfies the boundary valueproblem:

(3.1) ∂ta(up,q,ε) + divϕ(up,q,ε) = div(|∇up,q,ε|p−2∇up,q,ε

)+ ε∂t

(|∂tup,q,ε|q−2

∂tup,q,ε), (x, t) ∈ GT ,


(3.2) up,q,ε|Γ0= u0(x), up,q,ε|ΓT

= uT (x), up,q,ε|Γl= 0,

in a weak sense, see Definition 2.Here we assume that (x, t) ∈ GT = Ω × (0, T ), x ∈ Ω ⊂ Rd, Γ0 = Ω × t = 0,

ΓT = Ω× t = T, Γl = ∂Ω× [0, T ], p > 1, q > 1, ε > 0. Moreover, there is no dependencebetween two parameters p and q.

Here functions a and ϕ satisfy the following condition.Condition 1. Let a ∈ C2(R), a(0) = 0, ϕ(z) = (ϕ1(z), . . . , ϕd(z)), z ∈ R, ϕj ∈ C2(R),j = 1, . . . , d, ϕ(0) = 0. Function a(z) is not monotone and a′(z) does not equal to zeroidentically on intervals of positive measure.

Definition 2. Function up,q,ε ∈ L∞(GT )∩Lp(0, T ;W 1,p0 (Ω))∩Lq(Ω;W 1,q((0, T ))) is called

a weak solution to problem Πp,q,ε if the following assertions hold.

(EL.1) Let u ∈ L∞(GT )∩Lp(0, T ;W 1,p0 (Ω))∩Lq(Ω;W 1,q((0, T ))) be an extension of func-

tions u0 and uT into GT . Therefore, up,q,ε − u ∈ L∞(GT ) ∩ Lp(0, T ;W 1,p0 (Ω)) ∩

Lq(Ω;W 1,q0 ((0, T ))).

(EL.2) The following equality holds

(3.3a)

∫GT

(− a(up,q,ε)∂tφ−ϕ(up,q,ε) · ∇φ+ |∇up,q,ε|p−2∇up,q,ε · ∇φ

+ ε |∂tup,q,ε|q−2∂tup,q,ε∂tφ

)dxdt = 0

for every φ ∈ L∞(GT ) ∩ Lp(0, T ;W 1,p0 (Ω)) ∩ Lq(Ω;W 1,q

0 ((0, T ))).

Remark 1. We can reformulate (3.3a) in the following way:

(3.3b)

∫GT

(∂ta(up,q,ε)φ+ divϕ(up,q,ε)φ+ |∇up,q,ε|p−2∇up,q,ε · ∇φ

+ ε |∂tup,q,ε|q−2∂tup,q,ε∂tφ

)dxdt = 0.

Proposition 1. Under Condition 1, problem Πp,q,ε has at least one weak solution up,q,ε for

all u0, uT ∈ L∞(Ω) ∩W 1,p0 (Ω). Moreover, maximum principle

(3.4) ‖up,q,ε‖L∞(GT ) ≤M = max(‖u0‖L∞(Ω) , ‖uT ‖L∞(Ω)

).

and energy estimate

(3.5)

∫GT

(|∇up,q,ε|p + ε |∂tup,q,ε|q) dxdt ≤ C(4.20)

hold. The constant C(4.20) is defined in inequality (4.20) and does not depend on ε > 0.

3.2. Forward-backward p-parabolic equation.In the limit as ε→ 0+ equation (3.1) would be forward-backward p-parabolic equation

(3.6) ∂ta(u) + divϕ(u) = div(|∇u|p−2∇u

), (x, t) ∈ GT .

To formulate boundary conditions in the form of integral inequalities (see Remark 2), weneed boundary entropy-entropy flux triples (H,Q,Φ) (see section 2).

Definition 3. An entropy measure-valued solution to equation (3.6) is a measurable func-tion u : GT → R and a gradient Young measure νx,t satisfying the following conditions:


(FB.1) (Regularity) u ∈ L∞(GT ) ∩ Lp(0, T ;W 1,p0 (Ω)),

(3.7) ‖u‖L∞(GT ) ≤M ;

(FB.2) (Gradient Young measure) Here ν = νx,t is a probability measure for a.e. (x, t) ∈GT , and 〈νx,t,Sp(·)〉 is defined as a dual pairing of vector function Sp(ξ) = |ξ|p−2

ξwith a gradient Young measure νx,t, i.e.

(3.8) 〈νx,t,Sp(·)〉 :=

∫Rd

Sp(ξ) dνx,t(ξ);

(FB.3) (Entropy solution) Integral inequality

(3.9)

∫GT

(Q(u, k)∂tγ + Φ(u, k) · ∇γ − ∂1H(u, k)〈νx,t,Sp(·)〉 · ∇γ

− ∂21H(u, k) |∇u|p γ

)dxdt ≥ 0

holds for every boundary entropy-entropy flux triple (H,Q,Φ) and γ ∈ C∞0 (GT ),γ ≥ 0, k ∈ R;

(FB.4) (Incomplete boundary conditions) Essential limits

(3.10) esslimt→0+

∫Ω

(Q0(u(x, t), k) +Q0(u(x, t), u0(x))−Q0(k, u0(x))

)β(x) dx,


∫Ω

(Q0(u(x, t), k) +Q0(u(x, t), uT (x))−Q0(k, uT (x))

)β(x) dx

exist for every β ∈ L1(Ω), β ≥ 0 a.e. in Ω, Q0(z, k) = sign(z − k)(a(z)− a(k)), (seeExample 1 in section 2).

Remark 2. We have the following representation

(3.12) Q0(u(x, t), k) +Q0(u(x, t), u0(x))−Q0(k, u0(x)) =

2(a(k)− a(u(x, t))) if u(x, t) ≤ k ≤ u0(x),2(a(u0(x))− a(u(x, t))) if u(x, t) ≤ u0(x) ≤ k,0 if u0(x) ≤ u(x, t) ≤ k,0 if k ≤ u(x, t) ≤ u0(x),2(a(u(x, t))− a(u0(x))) if k ≤ u0(x) ≤ u(x, t),2(a(u(x, t))− a(k)) if u0(x) ≤ k ≤ u(x, t).

We have not proved yet that u = limε→0+

up,q,ε (q 6= 2) satisfies the following boundary

conditions

(3.13) esslimt→0+

∫Ω

(Q0(u(x, t), k) +Q0(u(x, t), u0(x))−Q0(k, u0(x))

)β(x) dx ≤ 0,


∫Ω

(Q0(u(x, t), k) +Q0(u(x, t), uT (x))−Q0(k, uT (x))

)β(x) dx ≥ 0.

In [9] it was proved that limε→0+

up,2,ε satisfies boundary conditions (3.13), (3.14). Moreover,

if we assume that function a(z) is increasing and a solution u has a trace on the boundary


∂GT , it follows from formula (3.12) and integral inequalities (3.13) and (3.14) that

(3.15) u|Γ0 = u0.

Under Condition 1, the latter equality may be violated.

In the following theorem it is asserted that an entropy measure-valued solution of equation(3.6) is a limit point of the set up,q,εε>0.

Theorem 1. Under Condition 1, there exists an entropy measure-valued solution to equation(3.6) such that

(3.16) u = limε→0+

up,q,ε

in Lp(0, T ;Lp(Ω)). Furthermore, gradient Young measure νx,t is associated with the set∇up,q,εε>0.

4. Proof of Proposition 1

To establish the existence of solution up,q,ε to problem Πp,q,ε, we use well-known resultsfor elliptic equations [12], [13].

We are going to deduce estimates (3.4), (3.5). Let us introduce the function

(4.1) uMp,q,ε = max(up,q,ε −M, 0) =

up,q,ε −M if up,q,ε > M

0 if up,q,ε ≤M, uMp,q,ε

∣∣∂GT

= 0,

(4.2) ∇uMp,q,ε =

∇up,q,ε if up,q,ε > M,

0 if up,q,ε ≤M,∂tu

Mp,q,ε =

∂tup,q,ε if up,q,ε > M,

0 if up,q,ε ≤M.

Putting φ = uMp,q,ε in (3.3a), we derive

(4.3)

∫GT

(∣∣∇uMp,q,ε∣∣p + ε∣∣∂tuMp,q,ε∣∣q) dxdt = I1 + I2,

where

(4.4) I1 :=

∫GT

a(up,q,ε)∂tuMp,q,ε dxdt, I2 :=

∫GT

ϕ(up,q,ε) · ∇uMp,q,ε dxdt.

Taking into account the properties of functions uMp,q,ε, ∇uMp,q,ε, ∂tuMp,q,ε, we have

(4.5) I1 =

∫GT

a(uMp,q,ε +M)∂tuMp,q,ε dxdt =

∫GT

∂t

(∫ uMp,q,ε

0

a(λ+M) dλ

)dxdt =

∫Ω

(∫ uMp,q,ε

0

a(λ+M) dλ

)dx

∣∣∣∣∣t=T

t=0

= 0,

(4.6) I2 =

∫GT

ϕ(uMp,q,ε +M) · ∇uMp,q,ε dxdt =

∫GT

div

(∫ uMp,q,ε

0

ϕ(λ+M) dλ

)dxdt = 0.

Hence, according to (4.3) we get

(4.7)

∫GT

(∣∣∇uMp,q,ε∣∣p + ε∣∣∂tuMp,q,ε∣∣q) dxdt = 0


and

(4.8) uMp,q,ε = 0 =⇒ up,q,ε ≤M.

Analogously, we obtain

(4.9) −up,q,ε ≤M and |up,q,ε| ≤M.

To derive estimate (3.5), we consider an extension u of functions u0, uT into GT such that

(4.10) u ∈ L∞(GT ) ∩ Lp(0, T ;W 1,p0 (Ω)) ∩ Lq(Ω;W 1,q((0, T ))), (up,q,ε − u)|∂GT

= 0.

In equation (3.3a) we put φ = up,q,ε − u:

(4.11) −∫GT

∂t(up,q,ε − u)a(up,q,ε) dxdt

+

∫GT

∇(up,q,ε − u) · ∇up,q,ε |∇up,q,ε|p−2dxdt =∫

GT

∇(up,q,ε − u) ·ϕ(up,q,ε) dxdt− ε∫GT

∂t(up,q,ε − u)∂tup,q,ε |∂tup,q,ε|q−2dxdt.

This reads in the following way

(4.12)

∫GT

(|∇up,q,ε|p + ε |∂tup,q,ε|q) dxdt = J1 + J2 + J3 + J4,

where

(4.13) J1 :=

∫GT

(|∇up,q,ε|p−2∇up,q,ε · ∇u+ ε |∂tup,q,ε|q−2

∂tup,q,ε∂tu)dxdt,

(4.14) J2 :=

∫GT

a(up,q,ε)∂tup,q,ε dxdt,

(4.15) J3 := −∫GT

a(up,q,ε)∂tu dxdt, J4 :=

∫GT

ϕ(up,q,ε) · (∇up,q,ε −∇u) dxdt.

Applying the Young inequality

yz ≤ κr′

r′yr′+κ−r

rzr, 1 < r <∞, r′ =

r

r − 1, y, z ≥ 0, κ ∈ (0, 1],

in several cases, we derive

(4.16) |J1| ≤∫GT

(δp′

p′|∇up,q,ε|p +

δ−p

p|∇u|p +

εδq′

1

q′|∂tup,q,ε|q +

εδ−q1

q|∂tu|q

)dxdt,

(4.17) |J3| ≤∫GT

(1

q|∂tu|q +

1

q′|a(up,q,ε)|q

′)dxdt,

(4.18) |J4| ≤∫GT

(δp2p|∇up,q,ε|p +

δ−p′

2

p′|ϕ(up,q,ε)|p

′+

1

p|∇u|p +

1

p′|ϕ(up,q,ε)|p

′

)dxdt,


where 1p + 1

p′ = 1, 1q + 1

q′ = 1. We evaluate the term J2 in the following way

(4.19) |J2| =

∣∣∣∣∣∣∫

Ω

(∫ up,q,ε(x,t)

0

a(λ) dλ

)dx

∣∣∣∣∣t=T

t=0

∣∣∣∣∣∣ ≤ 2 sup|z|≤M

∣∣∣∣∫ z

0

a(λ) dλ

∣∣∣∣ |Ω| .Gathering last estimates and choosing δ, δ1 and δ2 appropriately small and taking intoaccount (3.4), we find that

(4.20)

∫GT

(|∇up,q,ε|p + ε |∂tup,q,ε|q) dxdt ≤ C(p, q)

∫GT

(|∂tu|q + |∇u|p + |a(up,q,ε)|q

′

+ |ϕ(up,q,ε)|p′ )dxdt+ sup

|z|≤M

∣∣∣∣∫ z

0

a(λ) dλ

∣∣∣∣ |Ω| ≤ C(p, q)

∫GT

(|∂tu|q + |∇u|p

)dxdt

+ C(p, q, |Ω| , T )(

sup|z|≤M

|a(z)|q′+ sup|z|≤M

|ϕ(z)|p′+ sup|z|≤M

∣∣∣∣∫ z

0

a(λ) dλ

∣∣∣∣ ) =: C(4.20).

Estimates (3.4), (3.5) imply that the operator L defined by the formula

L(up,q,ε, ϑ) :=

∫GT

(−a(up,q,ε)∂t(ϑ−u)−ϕ(up,q,ε)·∇(ϑ−u)+|∇up,q,ε|p−2∇up,q,ε·∇(ϑ−u)

+ ε |∂tup,q,ε|q−2∂tup,q,ε∂t(ϑ− u)

)dxdt

is coercive in the space L∞(GT ) ∩ Lp(0, T ;W 1,p0 (Ω)) ∩ Lq(Ω;W 1,q(0, T )), that is

L(up,q,ε, up,q,ε ) =

∫GT

(|∇up,q,ε|p + ε |∂tup,q,ε|q) dxdt− J1 − J2 − J3 − J4 ≥

min

((1− δp

′

p′− δp2

p), ε(1− δq

′

1

q′)

)(∫GT

|∇up,q,ε|p dxdt+

∫GT

|∂tup,q,ε|q dxdt)−C(4.21),

where Ji are defined in (4.13)–(4.15) and

(4.21) C(4.21) =1 + δ−p

p

∫GT

|∇u|p dxdt+1 + εδ−q1

q

∫GT

|∂tu|q dxdt

+ |GT |( 1

q′sup|z|≤M

|a(z)|q′+

1 + δ−p′

2

p′sup|z|≤M

|ϕ(z)|p)

+ 2 |Ω| sup|z|≤M

∣∣∣∣∫ z

0

a(λ) dλ

∣∣∣∣ .Therefore, according to well known results (see [12, Theorem 9.2, Ch. IV], [13, Theorem

8.5]) we conclude that problem Πp,q,ε has at least one weak solution up,q,ε.


In this section we represent entropy measure-valued solutions in the form (3.16). Themain difficulty is that we have only compactness result on up,q,εε>0 in Lp(GT ) and onlyuniform boundedness of ∇up,q,εε>0 in Lp(GT ;Rd).


5.1. Relative compactness of up,q,εε>0.We are going to prove that an entropy measure-valued solution u of equation (3.6) is

represented as a limit point in Lp(0, T ;Lp(Ω)) of the set up,q,εε>0 as ε→ +0. Therefore,a gradient Young measure νx,t is associated with ∇up,q,εε>0.

From Definition 2 it follows that∫GT

∂twp,q,εφdxdt =

∫GT

(ϕ(up,q,ε) · ∇φ− |∇up,q,ε|p−2∇up,q,ε · ∇φ

)dxdt,

where wp,q,ε = a(up,q,ε) − ε |∂tup,q,ε|q−2∂tup,q,ε, φ ∈ Lp(0, T ;W 1,p

0 (Ω)). Therefore, we

conclude that ∂twp,q,ε ∈ Lp′(0, T ;W−1,p(Ω)) and for every φ ∈ Lp(0, T ;W 1,p

0 (Ω)) it followsthat

(5.1)

∣∣∣∣∫GT

∂twp,q,ε(x, t)φ(x, t) dxdt

∣∣∣∣ ≤ C(5.1) ‖φ‖Lp(0,T ;W 1,p0 (Ω)) ,

where

C(5.1) := sup|z|≤M

|ϕ(z)| |GT |p−1p + (C(4.20))

p−1p .

Let g(z) =∫ z

0(a′(τ))2 dτ . We take an arbitrary function ψ ∈ W s,p

0 (Ω), s ≥ [dp ] + 1, where

‖ψ‖L∞(Ω) ≤ C(Ω) ‖ψ‖W s,p0 (Ω). We have

(5.2)

∫Ω

(g(up,q,ε(x, t+ h))− g(up,q,ε(x, t))

)ψ(x) dx

=

∫ t+h

t

∫Ω

∂sg(up,q,ε(x, s))ψ(x) dxds =∫ t+h

t

∫Ω

a′(up,q,ε(x, s))∂sa(up,q,ε(x, s))ψ(x) dxds =∫ t+h

t

∫Ω

a′(up,q,ε(x, s))∂swp,q,ε(x, s)ψ(x) dxds

+

∫ t+h

t

∫Ω

a′(up,q,ε(x, s))∂s(ε |∂sup,q,ε(x, s)|q−2

∂sup,q,ε(x, s))ψ(x) dxds =∫ t+h

t

∫Ω


−∫ t+h

t

∫Ω

εa′′(up,q,ε(x, s)) |∂sup,q,ε(x, s)|q ψ(x) dxds

+

∫ t+h

t

∫Ω

∂s(ε |∂sup,q,ε(x, s)|q−2

∂sa(up,q,ε(x, s)))ψ(x) dxds =∫ t+h

t

∫Ω


−∫ t+h

t

∫Ω


+

∫Ω

ε(|∂tup,q,ε(x, t+ h)|q−2

∂ta(up,q,ε(x, t+h))−|∂tup,q,ε(x, t)|q−2∂ta(up,q,ε(x, t))

)ψ(x) dx.


We apply Theorem 5 from [14] to the set g(up,q,ε)ε>0. It follows from (5.1) that

(5.3)

∣∣∣∣∣∫ t+h

t

∫Ω


∣∣∣∣∣ ≤C(5.1) ‖a′(up,q,ε)ψ‖Lp(t,t+h;W 1,p

0 (Ω)) ≤C(5.1) max(A1,A2) ‖up,q,ε‖Lp(t,t+h;W 1,p

0 (Ω)) C(Ω) ‖ψ‖W s,p0 (Ω) → 0 as h→ 0+,

(5.4)

∣∣∣∣∣∫ t+h

t

∫Ω


∣∣∣∣∣ ≤A2

∫ t+h

t

∫Ω

ε |∂sup,q,ε(x, s)|q dxdsC(Ω) ‖ψ‖W s,p0 (Ω) → 0 as h→ 0+,

(5.5)∣∣∣ ∫

Ω

ε(|∂tup,q,ε(x, t+ h)|q−2

∂ta(up,q,ε(x, t+ h))

− |∂tup,q,ε(x, t)|q−2∂ta(up,q,ε(x, t))

)ψ(x) dx

∣∣∣ ≤∥∥∥ε( |∂tup,q,ε(·, t+ h)|q−2∂ta(up,q,ε(·, t+ h))− |∂tup,q,ε(·, t)|q−2

∂ta(up,q,ε(·, t)))∥∥∥Lq′ (Ω)

×

|Ω|1q C(Ω) ‖ψ‖W s,p

0 (Ω) → 0 as h→ 0+,

where A1 = sup|z|≤M

|a′(z)|, A2 = sup|z|≤M

|a′′(z)|, 1q + 1

q′ = 1. The limits (5.3) and (5.4) are

valid due to estimates (3.5) and absolute continuity of the Lebesgue integral. The latter

inequality and the corresponding limit are valid since the setε |∂tup,q,ε|q−2

∂ta(up,q,ε)ε>0

is relatively compact in Lq′(GT ).

From inequalities (5.3)–(5.5) and corresponding limits the following one-sided limit follows

(5.6) limh→0+

∫ T−h

0

‖g(up,q,ε(·, t+ h))− g(up,q,ε(·, t))‖W−s,p(Ω) dt = 0.

From estimate (3.5), boundedness of the domain Ω, the maximum principle (3.4) andCondition 1 it follows that

(5.7) g(up,q,ε)ε>0 ⊂ Lp(0, T ;W 1,p0 (Ω)) ∩ Lp(0, T ;Lp(Ω)).

From (5.6) and (5.7) (see [14, theorem 5]) it follows that the set g(up,q,ε)ε>0 is relativelycompact in Lp(0, T ;Lp(Ω)) = Lp(GT ). By the strict monotonicity of the function g the setup,q,εε>0 is relatively compact in Lp(GT ).

5.2. Entropy measure-valued solution (u, νx,t).We put φ(x, t) = ∂1H(up,q,ε(x, t), k)γ(x, t) in (3.3b) and integrate by parts:

(5.8)

−∫GT

(Q(up,q,ε, k)∂tγ + Φ(up,q,ε, k) · ∇γ − ∂2

1H(up,q,ε, k)(|∇up,q,ε|p + ε |∂tup,q,ε|q)γ

−∂1H(up,q,ε, k) |∇up,q,ε|p−2∇up,q,ε·∇γ−ε∂1H(up,q,ε, k) |∂tup,q,ε|q−2∂tup,q,ε∂tγ

)dxdt = 0,


where γ ∈ C∞0 (GT ), γ ≥ 0, see (FB.3) in Definition 3. It follows from Definition 1 andestimates (3.4) and (3.5) that the term ε∂2

1H(up,q,ε, k) |∂tup,q,ε|q is integrable and positivealmost everywhere on GT :

(5.9) −∫GT

(Q(up,q,ε, k)∂tγ + Φ(up,q,ε, k) · ∇γ − ∂2

1H(up,q,ε, k) |∇up,q,ε|p γ

−∂1H(up,q,ε, k) |∇up,q,ε|p−2∇up,q,ε·∇γ−ε∂1H(up,q,ε, k) |∂tup,q,ε|q−2∂tup,q,ε∂tγ

)dxdt ≤ 0.

Because of estimates (3.4) and (3.5) the set up,q,εε>0 is uniformly bounded in

Lp(0, T ;W 1,p0 (Ω)), the sequence ∇up,q,εl converges weakly to ∇u in Lp(GT ;Rd) as l → ∞

and εl → 0+. Moreover, there exists a gradient Young measure νx,t and we can expressthe following weak limit with the help of [15]:

∂1H(up,q,ε, k) |∇up,q,ε(x, t)|p−2∇up,q,ε(x, t) ∂1H(u, k)〈νx,t, Sp(·)〉 =

∂1H(u, k)

∫Rd

|ξ|p−2ξ dνx,t.

Let ∇u =∫Rd ξ dνx,t. Since

L(x, t, ξ) = ∂21H(u(x, t), k) |ξ|p γ(x, t)

is convex in ξ ∈ Rd, we have the following result (see [16, Theorem 4.3])

(5.10)

∫GT

∂21H(u, k) |∇u|p γ dxdt ≤

∫GT

∂21H(u, k)〈νx,t, |·|p〉γ dxdt ≤

lim infεl→+0

∫GT

∂21H(uεl , k) |∇up,q,εl |

pγ dxdt.

Moreover, we have

(5.11)

∣∣∣∣∫GT

ε∂1H(up,q,ε, k) |∂tup,q,ε|q−2∂tup,q,ε∂tγ dxdt

∣∣∣∣ ≤ε

1q

(∫GT

ε |∂tup,q,ε|q dxdt) q−1

q(∫

GT

|∂1H(up,q,ε, k)∂tγ|q dxdt) 1

q ≤

ε1q

(C(4.20)

) q−1q(∫

GT

|∂tγ|q dxdt) 1

q

sup|z|≤M

|∂1H(z, k)| → 0 as ε→ 0 + .

Since the set up,q,εε>0 is uniformly bounded in L∞(GT ) and it is relatively compact inLp(0, T ;Lp(Ω)) (see subsection 5.1), from this set we can select a subsequence up,q,εll∈Nwhich has a limit u ∈ L∞(GT )∩ Lp(0, T ;Lp(Ω)). Let function up,q,εl satisfy inequality (5.9)when ε = εl. It is easy to see that the function u satisfies the inequalities

‖u‖L∞(GT ) ≤M,

∫GT

|∇u|p dxdt ≤ C(4.20).

Therefore, a limit point of a subsequence up,q,εll∈N is an entropy measure-valued so-lution to equation (3.6). Moreover, a gradient Young measure νx,t is associated with asubsequence ∇up,q,εll∈N.


5.3. Essential limits (3.10) and (3.11).We are going to prove the existence of essential limits (3.10) and (3.11) with the help of

boundary entropy-entropy flux triples from Example 2 (see section 2).We put γ(t,x) = α(t)β(x) in inequality (3.9), where α ∈ C∞0 ((0, T )), β ∈ C∞0 (Ω), α ≥ 0,

β ≥ 0. Here we follow the idea represented in [11]. It can be shown that the function

fδ,w,k,β(t) =

∫Ω

Qδ,w(u(x, t), k)β(x) dx

belongs to BV ((0, T )). Moreover, similar results were proved in [8], [9]. Let⋃i

Ei = Ω. We

replace smooth function β(x) with a simple function∑i

βiχEi(x). Moreover, initial data u0

can be approximated by∑i

wiχEi(x). Therefore∣∣∣∣∣∫

Ω

Q0,u0(x)(u(x, t), k)β(x) dx−∑i

∫Ei

Qδ,wi(u(x, t), k)βi dx

∣∣∣∣∣→ 0

for a.e. t ∈ (0, T ) as∑i

wiχEi(x) ⇒

Ωu0(x),

∑i

βiχEi(x) ⇒

Ωβ(x), δ → 0+. The existence of

(3.11) is proved in the same way.

5.4. On Remark 2.It is important to note that it is impossible to deduce boundary conditions (3.13)–(3.14)

with the help of methods invented in [11]. Consider the functions s, ξK ∈ C0([0, T ]),introduced in [11]:

(5.12) s(t) = mint, T10, T − t, ξK(t) = 1− exp

(−s(t)K

), t ∈ [0, T ].

Due to the presence of ε∂t(|∂tup,q,ε|q−2

∂tup,q,ε)

in equation (3.1), we can not even formulatethe analogous lemma for up,q,ε when q 6= 2, q > 1.

Lemma 1. For any value of the positive parameter ε the inequality holds

(5.13) −∫GT

(Q(up,2,ε, k)∂tγ + Φ(up,2,ε, k) · ∇γ + εH(up,2,ε, k)∂2

t γ

− ∂1H(up,2,ε, k) |∇up,2,ε|p−2∇up,2,ε · ∇γ − ∂21H(up,2,ε, k) |∇up,2,ε|p γ

)ξKε

dxdt ≤

2ε

∫GT

H(up,2,ε, k)ξ′Kε∂tγ dxdt+A1

∫Ω

(H(u0(x), k)γ(x, 0) +H(uT (x), k)γ(x, T )

)dx,

where (H,Q,Φ) is an arbitrary boundary entropy-entropy flux triple, γ ∈ C∞0 (Ω×R) is non-negative test function, k ∈ R, M = max(‖u0‖L∞(Ω) , ‖uT ‖L∞(Ω)), A1 = max ‖a′‖C([−M,M ]),

Kε = ε/A1.

With the help of Lemma 1 proved in [9] we deduce

(5.14) −∫GT

(Q(u, k)∂tγ + Φ(u, k) · ∇γ − ∂1H(u, k)〈νx,t,Sp(·)〉 · ∇γ

− ∂21H(u, k) |∇u|p γ

)dxdt ≤ A1

∫Ω

(H(u0(x), k)γ(x, 0) +H(uT (x), k)γ(x, T )

)dx.


In comparison with (3.9), due to H0,u0(x)(u0(x), k) = H0,uT (x)(uT (x), k) = 0 a.e. in Ω,the right hand side of inequality (5.14) enables to define signs of essential limits (3.11) and(3.12). For details see subsection 5.3, also see [11], [8], [9].

Conclusion

In present paper we have proved only the existence of an entropy measure-valued solution(u, νx,t) to the forward-backward p-parabolic equation. Function u has been obtained in theform lim

ε→0+up,q,ε where up,q,ε is a weak solution to problem Πp,q,ε. We have not derived

boundary conditions in the case q 6= 2, q > 1, see Remark 2 in section 3.

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domains, North-Holland Mathematical Library, 69, Elsevier, Amsterdam, 2006.[14] J. Simon, Compact sets in the space Lp(0, T ;B), Annali di Matematica Pura ed Applicata 146, (1987),

no. 1, 65–96.

[15] D. Kinderlehrer, P. Pedregal, Gradient Young measures generated by sequences in Sobolev spaces, TheJournal of Geometric Analysis 4, (1994), 59–90.

[16] M.A. Sychev, A new approach to Young measure theory, relaxation and convergence in energy, Annalesde l’Institut Henri Poincare (C) Non Linear Analysis 16, (1999), no. 6, 773–812.

[17] P. Marcati, R. Natalini, Convergence of the pseudo-viscosity approximation for conservation laws,

Nonlinear Analysis: Theory, Methods & Applications 23, (1994), no. 5, 621–628.[18] A. Matas, J. Merker, The limit of vanishing viscosity for doubly nonlinear parabolic equations, Electronic

Journal of Qualitative Theory of Differential Equations 8, (2014), 1–14.

[19] O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier, F. Lenzen: Variational methods in imaging,Applied Mathematical Sciences, 167, Springer, New York, 2009.

[20] L. Ambrosio, N. Fusco, D. Pallara: Functions of bounded variation and free discontinuity problems,

Oxford University Press, New York, 2000.


Lavrentyev Institute of Hydrodynamics Siberian Branch of RAS, av. Ac. Lavrentyev 15, 630090Novosibirsk, Russia,

Novosibirsk State University, Pirogov av. 2, 630090 Novosibirsk, Russia,

E-mail address: kuznetsov [email protected]

CMAF-CIO, University of Lisbon, av. Prof. Gama Pinto 2, 1649-003 Lisbon, Portugal,E-mail address: [email protected]

JEPE Vol 2, 2016, p. 371-387

A BREATHER CONSTRUCTION FOR A SEMILINEAR CURL-CURL

WAVE EQUATION WITH RADIALLY SYMMETRIC COEFFICIENTS

MICHAEL PLUM AND WOLFGANG REICHEL

Dedicated to Prof. David Kinderlehrer on the occasion of his 75th birthday

Abstract. We consider the semilinear curl-curl wave equation

s(x)∂2t U +∇×∇× U + q(x)U ± V (x)|U |p−1U = 0 for (x, t) ∈ R3 × R.

For any p > 1 we prove the existence of time-periodic spatially localized real-valued

solutions (breathers) both for the + and the − case under slightly different hypotheses.Our solutions are classical solutions that are radially symmetric in space and decay

exponentially to 0 as |x| → ∞. Our method is based on the fact that gradient fields of

radially symmetric functions are annihilated by the curl-curl operator. Consequently, thesemilinear wave equation is reduced to an ODE with r = |x| as a parameter. This ODE

can be efficiently analyzed in phase space. As a side effect of our analysis, we obtain

not only one but a full continuum of phase-shifted breathers U(x, t + a(x)), where U isa particular breather and a : R3 → R an arbitrary radially symmetric C2-function.

1. Introduction

Real-valued breathers (i.e., time-periodic spatially localized solutions) of nonlinear waveequations in Rd × R have attracted attention of both physicists and mathematicians withthe sine-Gordon equation (1.2) being a prominent example. The phenomenon of existence ofbreathers is quite rare. In the context of semilinear scalar 1 + 1-dimensional wave equations(different from sine-Gordon) we are only aware of the example given in [10]. Breathersin discrete nonlinear lattice equations are more common, cf. [16] for a fundamental resultand [15] for an overview with many references. Complex valued time-harmonic breathersof the type u(x, t) = eiωtuuu(x) in wave equations with S1-equivariant nonlinearity are verywell studied objects, cf. [8], [20]. Such time-harmonic breathers are usually much easier toobtain, see Theorem 3 below, and they are the object of many papers in the context of thenonlinear Schrodinger equation, e.g. in the case of potentials with spatial periodicity [3],[18].

In this paper we consider the 3 + 1-dimensional semilinear curl-curl wave equation

(1.1)± s(x)∂2tU +∇×∇× U + q(x)U ± V (x)|U |p−1U = 0 for (x, t) ∈ R3 × R

2000 Mathematics Subject Classification. Primary: 35L71; Secondary: 34C25.Key words and phrases. semilinear wave-equation, breather, phase plane method.Received 28/10/2016, accepted 28/10/2016.

We gratefully acknowledge financial support by the Deutsche Forschungsgemeinschaft (DFG) throughCRC 1173.

371

372 MICHAEL PLUM AND WOLFGANG REICHEL

with p > 1. We will assume later that V, q, s : R3 → (0,∞) are positive, radially symmetricfunctions. We look for classical real-valued solutions U : R3×R→ R3 which are T -periodicin time and spatially exponentially localized, i.e., supR3×R |U(x, t)|eδ|x| <∞ for some δ > 0.We consider both the case of the coefficient +V (x) and −V (x) in front of the nonlinearity.In both cases we have existence results which differ in only one hypothesis. Concerningreal-valued breathers of (1.1) we are not aware of any other existence result. Our resultsare as follows. For a function f : R3 → R we say f(x) → 0 in the C2-sense as x → 0 iff(x),∇f(x), D2f(x)→ 0 as x→ 0.

Theorem 1. Suppose s, q, V : R3 → (0,∞) are radially symmetric C2-functions and let

T = 2π√

s(0)q(0) . Assume

(H1) T√

q(x)s(x) < 2π for all x ∈ R3 \ 0,

(H2)∣∣∣2π − T√ q(x)

s(x)

∣∣∣ 1p−1

→ 0 in the C2-sense as x→ 0,

(H3) supx∈R3

∣∣∣2π − T√ q(x)s(x)

∣∣∣ eδ(p−1)|x| <∞ for some δ > 0,

(H4) supx∈R3q(x)V (x) <∞.

Then there exists a T -periodic R3-valued breather solution U of (1.1)+ with the additionalproperty that supR3×R |U(x, t)|eδ|x| < ∞. The breather U generates a continuum of phase-shifted breathers Ua(x, t) = U(x, t + a(x)) where a : R3 → R is an arbitrary radially sym-metric C2-function.

Theorem 2. Suppose s, q, V : R3 → (0,∞) are radially symmetric C2-functions and let

T = 2π√

s(0)q(0) . Assume

(H1)’ T√

q(x)s(x) > 2π for all x ∈ R3 \ 0

and that (H2)–(H4) hold. Then there exists a T -periodic R3-valued breather solution U of(1.1)− with the additional property that supR3×R |U(x, t)|eδ|x| <∞. It generates a continuumof phase-shifted breathers Ua as described in Theorem 1.

The search for breather-solutions has spurred a lot of research in the area of nonlinearwave equations. One of the milestones was the discovery of the real-valued breather-family

u(x, t) = 4 arctan

(m sin(ωt)

ω cosh(mx)

), m, ω > 0, m2 + ω2 = 1

for the scalar 1 + 1-dimensional sine-Gordon equation

(1.2) ∂2t u− ∂2xu+ sinu = 0 in R× R,cf. [1]. The nonlinearity sinu is very special since perturbations of sinu – in general – do notpermit breather families, cf. [13], [9]. The situation is different for scalar 1 + 1-dimensionalnonlinear wave equations with x-dependent coefficients like

(1.3)± s(x)∂2t u− ∂2xu+ q(x)u± V (x)|u|p−1u = 0 for (x, t) ∈ R× R.Note that (1.3)± is a special case of (1.1)± for fields

U(x, t) =

00

u(x1, t)

A BREATHER CONSTRUCTION FOR A SEMILINEAR CURL-CURL WAVE EQUATION 373

since in this case divU = 0 and hence

∇×∇× U(x, t) =

00

−∂2x1u(x1, t)

.

For (1.3)−, the specific example of p = 3 and 1−periodic coefficient functions

s(x) = 1 + 15χ[6/13,7/14)(x), xmod 1

q(x) =

(13π

16

)2

−

(13 arccos((9 +

√1881)/100))

8

)2

− ε2 s(x),

V (x) = 1

given in [10] allowed for breather-solutions with minimal period 3213 for all ε ∈ (0, ε0]. This

remarkable result relies on tailoring the spectrum of −y′′ = λs(x)y and the use of spatialdynamics, center-manifold reduction and bifurcation theory. In a subsequent paper [11]methods of inverse spectral theory were developed that may allow in the future to generalizethe above specific example to a bigger class of coefficient functions.

The breather construction of [10] strongly exploits the structure of spatially varyingcoefficients in (1.3)−. Also in our present paper we make heavy use of the particular spatialdependence of the coefficients s(x), q(x) and V (x) in (1.1)±. What is even more importantis the particular property of the curl-operator to annihilate gradient fields. This enables usto construct gradient field breathers by ODE-techniques.

Let us point out that in our two main theorems we prove the existence of R3-valuedbreathers of (1.1). Sometimes, monochromatic complex-valued waves of the type u(x, t) =eiωtuuu(x) are also called breathers provided uuu decays to 0 at±∞. For such waves the nonlinearhyperbolic problem (1.3)± reduces to a nonlinear ODE problem for uuu:

−uuu′′ + (q(x)− ω2s(x))uuu± V (x)|uuu|p−1uuu = 0 on R

Therefore, as one might expect, many results on the existence of exponentially decayingnon-trivial solutions are known, e.g. in the case of periodic potentials [3], [18]. Also for thevector-valued wave equation (1.1) one can prove the existence of C3-valued breathers of the

type e2πT itU(x) under various assumptions on the coefficients, cf. [4], [5], [6], [7], [12], [14],

[17]. However, as the next result shows, it is remarkable that exactly the same assumptionsas in Theorem 1, Theorem 2 also lead to the existence of C3-valued monochromatic radially-symmetric breathers. This shows that (H1)–(H4) and (H1)’, (H2)–(H4) are in some sensenatural assumptions. We are, however, aware of the fact that neither the hypotheses ofTheorem 1, Theorem 2 nor the hypotheses of Theorem 3 are necessarily necessary for theexistence of breathers.

Theorem 3. Suppose that s, q, V : R3 → (0,∞) are radially symmetric C2-functions such

that (H1)–(H4) or (H1)’, (H2)–(H4) hold respectively, and T = 2π√

s(0)q(0) . Then there exists

a continuum of T -periodic C3-valued monochromatic breather solutions U(x, t) = e2πT itUUU(x)

of (1.1)±, respectively, with the property supR3 |UUU(x)|eδ|x| <∞.

Let us finally mention that the interest in breathers in the context of curl-curl nonlinearwave equations stems from the search for optical breathers, i.e., time-periodic spatially


localized solutions of Maxwell’s equations in anisotropic materials where the permittivitydepends nonlinearly on the electromagnetic fields, cf. [2]. The nonlinear Maxwell problemfor the electric field amounts to a quasilinear-in-time curl-curl wave equation, which is muchharder to treat than the semilinear problem (1.1). In [19] a one-dimensional reduction ofthe nonlinear Maxwell problem was considered. Based on an approximation by the so-called extended nonlinear coupled mode system (xNLCME) the authors achieved resultsthat indicate the formation and persistence of spatially localized time-periodic polychromaticsolutions.

The paper is organized as follows. In Chapter 2 we prove our three main theorems.In order to keep the proofs simple and short we decided to transfer to the Appendix twoelementary but slightly lengthy expansions on the inverses of functions given by explicitintegral formulas.

2. Proof of the results

In the above theorems radially symmetric functions from R3 → R occur. For such func-tions we use the following notation: if f : R3 → R is a radially symmetric C2-function thenwe denote by f : [0,∞) → R with f(|x|) = f(x) its one-dimensional representative, which

has the properties f ∈ C2([0,∞), f ′(0) = 0. The proofs of the main results require somepreparations. We begin with an observation.

Lemma 1. Let ϕ : [0,∞) → R be a C2-function and let W : R3 \ 0 → R3 be given byW (x) := ϕ(|x|) x

|x| . Then W can be extended to a function

(i) W ∈ C1(R3) if and only if ϕ(0) = 0.(ii) W ∈ C2(R3) if and only if ϕ(0) = ϕ′′(0) = 0.

Proof. The function W continuously extends to R3 if and only if ϕ(0) = 0.

(i) The first derivatives of W on R3 \ 0 are given by

∂Wi

∂xj(x) =

(ϕ′(r)− ϕ(r)

r

)xixjr2

+ϕ(r)

rδij

where r = |x|. For the limit as x → 0 to exist one again needs ϕ(0) = 0. In this case we

know that ϕ(r)/r → ϕ′(0) as r → 0 and ϕ′(r) − ϕ(r)r → 0 as r → 0. This shows that the

first partial derivatives of W on R3 \ 0 continuously extend to R3 if and only if ϕ(0) = 0.

(ii) The second derivatives of W on R3 \ 0 are

∂2Wi

∂xj∂xk(x) =

(ϕ′′(r)− 3

ϕ′(r)r − ϕ(r)

r2

)︸︷︷︸

→− 12ϕ′′(0) as r→0

xixjxkr3

+

(ϕ′(r)r − ϕ(r)

r2

)︸︷︷︸→ 1

2ϕ′′(0) as r→0

δikxj + δjkxi + δijxkr

Therefore the second partial derivatives of W on R3 \ 0 continuously extend to R3 if andonly if ϕ(0) = 0 and ϕ′′(0) = 0.

In the following we consider functions ψ = ψ(r, t) from [0,∞) × R to R. We use the

notation ψ′(r, t) = ∂∂rψ(r, t) and ψ(r, t) = ∂

∂tψ(r, t). Under the assumptions (H1)–(H4) ofTheorem 1 or (H1)’, (H2)–(H4) of Theorem 2 we look for solutions U of (1.1)± of the formU(x, t) := ψ(|x|, t) x

|x| .


Lemma 2. Let ψ : [0,∞) × R → R be a C2-function with ψ(0, t) = ψ′′(0, t) = 0. ThenU(x, t) := ψ(|x|, t) x

|x| is a C2(R3 × R) function. It solves (1.1)± if and only if ψ satisfies

(2.1)± s(r)ψ + q(r)ψ ± V (r)|ψ|p−1ψ = 0 for r ≥ 0, t ∈ R.

Proof. By Lemma 1 the function U is a C2 function of the variables x and t; note thatalso ψ(0, t) = ψ(0, t) = 0. Moreover, by construction U is a gradient-field, i.e., U(x, t) =∇xΨ(|x|, t) where Ψ(r, t) =

∫ r0ψ(ρ, t) dρ. Hence ∇× U = 0 and thus the claim follows.

The next result is a direct consequence of the fact that (2.1)± is autonomous with respectto t and that r ≥ 0 plays the role of a parameter.

Lemma 3. Suppose U(x, t) = ψ(|x|, t) x|x| solves (1.1)±. Let a : R3 → R be a radially

symmetric C2-function. Then

Ua(x, t) := U(x, t+ a(x))

also solves (1.1)±. Hence, from one radially symmetric breather one can generate a contin-uum of different phase-shifted breathers.

The proof of Theorem 1 relies on rescaling (2.1)+ as follows: let us find solutions ψ(r, t)of (2.1)+ of the form

(2.2) ψ(r, t) = τ(r)y(σ(r)t).

Inserting this into (2.1) and comparing coefficients tells us that y has to solve

(2.3) y + y + |y|p−1y = 0

where

(2.4) σ(r) =

(q(r)

s(r)

)1/2

, τ(r) =

(q(r)

V (r)

) 1p−1

.

For the proof of Theorem 2 we use the same ansatz (2.2) and obtain that y has to solve

(2.5) y + y − |y|p−1y = 0

where σ(r) and τ(r) are chosen as in (2.4). Next we collect some results about the solutionsof (2.3) and (2.5).

Lemma 4. Define the function A+ : R2 → R by A+(ξ, η) := η2 + ξ2 + 2p+1 |ξ|

p+1. Then

A+ is a first integral for (2.3), i.e., every solution y of (2.3) satisfies A+(y, y) = const. = cfor some c ∈ [0,∞). Every orbit of (2.3) is uniquely characterized by the value c ∈ [0,∞)and every solution y on such an orbit is periodic with minimal period L(c) and maximalamplitude N(c) := maxt∈R |y(t)|. Then

(i) L,N ∈ C([0,∞)

)∩ C∞

((0,∞)

)and L

([0,∞)

)= (0, 2π], N

([0,∞)

)= [0,∞).

(ii) N(c) is strictly increasing in c with N ′ > 0 on (0,∞), N(c) ≤√c for all c > 0 and

limc→0N(c)√c

= 1, limc→∞N(c) =∞.

(iii) L(c) is strictly decreasing in c with L′ < 0 on (0,∞), limc→∞ L(c) = 0 and L(0) =2π.


Figure 1. Part of the phase plane of (2.3) for p = 3 with two periodic orbits.

(iv) M = L−1 : (0, 2π] → [0,∞) is in C∞((0, 2π)

)and has the following expansions as

s→ 2π− √M(s) =

√α(2π − s)

1p−1 (1 +O(2π − s)),√

M(s)′

= −√α

p− 1(2π − s)

2−pp−1 (1 +O(2π − s)),√

M(s)′′

=

√α(2− p)

(p− 1)2(2π − s)

3−2pp−1 (1 +O(2π − s))

for some constant α > 0.

Proof. Let us first verify all statements for N(c). The function N(c) is given implicitlythrough

N(c)2 +2

p+ 1N(c)p+1 = c

which provides the strict monotonicity, continuity and differentiability properties of N(c)

for c > 0. It also implies the inequality N(c) ≤√c and limc→0

N(c)√c

= 1, limc→∞N(c) =∞.

Now we prove the statements for L(c). We use the first integral

|y|2 + |y|2 +2

p+ 1|y|p+1 = c

to solve for y in all four quadrants of the phase-plane, cf. Figure 1. Together with the


defining equation for N(c) this yields

L(c) = 4

∫ N(c)

0

1√c− y2 − 2

p+1yp+1

dy

= 4

∫ 1

0

N(c)√c−N(c)2z2 − 2

p+1N(c)p+1zp+1dz(2.6)

= 4

∫ 1

0

1√1− z2 + 2

p+1N(c)p−1(1− zp+1)dz.

Now we find that L(c) has the asserted smoothness properties and is strictly decreasing withL′ < 0 on (0,∞), limc→∞ L(c) = 0 and

limc→0

L(c) = 4

∫ 1

0

1√1− z2

dz = 2π.

The fact that L′ < 0 on (0,∞) follows from N ′ > 0 on (0,∞) and (2.14) in the proof of

Lemma 6. This yields also that M = L−1 ∈ C∞((0, 2π)

). The expansions for

√M and its

derivatives can be found in Lemma 6 in the Appendix.

Lemma 5. Define the function A− : R2 → R by A−(ξ, η) := η2 + ξ2− 2p+1 |ξ|

p+1. Then A−is a first integral for (2.5), i.e., every solution y of (2.5) satisfies A−(y, y) = const. = c forsome c ∈ R. Every bounded orbit of (2.5) is uniquely characterized by the value c ∈ [0, p−1p+1 ]

and for c ∈ [0, p−1p+1 ) every solution y on such an orbit is periodic with minimal period L(c)

and maximal amplitude N(c) := maxt∈R |y(t)|. Then

(i) L,N ∈ C([0, p−1p+1 )

)∩C∞

((0, p−1p+1 )

)and L

([0, p−1p+1 )

)= [2π,∞), N

([0, p−1p+1 )

)= [0, 1).

(ii) N is strictly increasing in c with N ′ > 0 on (0, p−1p+1 ) , N(c) ≤√

p+1p−1c for all

c ∈ [0, p−1p+1 ) and limc→ p−1p+1

N(c) = 1, limc→0N(c)√c

= 1.

(iii) L(c) is strictly increasing in c with L′ > 0 on (0, p−1p+1 ), limc→ p−1p+1

L(c) = ∞ and

L(0) = 2π.(iv) M = L−1 : [2π,∞)→ [0, p−1p+1 ) is in C∞

((2π,∞)

)and has the following expansions

as s→ 2π+ √M(s) =

√α(s− 2π)

1p−1 (1 +O(s− 2π)),√

M(s)′

=

√α

p− 1(s− 2π)

2−pp−1 (1 +O(s− 2π)),√

M(s)′′

=

√α(2− p)

(p− 1)2(s− 2π)

3−2pp−1 (1 +O(s− 2π))

for the same constant α > 0 as in Lemma 4.

Proof. Besides the equilibrium (0, 0) there are two further equilibria (±1, 0) connected bytwo heteroclinic orbits. The first integral

|y|2 + |y|2 − 2

p+ 1|y|p+1 = c


Figure 2. Part of the phase plane of (2.5) for p = 3 with a periodic orbit(blue) and two heteroclinic connections (red).

leads to closed orbits for 0 < c < A−(±1, 0) = p−1p+1 and provided initial conditions are chosen

in the bounded component of the set A−1−([0, p−1p+1 )

), cf. Figure 2. The defining equation for

the function N(c) is

(2.7) N(c)2 − 2

p+ 1N(c)p+1 = c and N(c) < 1

which provides the strict monotonicity, continuity and differentiability properties of N(c)

for 0 < c < p−1p+1 . Since N(c) < 1 we obtain from (2.7) the inequality N(c) ≤

√p+1p−1c, the

fact that N ′ > 0 on (0, p−1p+1 ), and limc→0N(c)√c

= 1. Moreover, N(c) → 1 as c p−1p+1 . This

completes the statements on N(c).

Now we turn to L(c). This time (2.7) together with the first integral yields

L(c) = 4

∫ N(c)

0

1√c− y2 + 2

p+1yp+1

dy

= 4

∫ 1

0

N(c)√c−N(c)2z2 + 2

p+1N(c)p+1zp+1dz(2.8)

= 4

∫ 1

0

1√1− z2 − 2

p+1N(c)p−1(1− zp+1)dz.

Clearly L(0) = 2π. Moreover L(c) has the asserted smoothness properties, and is strictlyincreasing since N(c) is strictly increasing for c ∈ [0, p−1p+1 ]. Because limc→ p−1

p+1N(c) = 1 and

1−z2− 2p+1 (1−zp+1) = (p−1)(1−z)2(1+o(1)) as z → 1 we see now that limc→ p−1

p+1L(c) =∞.

The fact that L′ > 0 on (0, p−1p+1 ) follows from N ′ > 0 on (0,∞) and (2.16) in the proof of

Lemma 6. This yields also that M = L−1 ∈ C∞((2π,∞)

). The expansions for

√M and its

derivatives can be found in Lemma 7 in the Appendix.


Proof of Theorem 1: We begin by choosing a C2-curve γ : [0,∞) → R2 in phase spacesuch that A+(γ(c)) = c2, where A+ is the first integral from Lemma 4. Such a curve is e.g.given by γ(c) = (0, c). There is a continuum of other possible choices of γ. The choice ofγ actually only selects a particular member of the continuum of phase-shifted breathers asdescribed in Lemma 3 (we will comment on this aspect at the end of the proof).

Let us denote by y(t; c) the solution of (2.3) with(y(0; c), y(0; c))

)= γ(c). Then y :

R × [0,∞) → R is a C2-function and y(t; c) is L(c2)-periodic in the t-variable. Now wedefine the solution ψ of (2.1)+ by

(2.9) ψ(r, t) := τ(r)y(σ(r)t; c) with σ(r) =

(q(r)

s(r)

)1/2

, τ(r) =

(q(r)

V (r)

) 1p−1

,

see (2.2), (2.4). The requirement of T -periodicity of ψ in the t-variable tells us how to choosec as a function of the radial variable r ∈ [0,∞), i.e.,

g(r) := σ(r)T!= L(c2).

Recall from Lemma 4 the definition M = L−1 and that M : (0, 2π]→ R is strictly decreasingand C∞ on (0, 2π). Now

(2.10) c(r) =√M(g(r))

has to be inserted into (2.9). Note that the assumption (H1) of Theorem 1 guarantees thatc(r) is well-defined and C2 on (0,∞). Next we show that ψ(r, t) tends to 0 as r → 0 and iseven exponentially decaying to 0 as r →∞. First note the estimate

|ψ(r, t)| ≤(q(r)

V (r)

) 1p−1

N(c(r)2)

≤(q(r)

V (r)

) 1p−1

︸︷︷︸≤B

c(r) by assumption (H4) and Lemma 4(ii)

≤ B√M(g(r)).

By assumptions (H2) and (H3) of Theorem 1 the argument of M in the above inequalitytends to 2π as r →∞ and as r → 0. By Lemma 4(iv) we have the estimate

(2.11) |ψ(r, t)| ≤ B√α (2π − g(r))

1p−1 O(1) as r →∞ and as r → 0.

Assumption (H3) of Theorem 1 and (2.11) yield |ψ(r, t)| ≤ C exp(−δr) for r ≥ 0 whichproves the exponential decay of U(x, t) = ψ(|x|, t) x

|x| as |x| → ∞.

Next we see that (2.11) and (H2) imply ψ(0, t) = 0. In order to apply Lemma 2 it remainsto prove ψ ∈ C2([0,∞) × R) and that ψ′′(0, t) = 0. For this we compute from (2.10) that


c(r) =√α(2π − g(r))

1p−1O(1)→ 0 as r → 0. Furthermore (2.10) implies

c′(r) =√M′(g(r))g′(r)

= −√α

p− 1(2π − g(r))

2−pp−1O(1)g′(r) by Lemma 4(iv)

=√α(

(2π − g(r))1p−1

)′O(1)

= o(1) as r → 0 by assumption (H2).

Likewise

c′′(r) =√M′′(g(r))g′(r)2 +

√M′(g(r))g′′(r)

=

√α(2− p)

(p− 1)2(2π − g(r))

3−2pp−1

(1 +O(2π − g(r))

)g′(r)2

−√α

p− 1(2π − g(r))

2−pp−1(1 +O(2π − g(r))

)g′′(r)

=√α(

(2π − g(r))1p−1

)′′︸︷︷︸

=:T1

+O(1)

√α(2− p)

(p− 1)2(2π − g(r))

2−pp−1 g′(r)2︸︷︷︸

=:T2

−O(1)

√α

p− 1(2π − g(r))

1p−1 g′′(r)︸︷︷︸

=:T3

.

The term T1 converges to 0 as r → 0 by assumption (H2). Recall that g is a C2-functionon [0,∞). The term T3 converges to 0 since g(r) → 2π as r → 0 and g′′ is bounded near

0. And since T2(r) = (1 − p)(

(2π − g(r))1p−1

)′g′(r) with g′ being bounded near 0 we see

that (H2) also implies T2(r) → 0 as r → 0. This shows that c′′(r) → 0 as r → 0. Hence ccan be extended to a function c ∈ C2

([0,∞)

)with c(0) = c′(0) = c′′(0) = 0. Having this

and recalling ψ(r, t) = τ(r)y(σ(r)t, c(r)) we see that ψ ∈ C2([0,∞) × R). Hence we maycompute

ψ′(r, t) = τ ′(r)y(σ(r)t, c(r)) + τ(r)y(σ(r)t, c(r))σ′(r)t+ τ(r)∂y

∂c(σ(r)t, c(r))c′(r)

and

ψ′′(0, t) =τ ′′(0) y(σ(0)t, c(0))︸︷︷︸=0

+2τ ′(0) y(σ(0)t, c(0))︸︷︷︸=0

σ′(0)t+ 2τ ′(0)∂y

∂c(σ(0)t, c(0)) c′(0)︸︷︷︸

=0

+ τ(0) y(σ(0)t, c(0))︸︷︷︸=0

σ′(0)2t2 + τ(0) y(σ(0)t, c(0))︸︷︷︸=0

σ′′(0)t

+ 2τ(0)∂y

∂c(σ(0)t, c(0))σ′(0)t c′(0)︸︷︷︸

=0

+τ(0)∂2y

∂c2(σ(0)t, c(0)) c′(0)2︸︷︷︸

=0

+ τ(0)∂y

∂c(σ(0)t, c(0)) c′′(0)︸︷︷︸

=0

=0,


where we have used y(·, 0) = 0, y(·, 0) = 0, y(·, 0) = 0. By Lemma 2 this implies thatU ∈ C2(R3 × R). The asserted continuum of solutions is now given by Lemma 3. Thisfinishes the proof of Theorem 1.

Now we will comment on the choice of the initial curve γ(c) = (0, c) which led to thesolution family y(t; c) such that (y(0; c), y(0; c)) = γ(c). Our objective was to determinesome C2-curve such that A+(γ(c)) = c2. The particular choice γ(c) = (0, c) is convenientbut arbitrary. Let us explain other possible choices of γ. E.g. take

γ(c) :=(y(b(c); c), y(b(c); c)

)for an arbitrary function b ∈ C2([0,∞);R). Clearly, A+(γ(c)) = A+ (y(b(c); c), y(b(c); c)) =c2 since A+ is a first integral of (2.3). With the new curve γ we can define a new solutionfamily y(t; c) through the initial conditions(

y(0; c), ˙y(0; c))

= γ(c)

By uniqueness of the initial value problem the new and old solution families have the simplerelation

y(t; c) = y(t+ b(c); c).

In order to see the effect of the choice of the new curve let us compare the solutions U , Ugenerated by γ, γ, i.e.,

U(x, t) = τ(r)y(σ(r)t; c(r))x

|x|,

where c(r) =√L−1(σ(r)). Likewise

U(x, t) = τ(r)y(σ(r)t; c(r))x

|x|

= τ(r)y(σ(r)t+ b(c(r)); c(r))x

|x|= U(x, t+ a(r)),

where a(r) = b(c(r))/σ(r) is a C2-function on [0,∞). Hence, this different choice of theinitial curve led to a phase-shifted breather as already explained in Lemma 3.

Proof of Theorem 2: Again we choose a C2-curve γ : [0, p−1p+1 )→ R2 in phase space such

that A−(γ(c)) = c. Now A− is the first integral from Lemma 5. As before, such a curveis e.g. γ(c) = (0, c). The fact that other choices of γ are also possible and just lead toa phase shift as shown in Lemma 3 has already been explained at the end of the proof ofTheorem 1. We denote by y(t; c) the solution of (2.5) with

(y(0; c), y(0; c))

)= γ(c). Then

y : R× [0, p−1p+1 )→ R is a C2-function and y(t; c) is L(c)-periodic in the t-variable. A solution

ψ of (2.1)− is then defined by

(2.12) ψ(r, t) := τ(r)y(σ(r)t; c) with σ(r), τ(r) as previously.

The condition of T -periodicity of ψ in the t-variable is the same as before and requires

g(r) := σ(r)T!= L(c2).

Now the inverse M = L−1 is defined on [2π,∞) → R as a continuous, strictly increasingfunction which is C∞ on (2π,∞), cf. Lemma 5. Assumption (H1)’ of Theorem 2 guaranteesthat

c(r) =√M(g(r))


is well-defined and C2 on (0,∞). Inserting c(r) into (2.12) yields a T -periodic solutionψ(r, t) of (2.1)−. We proceed via the estimate

|ψ(r, t)| ≤(q(r)

V (r)

) 1p−1

N(c(r)2)

≤(q(r)

V (r)

) 1p−1

︸︷︷︸≤B

√p+ 1

p− 1c(r) by assumption (H4) and Lemma 5(ii)

= B

√p+ 1

p− 1

√M(g(r).

As before, (H2) and (H3) imply that the argument of M in the above inequality tends to2π as r →∞ and as r → 0. Making use of the estimate in Lemma 5(iv) we obtain

(2.13) |ψ(r, t)| ≤ B√p+ 1

p− 1

√α (g(r)− 2π)

1p−1 O(1) as r →∞ and as r → 0.

As before assumption (H3) leads to the exponential decay of U(x, t) as |x| → ∞. Similarly

to the proof of Theorem 1 the expansions of√M ,√M′

and√M′′

and (H2) imply c′(0) =c′′(0) = 0 which leads in an identical way as before to ψ′′(0, t) = 0 and thus U ∈ C2(R3 ×R).

Proof of Theorem 3: We use the ansatz U(x, t) = ϕ(|x|)ei 2πT t x|x| . According to Lemma 2

it represents a T -periodic breather if ϕ : [0,∞)→ R is a C2-solution of

−(

2π

T

)2

s(r) + q(r)± V (r)|ϕ(r)|p−1 = 0 with ϕ(0) = ϕ′′(0) = 0

which exponentially decays to zero at ∞. This can be satisfied for

ϕ(r) :=

[±

((2π

T

)2s(r)

q(r)− 1

)q(r)

V (r)

] 1p−1

.

The assumptions (H1), (H1)’ guarantee that ϕ is well-defined. By (H3), (H4) it is exponen-tially decreasing as r → ∞ and by (H2) we see that ϕ(0) = ϕ′′(0) = 0 so that U(x, t) is aclassical solution of (1.1)± on R3 × R.


Appendix

Lemma 6 (Expansion of M = L−1 for (2.3)). M : (0, 2π] → [0,∞) is C∞ on (0, 2π) andhas the following expansions as s→ 2π−

M(s) = α(2π − s)2p−1 (1 +O(2π − s)),√

M(s) =√α(2π − s)

1p−1 (1 +O(2π − s)),

M ′(s) = − 2α

p− 1(2π − s)

3−pp−1 (1 +O(2π − s)),√

M(s)′

= −√α

p− 1(2π − s)

2−pp−1 (1 +O(2π − s)),

M ′′(s) =2α(3− p)(p− 1)2

(2π − s)4−2pp−1 (1 +O(2π − s)),

√M(s)

′′=

√α(2− p)

(p− 1)2(2π − s)

3−2pp−1 (1 +O(2π − s))

for some constant α > 0.

Proof. Let us begin by recalling from (2.6) that

L(c) = F

(2

p+ 1N(c)p−1

),

where

F (w) = 4

∫ 1

0

1√1− z2 + w(1− zp+1)

dz

= 4

∫ 1

0

1√1− z2

√1 + wκ(z)

dz with κ(z) =1− zp+1

1− z2and w ≥ 0.

Since κ is a continuous and positive function on [0, 1] we find that F ∈ C∞[0,∞), F (0) = 2π,F (∞) = 0 and F is strictly decreasing and convex with

F ′(w) = −2

∫ 1

0

κ(z)√1− z2(1 + wκ(z))

32

dz < 0,(2.14)

F ′′(w) = 3

∫ 1

0

κ2(z)√1− z2(1 + wκ(z))

52

dz > 0 for w ∈ [0,∞).(2.15)

Thus F−1 ∈ C∞((0, 2π]).

Our objective is to study L−1. Recall from the defining equation for N(c) that forw = 2

p+1N(c)p−1 one has the relation

c = N(c)2 +2

p+ 1N(c)p+1 =

(p+ 1

2

) 2p−1 (

w2p−1 + w

p+1p−1

)=: Φ(w).

This leads to the representation

L(c) = F (Φ−1(c)) and M = L−1 = Φ F−1.


Via Taylor-approximation with α = −1/F ′(0) > 0, β = F ′′(0) > 0 we obtain as s→ 2π−

F−1(s) = (F−1)′(2π)(s− 2π) +O((2π − s)2) = α(2π − s)(1 +O(2π − s)),

(F−1)′(s) =1

F ′(F−1(s))= −α(1 +O(2π − s)),

(F−1)′′(s) = −F ′′(F−1(s)

)(F ′(F−1(s))

)3 = βα3(1 +O(2π − s)).

Hence as s→ 2π− we obtain

M(s) = Φ(F−1(s)) = α(2π − s)2p−1 (1 +O(2π − s))

for α =(p+12

) 2p−1 α

2p−1 > 0,

M ′(s) = Φ′(F−1(s))(F−1)′(s)

=

(p+ 1

2

) 2p−1

(2

p− 1(F−1(s))

3−pp−1 +

p+ 1

p− 1(F−1(s))

2p−1

)(−α)(1 +O(2π − s))

=−2α

p− 1(2π − s)

3−pp−1 (1 +O(2π − s))

and

M ′′(s) =Φ′′(F−1(s))(

(F−1)′(s))2

+ Φ′(F−1(s))(F−1)′′(s)

=

(p+ 1

2

) 2p−1

(2(3− p)(p− 1)2

(F−1(s))4−2pp−1 +

2(p+ 1)

(p− 1)2(F−1(s))

3−pp−1

)α2(1 +O(2π − s))

+O((2π − s)

3−pp−1)

=2α(3− p)(p− 1)2

(2π − s)4−2pp−1 (1 +O(2π − s)).

The expansions for√M,√M′

= M ′

2√M

and√M′′

= 12M3/2

(M ′′M − 1

2 (M ′)2)

follow directly

from the expansions for M,M ′,M ′′.

Lemma 7 (Expansion of M = L−1 for (2.5)). M : [2π,∞) → [0, p−1p+1 ) is C∞ on (2π,∞)

and has the following expansions as s→ 2π+

M(s) = α(s− 2π)2p−1 (1 +O(s− 2π)),√

M(s) =√α(s− 2π)

1p−1 (1 +O(s− 2π)),

M ′(s) =2α

p− 1(s− 2π)

3−pp−1 (1 +O(s− 2π)),√

M(s)′

=

√α

p− 1(s− 2π)

2−pp−1 (1 +O(s− 2π)),

M ′′(s) =2α(3− p)(p− 1)2

(s− 2π)4−2pp−1 (1 +O(s− 2π)),

√M(s)

′′=

√α(2− p)

(p− 1)2(s− 2π)

3−2pp−1 (1 +O(s− 2π))


with the same constant α > 0 as in Lemma 6.

Proof. Let us begin by recalling from (2.8) that

L(c) = F

(2

p+ 1N(c)p−1

),

where

F (w) = 4

∫ 1

0

1√1− z2 − w(1− zp+1)

dz

= 4

∫ 1

0

1√1− z2

√1− wκ(z)

dz with κ(z) =1− zp+1

1− z2.

Since κ is a continuous function on [0, 1] which takes values only in [1, p+12 ] we find that

F (w) is well defined for w ∈ [0, 2p+1 ), F ∈ C∞[0, 2

p+1 ) and F (0) = 2π. Moreover, the

Taylor-expansion of κ at 1 yields

κ(z) =p+ 1

2+p2 − 1

4(z − 1)(1 + o(1)) as z → 1−

so that limw→ 2p+1

F (w) =∞. Finally, F is strictly increasing and convex with

F ′(w) = 2

∫ 1

0

κ(z)√1− z2(1− wκ(z))

32

dz > 0,(2.16)

F ′′(w) = 3

∫ 1

0

κ2(z)√1− z2(1− wκ(z))

52

dz > 0 for w ∈ [0,2

p+ 1)(2.17)

and hence F−1 ∈ C∞([2π,∞)

).

Our objective is to study L−1. Recall from the defining equation for N(c) that forw = 2

p+1N(c)p−1 one has the relation

c = N(c)2 − 2

p+ 1N(c)p+1 =

(p+ 1

2

) 2p−1 (

w2p−1 − w

p+1p−1

)=: Φ(w).

This leads to the representation

L(c) = F (Φ−1(c)) and M = L−1 = Φ F−1.

Via Taylor-approximation and α := 1/F ′(0) > 0, β := F ′′(0) > 0 having the same values asin the proof of Lemma 6 we obtain as s→ 2π+

F−1(s) = α(s− 2π)(1 +O(s− 2π)),

(F−1)′(s) = α(1 +O(s− 2π)),

(F−1)′′(s) = −βα3(1 +O(s− 2π)).

Hence as s→ 2π+ we obtain

M(s) = Φ(F−1(s)) = α(s− 2π)2p−1 (1 +O(s− 2π))


for α =(p+12

) 2p−1 α

2p−1 > 0,

M ′(s) = Φ′(F−1(s))(F−1)′(s)

=

(p+ 1

2

) 2p−1

(2

p− 1(F−1(s))

3−pp−1 − p+ 1

p− 1(F−1(s))

2p−1

)α(1 +O(s− 2π))

=2α

p− 1(s− 2π)

3−pp−1 (1 +O(s− 2π))

and

M ′′(s) =Φ′′(F−1(s))(

(F−1)′(s))2

+ Φ′(F−1(s))(F−1)′′(s)

=

(p+ 1

2

) 2p−1

(2(3− p)(p− 1)2

(F−1(s))4−2pp−1 − 2(p+ 1)

(p− 1)2(F−1(s))

3−pp−1

)α2(1 +O(s− 2π))

+O((s− 2π)

3−pp−1)

=2α(3− p)(p− 1)2

(s− 2π)4−2pp−1 (1 +O(s− 2π)).

As before, the expansions for√M,√M′

= M ′

2√M

and√M′′

= 12M3/2

(M ′′M − 1

2 (M ′)2)

follow directly from the expansions for M,M ′,M ′′.

References

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M. Plum

Institute for Analysis, Karlsruhe Institute of Technology (KIT),D-76128 Karlsruhe, Germany


URL: www.math.kit.edu/~plum

W. Reichel

Institute for Analysis, Karlsruhe Institute of Technology (KIT),D-76128 Karlsruhe, Germany


URL: www.math.kit.edu/~reichel

JEPE Vol 2, 2016, p. 389-413

GRAIN BOUNDARY MIGRATION WITH THERMAL GROOVING

EFFECTS: A NUMERICAL APPROACH

VADIM DERKACH, AMY NOVICK-COHEN, AND ARKADY VILENKIN

Abstract. Grain boundary migration in the presence of thermal grooving effects play

a critical role in the stability of the thin polycrystalline films used in numerous techno-logical applications. We present a computational framework for simulating this motion

which relies on a physical model due to Mullins [33, 26], according to which the grain

boundaries and the external surfaces are governed by mean curvature motion and sur-face diffusion, respectively, and along the thermal grooves where the grain boundaries

and the exterior surfaces couple, balance of mechanical forces, continuity of the chemical

potential, and balance of mass flux dictate the boundary conditions. By adopting anequi-spaced parametric description for the evolving surfaces [40, 13], the physical model

can be formulated as a coupled systems of ODEs and PDEs.

We propose a finite difference algorithm based on staggered grids for solving the re-sultant system, which we implement on a polycrystalline layer with columnar structure

containing three grains, assuming isotropy. Our algorithm, which is second order ac-

curate in space and first order accurate in time, conserves mass, dissipates energy, andsatisfies the predictions of Mullins[33], von Neumann-Mullins [32, 51] and Genin, Mullins,

Wynblatt [22], in appropriate limits. Effects such as pitting, hole formation, grain anni-hilation, wetting, and dewetting can be analyzed using our approach [15, 14, 17, 16].

1. Introduction

We report on a numerical algorithm designed to study simultaneous grain boundarymigration and surface topography evolution in polycrystalline films with columnar structure,which is particularly relevant for studying thin polycrystalline films. Simultaneous grainboundary migration and surface topography evolution often strongly affect the thermalstability of thin polycrystalline films, which are of high technological importance, since attemperatures which enable some surface diffusion, grooves form on the exterior surface wheregrain boundaries emerge, and these grooves can penetrate down to the substrate, leading tohole formation and sometimes eventually to full agglomeration [48].

2010 Mathematics Subject Classification. Primary: 06B10; Secondary: 06D05.Key words and phrases. numerical methods, finite difference schemes, microstructure, surface diffusion,

geometric motions, polycrystalline films.Received 05/11/2016, accepted 08/12/2016.

The authors acknowledge the support of the Israel Science Foundation (Grants # 752/10, # 1200/16).This work was also supported by the Technion’s Russell Berrie Nanotechnology Institute (RBNI) throughthe NEVET program and through a Capital Equipment Usage Grant.

389

Over recent years, there has been a plentitude of studies devoted to modeling microstruc-tural evolution in polycrystalline materials, taking into account a large spectrum of ef-fects. The majority have focused on the microstructural evolution within the interior ofthe specimen, with recent emphasis on their statistical properties [32, 23, 44, 5, 4, 49].Microstructural evolution of the exterior surface topology has also been studied, thoughto a lesser extent; there have been studies focusing on facetting, anisotropy, and stability,with some emphasis on grooving, traveling wave solutions, stagnation, and self-similarity[34, 22, 11, 28, 1, 42]. The number of studies focusing on the coupled motion of the twoeffects is markedly less abundant. While the analytic framework for the coupled motion hasyet to become fully developed, numerical methods have been proposed and implemented,certain bicrystalline geometries, with special initial and boundaries conditions, have beentreated in depth, and various phenomena, such as grooving, migration, and pitting, havereceived attention [34, 50, 26, 28, 38, 31, 52]. Recent experiments have reported on thepossibility of accompanying phenomena such as mazing [43], which for simplicity we do notattempt to model.

In the present paper we outline and test an algorithm which was developed to followthe evolution of coupled surface and grain boundary phenomena in realistic polycrystallinemicrostructures. In earlier papers, we used this algorithm to describe simultaneous grainboundary migration and grooving in a triangular bamboo three grain geometry in a thinpolycrystalline specimen, which can exhibit hole formation and grain annihilation [15] aswell as stabilization of hexagonal arrays [17]. Here, we present a detailed discussion of ournumerical algorithm, which is based on a parametric description of the evolving surfaces,with equidistributed grid points on a staggered grid. Our approach generalizes the primarily2D algorithm proposed by Pan and Wetton [40, 39, 10, 41]. Numerical methods, such aslevel set methods, thresholding dynamics, and MBO methods [45, 3, 19, 29], which havebeen widely utilized in simulating microstructural evolution, are not directly applicable forthe coupled motions considered here. Potentially phase field methods based on [37], couldbe implemented here, but this is also not straightforward. While finite element algorithmshave been proposed and analysed, and could be potentially useful for geometries such as weconsider [47, 8, 7, 6]; however realistic multi-grain geometries in R3 governed by the coupledmotions have so far received only very limited attention.

The outline of the paper is as follows. In Section 2, we present the physical model, thegeometry to be considered, and the governing laws of motion. In Section 3, we explain theparametric descriptions with equi-spacing which we adopt to describe the various evolvingsurfaces, and outline the resultant parametric problem formulation. Section 4 contains adetailed account of our algorithm. Section 5 provides some results pertaining to pitting andgrain annihilation, energy dissipation and mass conservation. Our results are compared withthe small slope approximation predictions of von Neumann-Mullins [32, 51], Mullins [33],and Genin, Mullins, Wynblatt [22]. Conclusions are given in Section 6.

2. The physical model: geometry and laws of motion

We focus on an idealized three grain system, which can be extended by reflection to yield afive grain system, see Fig. 1. The grains in the system are assumed to be of the same material,but to have different crystalline lattice orientations. For simplicity, we assume isotropy andneglect the possible effects of elasticity, evaporation, defects, and volume diffusion. Ourthree grain system contains three grain boundaries, three exterior surfaces and five bounding

390

(a) (b)

Figure 1. Sketch of (a) the three grain system, and (b) the resultant fivegrain system geometry.

planes. The three grain boundaries are assumed to evolve by mean curvature motion [32, 33],

(2.1) Vn = AH,

and the three exterior surfaces are assumed to evolve by surface diffusion motion [33, 34],

(2.2) Vn = −B4sH.

In (2.1)–(2.2), Vn denotes the normal velocity of the evolving surface, H denotes its meancurvature, and 4s is the Laplace–Beltrami operator, known also as the “surface Laplacian.”In (2.1), A is a kinetic coefficient known as the reduced mobility, and in (2.2), B is a kineticcoefficient known as the surface diffusion coefficient. Mirror symmetry is assumed withrespect to the five bounding planes, which are assumed to be stationary.

Boundary conditions need to be prescribed where an evolving surface intersects anotherevolving surface or a bounding plane. The intersection of a grain boundary with two exteriorsurfaces defines a “triple junction line,” known also as a “thermal groove” or a “grooveroot,” and the intersection of three grain boundaries will be referred to as an “internal triplejunction line.” The intersection of a grain boundary or an exterior surface with a boundingplane will be referred to as an “exterior free boundary.” The intersection of three grooveroots with an internal triple junction line defines a “quadruple point,” and the intersectionsof two exterior free boundaries define “corner points.”

Along groove roots, we impose i) “persistence;” namely that the grain boundary and theexterior surfaces remain attached, ii) balance of mechanical forces known also as Young’slaw, which is an isotropic version of Herring’s law [24, 25], namely that

(2.3) ~τ i · ~τ j = cos(θ), ~τ i · ~τk = ~τ j · ~τk = − cos(θ/2),

where ~τ i, ~τ j , and ~τk are orthogonal to the groove root and correspond to the unit tangentsto two exterior surfaces and to a grain boundary surface, respectively, which intersect alongthe groove root, and θ, the “dihedral angle,” is defined as

(2.4) θ = π − 2 arcsin(m/2), m = γgb/γext,

where γgb and γext are the energy/area of the grain boundary and of the exterior surfaces,respectively. iii) continuity of the chemical potential, and iv) balance of mass flux. We shall

391

be assuming that the chemical potential is proportional to the mean curvature and thatmass flux is proportional to the gradient of the chemical potential, see [34, 26, 27]. Alongthe internal triple junction line, which we shall assume to be unique we impose “persistence”as well as

(2.5) ~τ i · ~τ j = cos(2π/3),

where ~τ i, ~τ j are unit vectors which are orthogonal to the internal triple junction line andwhich are tangent to one of the two intersecting grain boundary surfaces. We refer also to(2.5) as Young’s law, since taking m = 1 in (2.4) because the energy/area of the various grainboundaries are equal, we obtain that 2π/3 = π−2 arcsin(1/2); then cos(2π/3) = − cos(π/3),so that (2.3) is satisfied

Along the exterior free boundaries, where an exterior surface intersects a bounding plane,we impose “persistence,” mirror symmetry, and zero mass flux. Along the free boundarieswhere a grain boundary intersects a bounding plane, we impose a “persistence” and mirrorsymmetry. The boundary conditions at the quadruple junction and the corner points willbe described in detail in Section 3.5. Initial conditions are prescribed in accordance withFig. 2.

Since (2.1)–(2.2) prescribe only the normal velocities of the surfaces, the governing equa-tions may be formulated as an evolutionary problem via various equivalent descriptions[21]. We adopt an approach based on equi-spaced parametric representations of the varioussurfaces, which yield equi-distributed grid points when solved using finite differences andghost points. While alternatively we might prescribe the tangential velocities, at least inR2 equi-distribution of the grid points was noted to have an overall stabilizing effect on theresultant algorithms [40, 9]. Our method generalizes the methodology proposed by Pan &Wetton [40, 39], which was implemented primarily for simple geometries in R2, and wasinfluenced also by earlier works such as [3].

3. The parametric problem formulation

3.1. Parametric representations with equi-spacing. The evolving surfaces, which wedenote by Si, i ∈ 1, . . . , 6, will be assumed to be representable by parametric hypersur-faces, Xi : [0, 1]2 × [0, T ]→ R3, namely

(3.1) Xi(α, β, t) =(xi(α, β, t), yi(α, β, t), zi(α, β, t)

), 0 ≤ α, β ≤ 1, 0 ≤ t ≤ T,

for some T > 0, where the Cartesian coordinates (x, y, z) are indicated in Fig. 1. Ratherthan specifying somewhat arbitrary tangential velocities, we impose equi-spacing of the(α, β) parametrization, namely that ‖Xα‖α = ‖Xβ‖β = 0, which implies that

(3.2) 〈Xα, Xαα〉 = 〈Xβ , Xββ〉 = 0,

where X represents one of the six surfaces, Xi. We shall be implicitly assuming sufficientlyregular and non-degeneracy, ‖Xα‖, ‖Xβ‖ > 0, of the surfaces Xi up to some time T .

The orientation of parameterizations for the various surfaces are indicated in Fig. 2,where Si, i ∈ Ψext := 1, 2, 3, refer to exterior surfaces and Si, i ∈ Ψgb := 4, 5, 6,refer to grain boundaries. The vectors −→τ iα , −→τ iβ denote tangent vectors to the surface Si,

are given in (3.6) and (3.7), respectively, and satisfy −→τ iα ⊥ Xiα and −→τ iβ ⊥ Xi

β . FromFig 2, one can see the 3 triple junction lines where two exterior surfaces, Sp, Sq, intersectwith the grain boundary surface S`, for p, q, ` ∈ Ψthermal :=

1, 2, 4; 2, 3, 5; 3, 1, 6

;

392

(a) (b)

Figure 2. The system of 3 grains. The exterior surfaces, Si, i ∈ Ψext :=1, 2, 3 as well as the grain boundary surfaces, Si, i ∈ Ψgb := 4, 5, 6are indicated in (a), as are the bounding planes IIi, i = 1, . . . , 5,

and the respective unit normals to these bounding planes, ~N i, i =1, . . . , 5. The quadruple junction (QJ) is indicated as It, and Ψcorner :=IIt, IIIt, IV t, Ib, IIb, IIIb, IV b, CI , CII , CIII refer to the remaining cor-ner points. In (b), on each surface Si, i ∈ Ψext ∪Ψgb, the orientation of theα–β parameterization is indicated, as are the various tangents to these sur-

faces, ~τ iα

, ~τ iβ

, i = 1, . . . , 6; here ~τ iα

and ~τ iβ

correspond to the tangents toSi in the direction of the α−parameterization and the β-parameterization,respectively, see text.

along these triple junction lines, thermal grooves will form in our system when m > 0.There is also one internal triple junction line where the 3 grain boundary surfaces S4, S5,S6 intersect. Moreover, there are 5 bounding planes of symmetry, indicated by Πi, i =1, . . . 5 in Fig 2, where Π5 refers to the “midplane” or base plane of the system located

at z = 0; the unit normal vectors−→N i to Πi are also indicated. In Fig 2, the location

of the quadruple junction (QJ) is indicated by It, and the other corner points ΨI, C :=IIt, IIIt, IV t, Ib, IIb, IIIb, IV b, CI , CII , CIII

are also indicated.

3.2. Equations governing the grain boundaries and exterior surfaces. In order toexpress (2.1)–(2.2) in terms of the parametric surfaces, Xi, let S denote one of the evolvingsurfaces whose parametrization is given by X. Then its normal velocity can be expressed as

(3.3)−→Vn = 〈Xt,

−→n 〉 .

where −→n =Xα×Xβ‖Xα×Xβ‖ is a unit normal to S. Its mean curvature H can be expressed as [20]

(3.4) H =1

2

⟨〈Xβ , Xβ〉 Xαα − 2 〈Xα, Xβ〉 Xαβ + 〈Xα, Xα〉 Xββ

〈Xα, Xα〉〈Xβ , Xβ〉 − 〈Xα, Xβ〉2,−→n

⟩,

The Laplace–Beltrami operator 4s may be defined as 4s = ∇s · ∇s where ∇s = ∇ −−→n ∂−→n , [20]. If P (α, β) is a smooth field is defined on the parametric surface X = X(α, β),

393

then (see [20])

∇sP =〈Xβ , Xβ〉 Xα − 〈Xα, Xβ〉 Xβ

〈Xα, Xα〉〈Xβ , Xβ〉 − 〈Xα, Xβ〉2Pα +

〈Xα, Xα〉 Xβ − 〈Xα, Xβ〉 Xα

〈Xα, Xα〉〈Xβ , Xβ〉 − 〈Xα, Xβ〉2Pβ ,

and straightforward technical calculations yield that

4sH =〈Xβ , Xβ〉 Hαα − 2 〈Xα, Xβ〉 Hαβ + 〈Xα, Xα〉 Hββ

〈Xα, Xα〉〈Xβ , Xβ〉 − 〈Xα, Xβ〉2+⟨

〈Xα, Xα〉 Xββ − 2 〈Xα, Xβ〉 Xαβ + 〈Xβ , Xβ〉 Xαα, | 〈Xβ , Xβ〉 |1/2−→τ β⟩

(〈Xα, Xα〉〈Xβ , Xβ〉 − 〈Xα, Xβ〉2

)3/2Hα+

⟨〈Xα, Xα〉 Xββ − 2 〈Xα, Xβ〉 Xαβ + 〈Xβ , Xβ〉 Xαα, | 〈Xα, Xα〉 |1/2−→τ α

⟩(〈Xα, Xα〉〈Xβ , Xβ〉 − 〈Xα, Xβ〉2

)3/2Hβ ,

(3.5)

where

−→τ α =Xα 〈Xα, Xβ〉 −Xβ 〈Xα, Xα〉

| 〈Xα, Xα〉 |1/2 · | 〈Xα, Xα〉〈Xβ , Xβ〉 − 〈Xα, Xβ〉2 |1/2,(3.6)

−→τ β =Xβ 〈Xα, Xβ〉 −Xα 〈Xβ , Xβ〉

| 〈Xβ , Xβ〉 |1/2 · | 〈Xα, Xα〉〈Xβ , Xβ〉 − 〈Xα, Xβ〉2 |1/2.(3.7)

Since the exterior surfaces Si, i ∈ Ψext evolve by surface diffusion (2.2) and the grainboundaries Si, i ∈ Ψgb evolve by mean curvature motion (2.1), we get that

(3.8)⟨Xit ,−→n i⟩

= −4sHi, i ∈ Ψext := 1, 2, 3,

(3.9)⟨Xit ,−→n i⟩

= Hi, i ∈ Ψgb := 4, 5, 6,

where explicit expressions for−→n , H, and 4sH were prescribed in (3.4)–(3.5).

3.3. Conditions along triple junctions lines. As noted our system contains three ther-mal grooves where two exterior surfaces, Sp, Sq, intersect a grain boundary S`, for p, q, ` ∈Ψthermal := 1, 2, 4; 2, 3, 5; 3, 1, 6. From Fig 2, we see that each thermal groove pos-sesses natural parametrizations with respect to α along Sp and S` and with respect to βalong Sq. Recalling that the conditions to be imposed include persistence, Young’s law,continuity of the chemical potential, continuity of mass flux as well as equi-spacing of theparametrization, we obtain that

Xp = Xq = X`, (persistence)(3.10) ⟨−→τ `α,−→τ pα⟩ =⟨−→τ `α,−→τ qβ⟩ = cos (π − θ/2), (Young’s law)(3.11)

〈Xpα, X

pαα〉 = 0, (equi-spacing)(3.12)

Hp = Hq, (continuity)(3.13) ⟨∇sH1,−→τ pα

⟩+⟨∇sHq,−→τ qβ

⟩= 0, (balance of mass flux),(3.14)

where θ is the dihedral angle, see (2.4). Expressions for H, ∇sH, −→τ α, −→τ β were givenin (3.4)-(3.7).

394

Along the internal triple junction line, where the grain boundaries S4, S5, S6 intersect,there is a natural parametrization with respect to β. Here persistence, Young’s law, andequi-spacing imply that

X4 = X5 = X6, (persistence)(3.15) ⟨−→τ 4β ,−→τ 5β⟩

=⟨−→τ 4β ,−→τ 6β

⟩= cos (2π/3), (Young’s law)(3.16) ⟨X4β , X

4ββ

⟩= 0, (equi-spacing).(3.17)

3.4. Conditions along the exterior free boundaries. Along the intersections of theexterior surface Sp with the bounding planes Πq and Π`, for p, q, ` ∈ Ψfb where Ψfb :=1, 4, 1; 2, 2, 3; 3, 3, 4

, we impose the conditions⟨

Xp − rq0,−→N q⟩

=⟨Xp − r`0,

−→N `⟩

= 0, (attachment)(3.18) ⟨−→n p,−→N q⟩

=⟨−→n p,−→N `

⟩= 0, (symmetry)(3.19)

〈Xpα, X

pαα〉 =

⟨Xpβ , X

pββ

⟩= 0, (equi-spacing)(3.20) ⟨

∇sHp,−→N q⟩

=⟨∇sHp,

−→N `⟩

= 0, (zero mass flux).(3.21)

Note that

(3.22)⟨r − ri0,

−→N i⟩

= 0,

prescribes the bounding planes Πi, i = 1, . . . 5, where−→N i is a unit normal vector to Πi and

ri0 is a point on Πi, which in our numerics will be chosen to be bounded away from theevolving surfaces.

Along the intersection of grain boundary Sp, for p = Ψgb, with Π5, we impose⟨Xp − r5

0,−→N 5⟩

= 0, (attachment)(3.23) ⟨−→n p,−→N 5⟩

= 0, (symmetry)(3.24)

〈Xpα, X

pαα〉 = 0, (equi-spacing).(3.25)

Similarly along the intersections of S4 with Π1 and Π2, we set⟨X4 − r1

0,−→N 1⟩

=⟨X4 − r2

0,−→N 2⟩

= 0, (attachment)(3.26) ⟨~n4, ~N1

⟩=⟨~n4, ~N2

⟩= 0, (symmetry)(3.27) ⟨

X4β , X

4ββ


and along the intersections of Sp with Π` for p, ` ∈5, 3; 6, 4

,⟨

Xp − r`0,−→N `⟩

= 0, (attachment)(3.29) ⟨−→n p,−→N `⟩

= 0, (symmetry)(3.30) ⟨Xpβ , X

pββ


395

Although the conditions of the form⟨−→n ,−→N⟩ = 0 could have been replaced by conditions of

the form⟨−→τ ,−→N⟩ = 1, the former turned out to be easier to handle numerically.

3.5. Conditions at the quadruple junction and at the corner points. In this subsec-tion, we describe the conditions imposed at the quadruple junction and at the other cornerpoints, ΨI C :=

It, IIt, IIIt, IV t, Ib, IIb, IIIb, IV b, CI , CII , CIII

, see in Fig 2. The bal-

ance of mass flux conditions will be discussed in detail in Section 3.6.

Conditions at It, the quadruple junction (QJ). At It we set

X1 = X2 = X3 = X4 = X5 = X6, (persistence)(3.32) ⟨X4β

‖X4β‖,Xjα

‖Xjα‖

⟩= cos(φ), j = 1, 2, 3, (angle condition)(3.33)

H1 = H2 = H3, (continuity)(3.34)3∑i=1

∫Γi

−→n Γi · ∇sHi dsi = 0, (balance of mass flux).(3.35)

The angle condition at the quadruple junction follows from assuming that the angle con-ditions prescribed by Young’s law hold smoothly up to the quadruple junction; this inparticular implies that cos(φ) = − 1√

3m√

4−m2, see [14]. The curve Γi which appears above as

well as in the balance of mass flux conditions which follow, corresponds to a smooth curvewhich normally intersects the bounding thermal grooves, see Section 3.6, and si correspondsto an arc-length parametrization of Γi.

Conditions at Ib. At Ib, where S4, S5, S6 and Π5 meet, we set

X4 = X5 = X6 (persistence)(3.36) ⟨X4 − r5

0,−→N 5⟩

= 0, (attachment)(3.37) ⟨X4α

‖X4α‖,X5α

‖X5α‖

⟩=

⟨X4α

‖X4α‖,X6α

‖X6α‖

⟩= cos

(2

3π

), (Young’s law).(3.38)

Conditions at IIb. The point IIb, where S4, Π1, Π2, and Π5 meet, is stationary, hence

(3.39)⟨X4 − r1

0,−→N 1⟩

=⟨X4 − r2

0,−→N 2⟩

=⟨X4 − r5

0,−→N 5⟩

= 0, (attachment).

Conditions at IIIb and IV b. At IIIb, where S5, Π3, Π5, intersect, we impose⟨−→n 5,−→N 3 +

−→N 5⟩

= 0, (symmetry)(3.40) ⟨X5 − r3

0, N3⟩

=⟨X5 − r5

0,−→N 5⟩

= 0. (attachment).(3.41)

We should have liked to require that⟨−→n 5,

−→N 3⟩

=⟨−→n 5,

−→N 5⟩

= 0; however looking at

the conditions imposed along the intersection of Sp with Π5 for p = Ψgb, and along the396

intersections of Sp with Π` for p, ` ∈5, 3; 6, 4

, this appears to be overly restrictive.

The conditions at IV b are analogous.

Conditions at IIt. At IIt, where S1, S2, S4 and Π1, Π2 intersect, we impose

X1 = X2 = X4, (persistence)(3.42) ⟨X4α

‖X4α‖,−→N 1 ×

−→N 2

⟩= 0, ( symmetry)(3.43) ⟨

X4 − r10,−→N 1⟩

=⟨X4 − r2

0,−→N 2⟩

= 0, (attachment)(3.44)

H1 = H2, (continuity)(3.45)2∑i=1

∫Γi

−→n Γi · ∇sHi dsi = 0, (balance of mass flux).(3.46)

Condition (3.43) constitutes a seemingly ”minimal” mirror symmetry condition.

Conditions at CI , CII and CIII . At CI , where S1, Π1, and Π4 intersect, we set⟨−→n 1,−→N 1 +

−→N 4⟩

= 0, (symmetry)(3.47) ⟨X1 − r1

0,−→N 1⟩

=⟨X1 − r4

0,−→N 4⟩

= 0, (attachment)(3.48) ∫Γ1

−→n Γ1

· ∇sH1 ds1 = 0, (zero mass flux)(3.49)

where (3.47) has been prescribed in analogy with (3.40). At CII and at CIII , where S2,Π2, Π3, and S3, Π3, Π4, respectively, intersect, analogous conditions are prescribed.

Conditions at IIIt and IV t. At IIIt where S2, S3, S5 and Π3 intersect, we set

X2 = X3 = X5, (persistence)(3.50) ⟨−→n 5,−→N 3⟩

=⟨X5α, X

5β

⟩= 0, (symmetry)(3.51) ⟨

X5 − r30,−→N 3⟩

= 0, (attachment)(3.52)

H2 = H3, (continuity)(3.53) ∑i∈2,3

∫Γi

−→n Γi · ∇sHi dsi = 0. (balance of mass flux)(3.54)

At IV t, where S1, S3, S6 and Π4 intersect, analogous conditions are imposed.

3.6. Approximation of the integral in the mass flux conditions. In Section 3.5,balance of mass flux at the quadruple junction and at the top corner points was prescribedin terms of a condition which included integrals of the form

(3.55)

∫Γ

−→n Γ · ∇sH ds,

397

where Γ is a curve lying on the exterior surface S which is bounded on either end by boundingplanes or by thermal grooves. We now prescribe more precisely the curve Γ, as well as our

Figure 3. A sketch of relationship between Γαβ , Γ and X (α, β, t).

approximation of the integral (3.55), near the point X (0, 0, t). See Fig. 3. The treatmentat the other corner points is analogously, though some care must be taken in defining thedirections of the various normals and tangents when summing such integrals.

We take Γ = Γ(s), where s is an arc-length parametrization, to be a smooth curvewhich lies on the exterior surface X(α, β, t) near X (0, 0, t), with preimage Γαβ , so that

ΓαβX(α, β, t)−−−−−−→ Γ. We require Γ to be pinned at either end at distance 1

2hα,12hβ , respectively,

from the corner in its preimage. Hence

(3.56) Γ (0) = X

(hα2, 0, t

), Γ (LΓ) = X

(0,hβ2, t

),

where LΓ denotes the length of Γ. Moreover Γ is taken to orthogonally intersect the boundingplane or thermal groove which lies at each end. For fixed t, let Γαβ be described in termsof polar coordinates, see Fig. 3, so that

(3.57) Γαβ (θ) =(α (θ) , β (θ)

)=(r(θ) cos θ, r(θ) sin θ

), θ ∈

[0,π

2

].

Then from (3.56)–(3.57), (α(0), β(0)) = (r(0), 0),(α(π2

), β(π2

))=(0, r

(π2

)), and

(3.58) r(0) =hα2, r

(π2

)=hβ2.

From (3.57), we get that (αθ(0), βθ(0)) = (rθ(0), r(0)),(αθ(π2

), βθ

(π2

))=(−r(π2

), rθ(π2

)).

Though it is a bit technical, the perpendicular intersection requirement implies that

(3.59) rθ(0) = −r(0)F(hα2 , 0

)E(hα2 , 0

) , rθ

(π2

)= r

(π2

) F(0,hβ2

)G(

0,hβ2

) ,where E = 〈Xα, Xα〉 , F = 〈Xα, Xβ〉, G = 〈Xβ , Xβ〉 . To be explicit, we take r (θ) tobe Hermit polynomial interpolant of the data (0, r (0)), (0, rθ (0)),

(0, r

(π2

)),(0, rθ

(π2

)),

398

namely that r(θ) = a + b θ + c θ2 + d θ2(θ − π

2

), where a = hα

2 , b = −hα2F(hα2 , 0)E(hα2 , 0)

, c =

2π2

[hβ − hα

(1− π

2

F(hα2 , 0)E(hα2 , 0)

)], d = 2

π2

[hβ

(F(

0,hβ2

)G(

0,hβ2

) − 4π

)− hα

(F(hα2 , 0)E(hα2 , 0)

− 4π

)].

Based on the above prescription, we get if ∇sH = O(1) that by the trapezoidal rule

(3.60)

∫Γ

−→n Γ · ∇sH ds =(−→n Γ · ∇sH

∣∣s=0

+ −→n Γ · ∇sH∣∣s=LΓ

) LΓ

2+O

(L3

Γ

)=

=

Xα

‖Xα‖· ∇sH

∣∣∣∣(hα2 , 0)

+Xβ

‖Xβ‖· ∇sH

∣∣∣∣(0,hβ2

) LΓ

2+O

(L3

Γ

),

.

By [18], LΓ =

π2∫0

√E r2

θ + 2F rθ r + G r2θ, and again by the trapezoidal rule,

(3.61)

LΓ ≈π

4

(√E r2

θ + 2F rθ r + G r2

∣∣∣∣θ=0

+√

E r2θ + 2F rθ r + G r2

∣∣∣∣θ=π

2

)=

=π

6

hα √EG− F2

E

∣∣∣∣∣(hα2 , 0)

+ hβ

√EG− F2

G

∣∣∣∣∣(0,hβ2

) .

From the definition of r (θ) and assuming thatF(hα2 , 0, t)E(hα2 , 0, t)

,F(

0,hβ2 , t

)G(

0,hβ2 , t

) are bounded, we

may conclude that r (θ) = O(max (hα, hβ)). Assuming moreover that rθr , E, F, G, are

bounded for all θ ∈[0, π2

], we get that LΓ = O(max (hα, hβ)) and that the approximations

appearing in (3.60)–(3.61) are O([max (hα, hβ)]3

). While the algorithm was implemented in

accordance with the discussion above, by applying the trapezoid rule once directly to (3.55),we could have mildly improved the approximation.

3.7. The governing equations. Combining the equations and conditions outlined in Sec-tions 3.1–3.6, we obtain the following problem formulation for the evolving equi-spacedparametric surfaces

(3.62)

〈Xpt ,−→n p〉 =4sHp

〈Xpα, X

pαα〉 =

⟨Xpβ , X

pββ

⟩=0

, p ∈ Ψext = 1, 2, 3

〈Xqt ,−→n q〉 =Hq

〈Xqα, X

qαα〉 =

⟨Xqβ , X

qββ

⟩=0

, q ∈ Ψgb := 4, 5, 6

Boundary Conditions

Initial Conditions399

where for Hp for p ∈ Ψext,

H =1

2

⟨⟨Xpβ , X

pβ

⟩Xpαα − 2

⟨Xpα, X

pβ

⟩Xpαβ + 〈Xp

α, Xpα〉 X

pββ

〈Xpα, X

pα〉⟨Xpβ , X

pβ

⟩−⟨Xpα, X

pβ

⟩2 ,−→n p⟩

and Hq is similarly defined for q ∈ Ψgb. In our simulations, the initial conditions were chosenin accordance with Fig. 2.

4. Numerical algorithm

In this section, we present our numerical algorithm for solving the problem formulatedin (3.62). The algorithm uses staggered grids and finite difference approximations, whichreduce the system to a DAE (differential and algebraic equation) system in time, which isthen solved implicitly using Newton iterations.

4.1. The staggered grids. Let X : [0, 1]2 × [0, T ] → R2 represent one of the evolvingsurfaces, Xi = Xi(α, β, t), i ∈ 1, . . . , 6. Each surface X is spatially approximated on astaggered grid, containing “interior points,” “ghost points,” and “groove points,” see Fig. 4.Let Xk

i,j = X(αi, βj , tk) denote the approximation of X at staggered grid point (i, j) at time

tk ∈ [0, T ], where αi = (i − 1/2)hα, βj = (j − 1/2)hβ , i ∈ 0, 12 , 1, . . . , N,N + 1

2 , N + 1,j ∈ 0, 1

2 , 1, . . . ,M,M + 12 ,M + 1, where hα = N−1, hβ = M−1 reflect the respective

interior grid spacings. The indices (i, j) ∈ 1, N × 1, . . . ,M refer to interior points, and

α

β

0

0

1

1

2

2

3

3

4

4

i-2

i-2

i-1

i-1

i

i

i+1

i+1

i+2

i+2

N-3

N-3

N-2

N-2

N-1

N-1

N

N

N+1

N+1

0 0

1 1

2 2

3 3

4 4

j-2 j-2

j-1 j-1

j j

j+1 j+1

j+2 j+2

M-3 M-3

M-2 M-2

M-1 M-1

M M

M+1 M+1

ghost points

ghost−groove points

corner points

groove points

interior points

Figure 4. The staggered grid mesh. The interior points (•) and ghostpoints() are indicated. The groove points are indicted (♦,,∗), and thecorner points () and ghost-groove points (∗) are marked.

the indices (i, j) ∈ 0, N + 1×0, 1, . . . ,M,M + 1∪0, 1, . . . , N,N + 1×0,M + 1 referto ghost points. The indices with (i, j) ∈ 1

2 , N + 12×0,

12 , . . . ,M,M + 1

2 ,M + 1∪ (i, j) ∈ 1

2 , N + 12×0,

12 , 1, . . . ,M,M + 1

2 ,M + 1 refer to groove points. Within the set of groove

points, we refer to (i, j) ∈ 12 , N + 1

2 × 12 ,M + 1

2 as “corner points,” and we refer to400

(i, j) ∈ 12 , N + 1

2 × 0,M ∪ 0, N × 12 ,M + 1

2 “ghost-groove points.” The physicalboundaries of X, which correspond either to triple junction lines or to free boundary lines,are approximated using the groove points. This approach allows us to use centered finitedifferences at the interior points as well as at the groove points.

At each time step tk, the variables Xki,j at the interior and ghost points are updated, as

are Hki,j for k ∈ Ψext, the mean curvatures on the exterior surfaces.

4.2. Approximation of derivatives. Let(∂ X∂ t

)ki,j

denote the standard forward approxi-

mation of Xt at (αi, βj , tk) which has O(4tk) accuracy with 4tk = tk − tk−1, let(∂X∂α

)ki,j

,(∂X∂β

)ki,j

, denote the standard centered difference approximations which have O(

(hα)2)

and

O(

(hβ)2)

accuracy, respectively, and let(∂2X∂α2

)ki,j

,(∂2X∂β2

)ki,j

, denote the standard centered

difference approximations of the second order partial derivatives, whose respective accu-

racy is given by O(

(hα)2)

and O(

(hβ)2)

. Centered difference approximations of the sec-

ond mixed derivatives, which have O(

(hα)2

+ (hβ)2)

accuracy, are indicated by(∂2X∂α∂β

)ki,j.

Analogous notation is used to indicate the derivatives of H on the exterior surfaces.

4.3. Approximations along the triple junction lines and free boundary lines.Triple junction lines and exterior free boundary lines are approximated along groove pointsof the grids, see Fig. 4. Suppose Xk

i,j is known at the interior and ghost points, and wewish to approximate X at the groove points. Using the neighboring ghost point and three”collinear” interior points (or ghost points), and Taylor expanding

(4.1) Xki, 12

=5Xk

i,0 + 15Xki,1 − 5Xk

i,2 +Xki,3

16+O

((hβ)

4), i ∈ 0, 1, . . . , N,N + 1.

Similarly, X may be estimated at the other groove points which are not corner points. Usingthe above estimates, X can be estimated at the corner points. For example,

(4.2) Xk12 ,

12≈

5(Xk

12 ,0

+Xk0, 12

)+ 15

(Xk

12 ,1

+Xk1, 12

)− 5

(Xk

12 ,2

+Xk2, 12

)+(Xk

12 ,3

+Xk3, 12

)32

,

to O(

(hα)4

+ (hβ)4)

accuracy. Obtaining order fourth accuracy in (4.1)-(4.2) allows us to

maintain overall second order spatial accuracy in approximating the system (3.7).The derivatives ∂X

∂α , ∂X∂β along the groove can be approximated using second order centered

difference approximations with functional evaluations at the ghost points,(∂X

∂α

)ki,j

=Xki+ 1

2 ,j−Xk

i− 12 ,j

hα+O

((hα)

2),

(∂X

∂β

)ki,j

=Xki,j+ 1

2

−Xki,j− 1

2

hβ+O

((hβ)

2).

Similar approximations are used to estimate the mean curvature H and its derivatives alongthe triple junction lines and external free boundary lines on the exterior surfaces.

401

4.4. Numerical solution of the DAE system. Let n ∈ N and T > 0 be given, and letus consider the solution of the following general system of n DAE

(4.3) F (Y ′, Y, t) = 0, t ∈ (0, T ], Y (0) = Y0,

where Y (t) = (y1(t), . . . yn(t)), Y : [0, T ]→ Rn, and Y0 ∈ Rn is prescribed. Let Y k = Y (tk)denote an approximation of Y at time 0 ≤ tk ∈ [0, T ], for k = 0, 1, . . . , NTmax , NTmax ∈ N,with t0 = 0, and set 4tk = tk−tk−1. Applying the backward Euler method [2] to the systemgiven in (4.3), we obtain the following n× n system of nonlinear equations for Y k,

(4.4) F

(Y k − Y k−1

4tk, Y k, tk

)= 0, k = 1, . . . , NTmax , Y 0 = Y0.

Noting that F : Rn → Rn has n components, fi = fi

(Y k−Y k−1

4tk , Y k, tk

), i = 1, . . . , n, its

Jacobian with respect to Y k can be expressed as

(4.5)

∂f1

∂yk1· · · ∂f1

∂ykn...

. . ....

∂fn∂yk1

· · · ∂fn∂ykn

.

For k = 1, . . . , NTmax , we use Newton’s iteration method to find successively better approxi-mations to the zeroes (roots) of F (Y ) until an estimate of the local error is not greater thansome prescribed tolerance, 0 < ε 1. We chose ε = 10−11 to achieve accuracy comparableto the estimated accuracy in the approximation of the Jacobian matrix. Additionally, weset a tolerance bound of ε = 10−6 on F (Y ). See Algorithm 1 in Appendix.

4.5. Approximation of the Jacobian matrix. We approximate the Jacobian matrix(4.5) numerically, based on the central difference approximations

(4.6)∂Fi∂yj

=Fi (y + ε ej)− Fi (y − ε ej)

2 ε+O

(ε2),

where ej = (0, . . . ,

element j︷︸︸︷1 , . . . , 0) and ε is a small positive parameter. In [35] the suggestion

is made to chose ε = ε1/3M , where εM is the machine epsilon. Adopting this approach in our

double precision setting yields ε2/3M ≈ 10−11 accuracy in (4.6). See Algorithm 2 in Appendix.

4.6. A numerical algorithm. We outline below our numerical algorithm for solving thesystem (3.62). We consider Hp, the mean curvature of the exterior surfaces Sp, p ∈ Ψext asadditional variables. Accordingly we do not need to calculate the third and fourth derivativesof X with respect to α, β. We discretize each exterior surface using staggered grids, asindicated above, with N×N interior grid points. Similarly we discretize each grain boundarysurface using staggered grids with N ×M interior grid points.

We choose initial conditions in accordance with the configuration portrayed in Fig. 2. Attime tk, let (Xk

i,j)p and (Hk

i,j)p, p ∈ Ψext, denote the value of Xp and Hp at grid point (i, j),

and let (Xki,j)

q, q ∈ Ψgb, denote the value of Xq at grid point (i, j). Discretization of thegoverning equations requires approximation of the time derivatives of Xp, Xq, as well as ofthe first and second derivatives of Xp, Hp and Xq with respect to α and β at the interiorgrid points. Discretization of the boundary conditions requires approximations for Xp, Hp

402

and Xq and their first derivatives with respect to α and β along the groove points of thegrid.

The discretization of the governing equations and boundary conditions yields an ODAEsystem, which we indicate for simplicity by

(4.7) F (U ′, U, t) = 0, t ∈ (0, T ], U(0) = U0.

In (4.7), U = (u1, u2, . . . , u`, . . .) is taken to reflect a fairly natural ordering of the elements

(Xi,j)p =

((xi,j)

q, (yi,j)q, (zi,j)

q), (Hk

i,j)p, (Xi,j)

q =(

(xi,j)q, (yi,j)

q, (zi,j)q),

which are defined at the interior and ghost points of the various grids used in approximatingSp and Sq respectively. The vector U contains 3 (N + 2) (4N + M + 14) elements and doesnot include evaluations along the groove points, since the variables and equations at thegroove points are prescribed with the help of the interior and ghost points using (4.1)-(4.2).

We solve (4.7) in accordance with the discussion in Sections 4.4 and 4.5. See Algorithms 1

and 2 in the Appendix. The overall accuracy of our discretization is O(

(hα)2

+ (hβ)2)

in space and O(4t) in time. Although we do not prove convergence, we tested for self-consistency of our method by taking smaller time steps and grid partitions.

By considering the governing equations and boundary conditions, it can be seen thatchanges in the entries at some grid point (i, j) influence the values of the entries only insome small neighborhood of grid points around the grid point. By defining a neighborhoodfor each grid point and undertaking functional evaluations only in this neighborhood, therunning time for the algorithm could be reduced by partial parallelization.

Figure 5. Visualization of the nz = 163188 nonzero entries in the Jacobianmatrix for our system where m = 0.1, Lx = 30, Ly = 30, Lz = 0.25, with4t = 0.25, N = 20, M = 10.

403

The resultant Jacobian matrix, which we denote as A, is sparse and not symmetric, seeFig 5. We used the MATLAB UMFPACK [12] solver for solving Ax = b. With regardto software, the algorithm was implemented in ”C”, together with various extension andpackages, and the postprocessing was undertaken in MATLAB. The calculations of thefunction F and its Jacobian J were done in parallel using the MPI or ”Message PassingInterface” [30] library of extensions to ”C”. The simulations were run on the RBNI high-performance ”NANCO” computer of Technion Center for Computational Nanoscience andNanotechnology.

5. Numerical results and tests

5.1. Some numerical results. Simulations were undertaken for the initial conditions por-trayed in Fig 2, with Lx = Ly = 30, Lz = 0.25, m ∈M, whereM = 0.0, 0.001, 0.01, 0.05, 0.01,for 0 ≤ t ≤ 100 with N = 50, M = 25, 4t = 0.5. Note that the values Lx = Ly = 30,Lz = 0.25 are appropriate for modeling thin films, since the height to width ratio isLz/Lx = Lz/Ly = 1/120. We also tested our program with other values of Lx, Ly, Lz,such as Lx = Ly = Lz = 1, and with other values for N , M , 4t.

Results from a numerical simulation with m = 0.1 and Lx = Ly = 30, Lz = 0.25 can beseen in Fig. 6. As expected, formation of thermal grooves can be observed. It is not clearwhether annihilation of the corner grain will eventually occur here prior to break up of thethin film. Considerable pitting seems to be occurring at the quadruple point as well as atthe corner point, IIt, which is located above (x, y) = (0, 0).

In Fig. 7, the heights at the quadruple junction and at corner points from simulationswith m ∈M where M = 0.0, 0.001, 0.01, 0.05, 0.01, and Lx = Ly = 30, Lz = 0.25, can beseen. Due to symmetry, the behavior at the corner points IV t, CII , is the same as at IIIt,CI , respectively, and has not been portrayed. Note the monotone decrease in the height atthe quadruple point and at the corner points IIt, IV t, whereas the behavior of the heightat the corner points CI , CIII is nonmonotone.

5.2. Comparison with the von Neumann-Mullins law. Von Neumann and Mullinsdeveloped a formula [32, 51] for the evolution of the surface area, S(t), of a grain, which isembedded in R2 with n trijunctions along its perimeter, within a planar network of grainswith grain boundaries that evolve by mean curvature motion, Vn = Aκ, namely,

(5.1)dSdt

=Aπ

3(n− 6).

The “corner grain” in our system, whose exterior surface, S3, contains the corner point CIII ,may be viewed, by extending our three grain system by mirror symmetry, as one quarter ofan embedded grain with four trijunctions along its perimeter.

In our system, when m = 0 the exterior surfaces remain flat and the grain boundariesevolve by mean curvature motion, Vn = AH, where H is the average of the principle curva-tures, see (2.1),(3.4); under these circumstances, writing (5.1) in terms of our dimensionlessvariables, we find for the surface area of the corner grain, |S3|, that

(5.2)d|S3|dt

= −Aπ/12.

Integrating (5.2) yields that |S3|(t) = |S3|(0) − π t/12, which implies that the corner grainshould annihilate at time Ta = 12|S3|(0)/π. If 0 < m 1 and if the specimen is sufficiently

404

(a) (b)

(c) (d)

Figure 6. Simulation results for m = 0.1, Lx = Ly = 30, Lz = 0.25, attimes (a) t = 10, (b) t = 40, (c) t = 70, (d) t = 100. Pitting can be seen atthe quadruple junction as well as at the corner point, IIt.

thick, then (5.2) should still be roughly accurate. However break up may occur beforethe grain annihilates if the specimen is too thin, and (5.2), as an approximation, becomesinaccurate as m is increased.

In Fig. 8, the area of S3 is portrayed for m ∈M, whereM = 0.0, 0.001, 0.01, 0.05, 0.01,as a function of time. When m = 0.0, the surface area evolves in accordance with (5.2),and decreases linearly. Monotone decrease is seen for all values of m ∈ M; however, as mincreases, the decrease rate slows, implying that grain annihilation should take longer. Thiscorroborates the predictions in [26] that for relatively small thin specimens, groove formationshould slow grain boundary migration. It follows from the formula Ta = 12|S3|(0)/π thatwhen m = 0, annihilation of the corner grain in our simulation should occur at time Ta ≈859.4, and the results portrayed in Fig. 8 indicate that Ta > 859.4 when m > 0. Since the

405

Time

0 20 40 60 80 100

He

igh

t a

t Q

J (

It )

0

0.05

0.1

0.15

0.2

0.25

m=0.0

m=0.01

m=0.05

m=0.1

(a)

Time

0 20 40 60 80 100

He

igh

t a

t II

t

0

0.05

0.1

0.15

0.2

0.25

m=0.0

m=0.01

m=0.05

m=0.1

(b)

Time

0 20 40 60 80 100H

eig

ht

at

IIIt (

IVt )

0

0.05

0.1

0.15

0.2

0.25

m=0.0

m=0.01

m=0.05

m=0.1

(c)

Time

0 20 40 60 80 100

He

igh

t a

t C

I (C

II)

0.24

0.245

0.25

m=0.0

m=0.01

m=0.05

m=0.1

(d)

Time

0 20 40 60 80 100

He

igh

t a

t C

III

0.24

0.245

0.25

m=0.0

m=0.01

m=0.05

m=0.1

(e)

Figure 7. The heights at (a) the quadruple junction, at the corner points(b) IIt, (c) IIIt, and at the corner points (d) CI , (e) CIII , as functionsof time for m ∈ M where M = 0.0, 0.001, 0.01, 0.05, 0.01. Here Lx =Ly = 30, Lz = 0.25. In accordance with Fig. 6, the height at the quadruplejunction and at IIt, IIIt exhibit monotone decrease, whereas the heightsat CI , CIII appears to oscillate.

simulations reflected in Fig. 6 were undertaken for m = 0.1 with 0 ≤ t ≤ 100, it is notsurprising that annihilation was not seen.

406

Time

0 20 40 60 80 100

Are

a o

f S

3

200

205

210

215

220

225

m=0.0

m=0.01

m=0.05

m=0.1

von Neumann-Mullins

Figure 8. Comparison with the von Neumann-Mullins law. For m ∈ Mwhere M = 0.0, 0.001, 0.01, 0.05, 0.01, the area of S3, |S3|, decreasesmonotonically. When m = 0.0, the area of S3 decreases linearly in accor-dance with the von Neumann-Mullins law. Here surface diffusion appearsto slow the rate at which the smallest grain shrinks.

5.3. Verifying mass conservation and energy dissipation. Assuming sufficient regu-larity, our problem formulation can be shown to satisfy [14, 36]

(5.3)∑i∈Ψext

∫SiVn dS = 0,

where∫SiVn dS is the integral of the normal velocity over the surface Si, which implies

conservation of the total volume of our three grain system. Assuming that the density ofour 3 grain system, composed of 3 grains of the same (isotropic) material, is constant, (5.3)also implies mass conservation.

Our problem formulation can also be shown to satisfy [14, 36]

(5.4)d

dt

∑i∈Ψext

|Si|+m∑i∈Ψgb

|Si|

= −B∑i∈Ψext

∫Si|∇sHi|2 dS −mA

∑i∈Ψgb

∫|Hi|2 dS.

where |Si| is the surface area of surface Si. Equation (5.4) is an energy dissipation equality,prescribing the rate of dissipation of the (dimensionless) total energy in the system,

Esystem := ∑i∈Ψext

|Si|+m∑i∈Ψgb

|Si|

=1

γext

γext

∑i∈Ψext

|Si|+ γgb∑i∈Ψgb

|Si|,

where γext and γgb are the surface free energies of the exterior surfaces and the grain bound-aries, respectively, and m = γgb/γext.

In Fig. 9a, mass conservation of our algorithm is verified in that the relative variationin the total volume of the system, [V (t) − V (0)]/V (0) is ≈ 10−4, in accordance with theaccuracy of our numerical scheme. In Fig. 9b, we see that Esystem decreases in accordancewith (5.4) after a short initial transient. The initial transient is apparently due to the lackof compatibility of the initial conditions with the boundary conditions (Young’s law) alongthe thermal grooves.

407

Time

0 20 40 60 80 100

[ V

(t)

- V

(0)

] /

V(0

)×10

-4

-1.5

-1

-0.5

0

0.5

1

m=0.0

m=0.01

m=0.05

m=0.1

(a)

Time

0 20 40 60 80 100

Esyste

m

899.5

900

900.5

901

901.5 m=0.0

m=0.01

m=0.05

m=0.1

(b)

Figure 9. For Lx = Ly = 30, Lz = 0.25, m ∈ M, whereM = 0.0, 0.001, 0.01, 0.05, 0.01, (a) the relative total volume of thesystem, [V (t) − V (0)]/V (0), and (b) the (dimensionless) weighted sur-face area, Esystem, as functions of time, see (5.3)–(5.4). Note that|V (t) − V (0)|/|V (0)| < 10−4, and Esystem exhibits monotone decrease fol-lowing a short initial transient. After the initial transient, the variation in|V (t)− V (0)|/|V (0)| is also smaller.

5.4. Comparison with Mullins (1957) and Genin, Mullins, Wynblatt (1992).Mullins (1957) [33] and Genin, Mullins & Wynblatt (1992) [22] studied grooving and pittingphenomena based on a small slope linearized approximation of surface diffusion, which isvalid in the limit m → 0. More specifically, Mullins [33] considered the thermal grooveresulting from an initially flat grain boundary which perpendicularly intersected an initiallyflat exterior surface of infinite extent. Based on the small slope approximation, he estimatedthe height of the thermal groove to be given by

(5.5) hM (t) = h0 −m

23/4 Γ(5/4)√

4−m2t1/4.

where Γ(·) is the Gamma function. Genin, Mullins & Wynblatt [22] considered the pittingwhich occurs where two thermal grooves develop on an initially flat exterior surface of infiniteextent, which is bounded by two planar grain boundaries which are constrained to remainplanar, to perpendicularly intersect the exterior surface, and to intersect each other at aprescribed angle ψ. Based on the small slope approximation, they numerically showed theheight at the point of intersection to be given by

(5.6) hGMW (t, ψ) = h0 −mf(ψ)

23/4 Γ(5/4)√

4−m2t1/4.

where Γ(·) is the Gamma function and f(ψ) is a monotonically decreasing function of theangle ψ. They estimated f(ψ) for various values of ψ, and found that f(72) ≈ 2.5,f(108) ≈ 1.66, f(126) ≈ 1.4. Moreover they demonstrated analytically that f(90) = 2,and it follows from (5.5) that f(180) = 1. Using cubic spline interpolation based on theirdata, we estimated that f(120) ≈ 1.479.

408

Time0 50 100 150 200

Heightatcorner

points

0

0.05

0.1

0.15

0.2

0.25 QJ (It)IIt

IIIt (IVt)MullinsGenin et al.; ψ = 90o

Genin et al.; ψ = 120o

(a)

ln[ Time ]-2 0 2 4 6

ln[h0-h(t)]

-4

-3.5

-3

-2.5

-2

-1.5

-1QJ (It)IIt

IIIt (IVt)MullinsGenin et al.; ψ = 90o

Genin et al.; ψ = 120o

(b)

Figure 10. From simulations with m = 0.1. (a) The height at the variouscorner points, and (b) the natural logarithm of the normalized height atthe various corner points, as functions of time.

In Fig. 10a, we compare the evolution of the height at the quadruple junction, QJ, and theheight of the various corner points with the predictions of Mullins [33] and Genin, Mullins &Wynblatt [22]. One can clearly see that the height at IIt behaves roughly like hGMW (t, 90),the height at the quadruple junction QJ behaves roughly like hGMW (t, 120), and the heightat IIIt(IV t) behaves roughly like hM (t), see (5.5), (5.6). Taking into account the geometryof the system (see Fig. 2), these similarities are to be expected if nonlinear effects and finitesystem effects can be neglected. In Fig. 10b, we can see that the behavior of the heightsof all of the corner points is approximately proportional to t1/4; this behavior is reasonablesince m = 0.1 here, though at later times nonlinear effects and finite system effects shouldbecome more apparent.

6. Conclusions

We have developed a robust algorithm designed to follow the coupled evolution by surfacediffusion of the exterior surfaces with internal grain boundary migration by motion by meancurvature in systems containing a small number of grains. Our algorithm has been verifiedin several ways. We have shown that it conserves mass and dissipates energy to withinthe accuracy of the algorithm. It reproduces the predictions of the von Neumann-Mullinslaw [32, 51] in the limit m → 0. Moreover for 0 < m 1, it reproduces Mullins’ [33]classical grooving predictions at the grooves which form along the exterior boundaries ofthe domain, and at appropriately constrained trijunctions, it reproduces the predictions ofGenin, Mullins, Wynblatt [22] for ψ = 120.

The geometry treated in this paper is portrayed in Fig. 11a. Our numerical approach canbe readily adapted to treat other geometries, such as the geometries portrayed in Fig. 11.More specifically, the geometries in Figs. 11b, 11c were studied in [7], that in 11d was studiedin [14, 15, 17], and the geometries portrayed in Figs. 11e, 11f are reminiscent of geometriesdiscussed in [46]. The geometries in Fig. 11d, 11f are relevant in particular to the study ofpitting.

409

(a) (b) (c)

(d) (e) (f)

Figure 11. Our algorithm can be adapted to follow the evolution of thegrain systems portrayed above.

It should be possible to incorporate additional effects, such as elastic stresses or varioustypes of defects, by coupling our system of equations with additional fields. It should berather straightforward, though a bit technical, to include anisotropy. While these effects mayalso be incorporated within the framework of finite element methods [8], we emphasize thateven within the isotropic framework treated here, it appears that we are the first to be ableto undertake realistic 3D simulations of phenomena such as pitting, grain annihilation, holeformation, and more recently dewetting phenomena and stabilization of hexagonal arrays[15, 17, 16].

An eventual goal is to treat many grain systems, starting perhaps by considering nano-crystalline specimens which are several grains deep. In this context it may be possible touse our approach to describe the evolution of the exterior surfaces, coupled with a level setapproach to follow the evolution of the grain boundaries within the interior. If, using hybridmethods as suggested above or some other methods, we find new phenomena of interest, wemay adjust our algorithm to focus in locally on the details of the dynamics.

410

Appendix: Algorithms

Algorithm 1 Calculating an approximate solution to the discretized system

Input: U0, ε, NTmax , 4tkNTmax1 , Nmax

Output: Uk for k = 1, ..., NTmaxSet ε = 10−6

Set t = 0for k = 1, . . . , NTmax doV = Uk−1

Set t = t+4tkSet i = 1repeat

Calculate F(V−Uk−1

4tk , V, t)

Calculate the Jacobian J(V ), see Algorithm 2Solve J(V ) δ = −F (V ) (a linear system, Ax = b)Update V = V + δSet i = i+ 1

until ||F (V )|| ≤ ε or ||δ|| ≤ ε or i > Nmaxif ||F (V )|| > ε then

stop the algorithm . the desired tolerance cannot be achievedend ifSet Uk = V

end for

Algorithm 2 Approximating the Jacobian

Input: V , F , N , MOutput: Jacobian J

Set Length = 3N (4N + 3M)Set ε = 10−6

for i = 1, . . . , Length do

Set e = (0, . . . ,

element i︷︸︸︷1 , . . . , 0)

Calculate F (V + ε e)Calculate F (V − ε e)Set column i in matrix J to be F (V+ε e)−F (V−ε e)

2 εend for

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Department of Mathematics, Technion-IIT, Haifa 32000, Israel


Department of Mathematics, Technion-IIT, Haifa 32000, Israel


Racah Institute of Physics, The Hebrew University, Jerusalem 91904, IsraelE-mail address: [email protected]

413

JOURNAL OF ELLIPTIC AND PARABOLIC EQUATIONS

VOLUME 2, p. 1-413

CONTENTS

T. Huang - L. Liu - Y. Luo - C. Wang : Heat flow of extrinsic biharmonic maps from

a four dimensional manifold with boundary 1

L. Bandeira - P. Pedregal : The role of non-negative polynomials for rank-one convexity

and quasi convexity 27

S. Barile : Existence and multiplicity for quasi-critical fourth order quasilinear problems

with generalized vanishing potentials 37

F. Gmeineder : Symmetric–Convex Functionals of Linear Growth 59

L. M. De Cave - F. Oliva : On the regularizing effect of some absorption and singular

lower order terms in classical Dirichlet problems with L1 data 73

J. Naumann : Existence of weak solutions of an unsteady thermistor system with p-

Laplacian type equation 87

V. Ambrosio : infinitely many periodic solutions for a fractional problem under perturba-

tion 105

J.I. Dıaz - D. Gomez-Castro - C. Timofte : The effectiveness factor of reaction-diffusion

equations: homogenization and existence of optimal pellet shapes 119

G. Devillanova : Multiscale weak compactness in metric spaces 131

H. Guo - F. Hamel : Monotonicity of bistable transition fronts in RN 145

E. DiBenedetto - U. Gianazza - V- Vespri : Remarks on Local Boundedness and Local

Holder Continuity of Local Weak Solutions to Anisotropic p-Laplacian Type Equations 157

I. Birindelli - F. Demengel : Existence and regularity results for fully nonlinear operators

on the model of the pseudo Pucci’s operators 171

2

JOURNAL OF ELLIPTIC AND PARABOLIC EQUATIONS

VOLUME 2, p. 1-413

CONTENTS

D. Torrejon - M. Emelianenko - D. Golovaty : Continuous time random walk based

theory for a one-dimensional coarsening model 189

S. N. Antontsev - J.I. Dıaz : Finite speed of propagation and waiting time for local

solutions of degenerate equations in viscoelastic media or heat flows with memory 207

G. Cimatti : A boundary value problem for a nonlinear elliptic system relevant in general

relativity 217

G. C. G. Dos Santos - G. M. Figueiredo : Positive solutions for a class of nonlocal

problems involving Lebesgue and generalized spaces : scalar and system cases 235

J. Dolbeault - M. J. Esteban - M. Loss : Interpolation inequalities, nonlinear flows,

boundary terms, optimality and linearization 267

Y. Du - B. Lou - M. Zhou : Spreading and Vanishing for Nonlinear Stefan Problems in

High Space Dimensions 297

F. J. S.A. Correa - A. C. dos Reis Costa : Nonlocal Neumann Problem with Critical

Exponent from the Point of View of the Trace 323

M. Bildhauer - M. Fuchs - J. Muller - C. Tietz : On the solvability in Sobolev

spaces and related regularity results for a variant of the TV-image recovery model: thevector-valued case 341

S. N. Antontsev - I. V. Kuznetsov : Singular perturbations of forward-backward p-

parabolic equations 357

M. Plum - W. Reichel : A breather construction for a semilinear Curl-Curl wave equation

with radially symmetric coefficients 371

V. Derkach - A. Novick-Cohen - A. Vilenkin : Grain boundary migration with thermal

grooving effects: a numerical approach 389

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