Interpolation and partial differential equationsLech Maligranda, Lars Erik Persson, and John Wyller Citation: J. Math. Phys. 35, 5035 (1994); doi: 10.1063/1.530829 View online: http://dx.doi.org/10.1063/1.530829 View Table of Contents: http://jmp.aip.org/resource/1/JMAPAQ/v35/i9 Published by the American Institute of Physics. Additional information on J. Math. Phys.Journal Homepage: http://jmp.aip.org/ Journal Information: http://jmp.aip.org/about/about_the_journal Top downloads: http://jmp.aip.org/features/most_downloaded Information for Authors: http://jmp.aip.org/authors

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Interpolation and partial differential equations Lech Maligranda Department of Mathematics, L&e; University of Technology, S-95187 L&e;, Sweden

Lars Erik Persson Department of Mathematics, Lulei University of Technology, S-95187 Lule;, Sweden and Department of Mathematics, Narvik Institute of Technology, P.0. Box 385, N-8501 Narvik, Norway

John Wyller Department of Mathematics, Narvik University of Technology, P.0. Box 385, N-8501 Narvik, Norway

(Received 31 March 1994; accepted for publication 8 April 1994)

One of the main motivations for developing the theory of interpolation was to apply it to the theory of partial differential equations (PDEs). Nowadays interpolation theory has been developed in an almost unbelievable way {see the bibliography of Maligranda [Interpolation of Operators and Applications (I9261990), 2nd ed. (Lulei University, Lule& 1993), p. 1541). In this article some model examples are presented which display how powerful this theory is when dealing with PDEs. One main aim is to point out when it suffices to use classical interpolation theory and also to give concrete examples of situations when nonlinear interpolation theory has to be applied. Some historical remarks are also included and the relations to similar results are pointed out. .

1. INTRODUCTION

One of the main motivations for developing the theory of interpolation was to apply it to the theory of partial differential equations (PDEs). The original definitions in real interpolation theory worked out by Gagliardo, Lions, Peetre, etc. were usually given in terms of PDEs (the trace theorem). However, around 1960 it was discovered (by Peetre) that these interpolation spaces equivalently could be described (via the parameter theorem) by using the so-called K functional (see our Sec. II). From the 1960s until today the interpolation theory has been developed in an unbelievable way as an art in its own right (with separate conferences with hundreds of partici- pants, etc.). See, e.g., the books by Bergh-Lofstrom,’ TriebeL2 Krein-Petunin-Semenov,3 and especially the bibiliography of Maligranda.4 However, in our opinion, many genuine new appli- cations of this powerful theory have not yet been pointed out, and many times the original arguments for developing the theory have been forgotten or not emphasized.

In the present article we will point out the fact that the present state of the interpolation theory is a very powerful tool for obtaining qualitative information from the boundary data about the solutions of PDEs. We start out by employing classical linear interpolation theory to the linear wave equation. However, in many cases when studying nonlinear problems, we need powerful tools of modern nonlinear interpolation theory. We briefly discuss the latter theory and prove some new regularity results for the p-Laplace and the Korteweg-de Vries equations. Finally, we com- pare our results with other similar results and shortly discuss the possibilities of further developing and applying these ideas.

This article is organized as follows: In Sec. II we give a preliminary discussion on interpola- tion theory and PDEs including historical remarks, a description of (Lions-Peetre’s) real interpo- lation theory, and a brief description of the main ideas of this article. In Sec. III we prove and discuss an application of linear interpolation theory to a wave equation with variable coefficients (Theorem 1). In Sec. IV we present and discuss a useful interpolation theorem for nonlinear operators (Theorem 1). Section V is devoted to the discussion of the p-Laplace equation and we

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5036 Maligranda, Persson, and Wyller: Interpolation and PDEs

point out that the physical information inherited in the boundary data can be carried over to the corresponding solutions by using the nonlinear interpolation theory (see Theorems 1 and 2). In Sec. VI we do a similar investigation concerning the Korteweg-de Vries equation (see Theorem 1). Finally, Sec. VII is reserved for a final discussion and some concluding remarks, and some open questions which we hope are of interest for people working in both the fields of physics and pure mathematics and which probably must be tackled by using ideas from both these disciplines.

II. A PRELIMINARY DISCUSSION ON INTERPOLATION THEORY AND PDEs

A. The state of the art

The first interpolation theorem (the Riesz-Thorin interpolation theorem, see Ref. 1 p. 2 and Ref. 5) was proven by Riesz already in 1926. This theorem was the first step in creating the (Calderon) complex method ofinterpolation [.I@. In a similar way, the Marcinkiewicz interpolation theorem from 1939 (see Ref. 1, p. 9 and Ref. 6) inspired the (Lions-Peetre) real method of interpolation (.) 0,4 . Nowadays, there exist several methods of interpolation, but the real and complex methods are sufficient for most applications of linear problems. Furthermore, the inter- polation spaces at hand are described explicitely in almost all situations of practical importance (see, e.g., Refs. l-4) and, thus, the results can be given in familiar terms.

B. On (Lions-Peetre) real method of interpolation

. For the purposes of this article, it is sufficient to work with the real method, and in what follows we only describe this method. The starting point is to consider a Banach pair A = (A,, ,A I) consisting of two Banach spaces A,, and A,, which both are continuously embedded in a (large) Haussdorff topological vector space. For O< 8< 1 and 1 sqsm, the real interpolation space ~B,q=(Ao,A,)B,q consists of all a EA~+A, satisfying

(1 Om(t-eK(t,a))Y F ‘lq, 1

4cm I141i8,q=l1410,q=

sup(teeK( t,a)), q= m, r>o

where K(t,a)=K(t,a,Ao,A1) is the Peetre K functional defined by

It is easy to see that A,,, is an intermediate space with respect to the Banach couple A, i.e., that A,nA,CACA,+A, and A,,, is an interpolation constnrction in the following sense: For any linear operator T:A, + A, --f B. + B, which maps A0 boundedly into B. with norm MO and A, boundedly into B, with norm M, , T maps A B,q boundedly into B O,q with the norm less than M’-6MO

0 If A :‘C A0 , then it is easy to see that

A1=ABo.qO CA81 41 C Ao

provided that eo> 8, or that 0,= 8t and q o~q,. Here we use the convention that AO,q=AO and Al,q=AI.

For more details and proofs concerning this interpolation theory of linear operators, we refer to the original article7 of Peetre and the books in Refs. l-3. Moreover, concerning the definitions of the function spaces we are working with in this article, we refer to the books in Refs. l-3 and the classical book’ by Adams.

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Maligranda, Person, and Wyller: Interpolation and PDEs 5037

C. The main idea

Let P(x,t), x E tiCR”, t E R be a partial differential operator and consider the PDE

i

Pu=g u=f (2.1)

and the family of operators {T,}, 0 s t < ~0, where Tt maps the boundary value functionf(x) into the solution u(x,t) of (2.1), i.e.,

TJ’(x)=u(x,t).

Now, if these operators have the boundedness properties

for all f EAOflA,, where (Ao,A1) and (Bo,B1) are Banach pairs, then, by interpolation, we can control the boundedness in all “intermediate” cases, i.e.,

11 TJi (&.B,)Q-- 0 ~cl-eC~llfll~Ao.Ai)s.q (2.3)

for all fe A,flA i . Moreover, if we have some additional information about the operators T, (e.g., that they are Lipschitz, monotone, etc.), then we can extend the inequality (2.3) to hold even on the closure of A,flA, in (Ao,A,)e,q [when q< 00 this closure in fact coincides with (A, ,A 1) e,,].

However, if we have the explicit additional information that A t CA,, then the assumptions in Eq. (2.2) can be replaced by

ikflls,-ollfll/t,, f EAo,

llL%,~C,llfll~,, f EAI (2.4)

and the conclusion is that the inequality in (2.3) holds for every f from the spaces (Ao,A ,)e,, . Nowadays, we have concrete descriptions of the spaces (A0 ,A i) B,q in most cases of practical

importance. Therefore, this technique is very powerful and easy to use.

D. A concrete example

Assume that 1 <po, pi, qo, qi~m, and

IITfl~~qo~A~allfll~,o

and

It is well known that

1-e 8 1 (LPo’LP,)e,P~=LPo’ po +pi =pB

and, thus, by using real interpolation, we find that

(2.5)

(2.6)

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5038 Maligranda, Persson, and Wyller: Interpolation and PDEs

where

1 1-e I9 1 i-e 8 -=-+- 4s 40 41’ Pe=Po+i7

osesl.

Note that pe (qe) are real numbers between p. and pi (q. and ql). Remark: In many applications one of the estimates (2.5) and (2.6) can be taken as an energy

estimate (e.g., po=qo=2 and (~=l; cf. Theorem 1).

E. On nonlinear PDEs and the interpolation theory

We pronounce that several important physical phenomena can be modeled by means of non- linear PDEs. Such equations can be formulated in terms of nonlinear differential operators. Fortunately, nowadays we know fairly well that several operators of this type can be interpolated in a similar way as above. In practice the information in the extremal cases is usually known as norm estimates (of the “energy” type) and generalized variants of Holder, (semi-)Lipschitz, or sublinear conditions, respectively. Nowadays we have fairly good knowledge about interpolation results in such cases (see the review article in Ref. 9 and the references given there). This fact is of outstanding importance for several applications and is one of the main objectives for writing the present article (see, e.g., Sets. 5 and 6).

Ill. AN APPLICATION OF LINEAR INTERPOLATION THEORY TO A WAVE EQUATION WITH VARIABLE COEFFICIENTS

Consider the following Cauchy problem for the one-dimensional wave equation

Dot)= (4xb,kr)),, x E R t>O, L&~=O, XER,

utlrzo=f(x), XER. I (3.1)

Let u denote the solution of Eq. (3.1), and let T, be the evolution operator defined by TJ(x)=u(x,t).

Case 1: a(x) is constant [a(x) =c]. Then, by using Young’s inequality for convolution we find that

(3.2)

Case 2: Assume that a(x) is essentially bounded, i.e., that

O<ce2(x)~Cc~. (3.3)

The main goal of this section is to present a simple (interpolation) proof of the fact that Eq. (3.2) also holds in this case.

Theorem 1: If a(x) satisjes Eq. (3.3) and 2 c p GW, then the inequality (3.2) holds true. Proof For p = 2, we have the “energy” estimate

Ilu(.,t)llz=IITrfll~~fllfllz, Vf EL2.

If f E L”, then the Holder inequality yields

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Maligranda, Persson, and Wyller: Interpolation and PDEs 5039

Hence the operators Tt :L2 + L2 and T, :L” -+ L” are bounded. Then, by applying Riesz- Thorin’s theorem with p. = q. = 2, p 1 = q 1 = 00, and @= (p - 2)/p, we find that T, is a bounded operator from Lp into Lp, 2<p<a, such that

Remark: A characteristic feature for the n-dimensional case (n > 1) is that the estimate (3.2) is true only for p in the neighborhood of p = 2 even if a(x) = c (Case 1).

IV. A USEFUL INTERPOLATION RESULT FOR LOCALLY H6LDER OPERATORS

Many nonlinear differential operators of practical importance may be regarded as some type of locally Holder operators (see, e.g., our examples given in Sets. V and VI). In this section we will present and discuss an interpolation theorem of Maligranda” which deals with very general operators of this kind. First we need the following definition:

DeJnition I: Let O<KI and O<PGCQ. We say that the pair (A,,At) has a (0,~) approximate identity tf there is a family of continuous mappings &:A,, --+ At for O<tG, such that

2°11S,a-alle,p+t-ellS,a-~Il~o + 0 US tl0 for UEA~,~

and it is uniform on compact subsets of A,, . Example I (Bona and Scott”): The pair (L2(R),Hm(R)), w h ere m is a positive integel; has a

(f?,2) approximate identity. Theorem 1: (Maligrandat’, cf also Ref 12) Let AtCAo, BtCB,, and O<,~u<l, O-=+ ao,

cxt<m. Suppose that T is a mapping such that (i) T A,, + B. and, for a, bEA,,

llT~-Tbll~,~~(ll~~l,,,,llbll~,,,Ila-bli,,r)il~-bllA”~

and (ii) T: At --+ Bt and, for aeAt

where cp: R: -+ R is continuous and nondecreasing in each variable and $I R+ + R+ is con- tinuous and nondecreasing. If e>p or B=p and ps r, then T maps A ,sp into B,, and, for a EAe,p 7

where

a=(l-77)cfo+17q, p=ffq, h(t)=cp(t,2t,2t)‘-‘tr,k(2t)“.

In addition, assume that the pair (A,,At) has a (0,~) approximate identity {S,} for some @-p or 8=p and PSI; and that

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5040 Maligranda, Persson, and Wyller: Interpolation and PDEs

(iii) T is continuous as a map of At to B,. Then T is a continuous map from A,, to B,, . Remark I: The above theorem was previously obtained in the following special cases:

(a) ;=$ cp=+=const, and ao=al=a)=l [Lions 13-15 (Theorem 3.1) and Peetrei6 (Theorem . 7

(b) ~‘0 and ~(u,u,w)=C(U,~) [Tartar17 (Theorem 2)], (c) cp(u,u,w)=C(u+u) and %=a,=1 [Bona and Scott” (Theorem l)].

In the next two sections we will demonstrate how Theorem 1 can be used to obtain regularity information about the solutions of the p-Laplace equation and the Korteweg-de Vries equation.

V. A REGULARITY RESULT FOR THE pLAPLAClAN EQUATION

For 1 <p<m, let

uc~p(R:):g EU(R:),i= 1,2,...,n , I I

with the norm

llullwl*P= ( Ilull ;+, llfq) lip. Moreover, let W. ‘*P denote the closure of Bin W’@ and W- ‘G the dual space of Wh,p. Then the operator T: WA*” -+ W-‘@‘, as a dual mapping on WATp, defined by the formula

T-i & ( lglp-2 2) +IuI~-~u (5.1)

is strictly monotone,, h continuous and coercive, and therefore, by the Browder-Minty theorem, for every f E W- l,p there is a unique u E W. l*p such that

Tu=f. (5.2)

Our regularity result reads: Theorem 1: Let 2GpCm. IffE(W- ‘8p’ L2 fl W-“p’)e, then the solution u of the equation Tu =f

belongs to (W~p,L2(p-*‘)e,4 and jlz&,+~j@,-‘. ’ Proof The first step consists of proving that the inverse mapping T- ’ : W- lvp’ -+ WA3p, is a

Holder mapping of order a= ll(p - 1). Since p 32, we have

Hence, by putting a = u, b = u and a = dUlaXi, b = dVldXi, and integrating, one obtains

(Tu-Tu,u-u)= J

(Tu-Tu)(u-u)dX~22-pllU-u[IP, VU,UE wp.

Therefore, by the Holder inequality

22-J’~~~-u~~P~(T~-Tu,~-u)+-u~~~~Tu-T~~~.+.

and, thus

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Maligranda, Persson, and Wyller: Interpolation and PDEs 5041

or, equivalently

Second, we have

(5.3)

llT-‘fllL2(p-,)~llf112/2(p-1’, Vf e L2n W-‘J”. (5.4)

We prove this fact in the following way: For every ~-0, let u, be a solution of the equation

U,+&IU,IP-2u,=U.

Then (du,/dx,)[l+(p-1)~~~,~~-~]=duldx~, and uE, u,Iu,IP-~E Wh*p. Since

j --& (lgi” E) IUPU, dx= 1 igp-2 g & hIp-2%w

=(p-l)/ lglp-2 g: Iu,Ip-2 dx20 I

and

it follows that

U,Q,[l+&IU Ip+]=u E

lluslg2~,= 1 Iup-‘) dx= 1 Iu,Ip-2u,Iu,Ip-%, dx

s I IuIp-2uIu IP-2u dx E E

+- IuI~-~uIu,Ip-~u, dx+i j- -& ( lg1,-2 -$) Iu,I~-~u, dx

= I

Tu+,~P-~u, dx.

Then, by using Schwarz inequality, we get

or

which means that the sequence {us} is bounded in L2@‘). Moreover, since

11~-~,11~~=~111~,1~-2~~11~~=~11~~11~~~-~~ -+ 0 as E -+ 0, we have that u E L2(p-‘) and

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5042 Maligranda, Persson, and Wyller: Interpolation and PDEs

which is Eq. (5.3). Finally, by using Theorem 1 with a0 = a1 = ll(p - 1 ), we obtain

and the proof is complete. cl We close this section by stating the following complement to Theorem 1: Theorem 2: Let l<pS2. Iffe(W-t~P’,L2rlW-t~P’)~q,

belongs to ( Wip,L2n W-tSp’) then the solution of the equation Tu=f

~7 where y@(p-1)/r and r=max(I,q(p -I)/[?$2 -p) +p-I]). Proof (sketch): It can be proven that

and then, by using Theorem 1 with ao= 1 and my1 = ll(p - 1) and inequality (5.4), we get the result with the estimate

VI. A REGULARITY RESULT FOR THE KORTEWEG-DE VRIES EQUATION

Consider the initial-value problem for the Korteweg-de Vries equation

i

Ur+UU,+U,,=O, XER, t>O, u(x,O)=f(x), XER (6.1)

Remark: The Korteweg-de Vries equation can be derived from a system of hyperbolic equa- tions of the formI

g +A(U) g +Dc3) ; ,& ,U U=O, i i

(6.2)

where U is a n X 1 column vector, A(U) is an n X n matrix with distinct real eigenvalues, and Dc3) is a homogeneous matrix differential operator, by means of the reductive perturbation expansion, i.e., one looks for the influence of weak dispersion and nonlinearity on a single mode of the linearized version of Eq. (6.2) in the nondispersive limit. As examples of continuum-mechanical systems of the form (6.2) we mention a one-dimensional (1D) model for shallow water waves” and hydromagnetic waves.2o Thus the Korteweg-de Vries equation can be considered as an evo- lution equation for weakly nonlinear and dispersive waves, with applications to different continuum-mechanical systems.

Bona and Scott” proved that the problem (6.1) is well posed in all the Sobolev spaces HS for s 3 2. In the proof they used a simple extension of the nonlinear interpolation theorem of TartarI (cf. Remark I), which is a special case of our Theorem 1.

Theorem 1: If ss2 and fEM, then u~C(O,tc;Hs) and

where h,,,o:R+ --f R, is a continuous nondecreasing function. Moreover; the mapping u. + u from H” into C(O,t,;Hs) is continuous.

Proof (cf also Ref 11): By using some earlier results, Bona-Scott” proved the existence of continuous functions $,,, :R+ -+ R, , ma 1 such that

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Maligranda, Person, and Wyller: Interpolation and PDEs 5043

Moreover, for m> 1 the mapping T:Hm --+ C( O,t, ;H”) defined by Tf = u is continuous. We prove that if m>2 and m- I<s<m, then T maps H”-’ into C(O,t, ;L2) and there exists a continuous, nondecreasing function (P~,‘~:R+ 4 R+ such that

IITf-TgllC(0,lo;L2)~~pm,fg(llnlHm-l,llgll*m-l)llf-gllL2. (6.5)

For m>2, the mapping T:Hm-’ -t C(O,ro;Hm-‘) is continuous and therefore also continuous from H”‘-’ to C(O,t, ;I,*). Since T is continuous, it suffices to prove Eq. (6.5) forJ g belonging to some dense subset of Hm- ’ . Therefore, suppose f, g E H” and let w = u - u = Tf - Tg . Then w satisfies the initial-value problem

We get

i Wt+~(U+U)W]X+WXXX=O, WC&O) =f(x) -g(x).

; (I, w*(x,r)dx)=-2~R ww,,, dx-fR w[(u+u)wl, dx

= 2 I

w,w,, dx+ R s

R w,(u+u)w dx

= ~Rb+$L dx+; I,( u+u)(w*), dx= -; I

(u+u),w’ dx. R

Then, by using the Sobolev inequality

and, thus, according to Gronwall’s inequality

IML . )Il~2+(0~ * >11~2(P2,f(llfllH2MlH2)r

where c&a,b)=exp(&,(a)ar+ +,(b)br). Now, by taking the supremum over r E [0, to], we have

Hence, by using the interpolation Theorem 1 with

Ao=L2, B,=C(0,ro;L2),

A,=H”, B1=C(O,ro;H”),

m-l p=--jy 7 &Jr2

m’ r=p=2, (Y(j=ar=l

and the facts

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5044

and

Maligranda, Persson, and Wyller: Interpolation and PDEs

we obtain Eq. (6.3). Moreover, since the pair (L2,Hm) admits a (13,2) approximate identity (see Example l), it follows from Theorem 1 that T is a continuous mapping from HS to C(O,r,;HS) and the proof is complete. Cl

VI. FINAL DISCUSSION AND CONCLUDING REMARKS

A. Some general open questions

Roughly speaking, we can say that the existing nonlinear interpolation theory is of the type that an interpolation method for linear operators, e.g., the real method ()e,4 studied in this article, is also an interpolation method for a class of nonlinear operators. The following questions seem to be of great interest both from the point of view of pure mathematics and of applications:

Consider the subclass NO of nonlinear operators.

(a) Describe the biggest subclass such that the real method () B,q is an interpolation method for L U NO, where L is the class of linear operators (this class surely contains sublinear and Lipschitz operators, see Ref. 9).

(b) The same question applies to the complex and other methods. (c) Is it possible to make a new construction of interpolation spaces such that these spaces are

interpolation methods for a fixed class of nonlinear operators which cannot be covered by the existing interpolation theory of nonlinear operators (which is clearly modeled upon the classical constructions)?

Remark: If a suitable theory suggested in (c) exists, then we can cover applications for a new class of nonlinear PDEs and thus obtain further possibilities for concrete applications.

B. Sobolev or Sobolev-Besov regularity of solutions of variational inequalities via interpolation

Several articles are devoted to the Sobolev and Sobolev-Besov regularity of solutions to variational inequalities or the Cauchy problem of PDEs using interpolation of linear or nonlinear operators.

A variational inequality can be formulated as follows: Find u E K such that

(Tu-f,u-u)aO, Vu EK,

where K is a closed convex subset of a Banach space X, f is a fixed element of the dual space X*, and T:X -+ X* is a monotone or pseudomonotone operator. In some cases it is possible to approximate the solutions of variational inequalities with solutions of operator equations and get a priori estimates. Then we can use the interpolation theorem of locally Holder operators to prove the regularity of the solutions of variational inequalities. For more details see Lions,‘3-‘5 Tartar,17 Boccardo,*’ Brezis,** Chipot,23 Lions-Magenes,24 and Tarta~*~

Maligranda,t*

C. Further regularity results for solutions of PDEs via interpolation

Interpolation techniques were used by many authors in the proof of the regularity of the initial-value problems for PDEs:

(i) For the wuve equarion

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.

Maligranda, Persson, and Wyller: Interpolation and PDEs 5045

u,,-Au=O, XER",

~(x,o)=g(x),

u,W)=f(x)

it was investigated by Peral and Perzan.27 For the Klein-Gordon equation

i

u,,-A,++u=O, XER~

u(x,O)=O,

4-d>=f(x)

it was done by Marshall-Strauss-Wainger,28 and for the nonlinear wave equation

u,,-A,u+p(u)=O

by Lions,13 Pecher,29 Saut,30 and Simon.3’ (ii) For the generalized hear conduction equation

i

u,= (a(x,~h,), , u(x,O)=f(x)

it was done by Da Prato-Grisvard.32 (iii) For the generalized Burgers equation

i

u,+ cp(u),=O, x E R, r>O,

u(x,O)=f(x), xeR

it was done by De Vore-Lucier.33 [Note that when C&U) = u2, we have the Burgers equation.] (iv) For the nonlinear Schriidinger equation (NSE)

u,+cp(lul*)u-iAu=O

see Abdelouhab,34 Ponce-Urbina,35 and Reed-Simon.36 (v) For the Boltzmarm equation, see Gustafsson.37 For the Navier-Stokes equations and

related problems, see Lions,14 Borchers-Miyakawa,s8 and Kozono-Sohr.39

D. A final conclusion

We claim that linear and nonlinear interpolation theory have now been developed to a stage where they can be used as very powerful tools for the purpose of predicting the regularity of the solutions of PDEs from their boundaries and/or initial data. We are convinced that many new beautiful applications are waiting behind the door which we tried to open. This is the real scope of this article.

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Berlin, 1978). 3S. G. Krein, Yu. I. Petunin, and E. M. Semenov, Interpolation of Linear Operators (American Mathematical Society,

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