Time-vanishing properties of solutions of some degenerate ...

Time-vanishing properties of solutions of some

degenerate parabolic equations with strong

absorption

Y. Belaud

Laboratoire de Mathematiques et Physique Theorique,CNRS ESA 6083,

Faculte des Sciences et Techniques,Universite Francois Rabelais, 37200 Tours

e-mail : [email protected]

Abstract

Let Ω be an open bounded subset of IRN and b a measurable nonnegativefunction in Ω. We study the time compact support property for ∂tu −∆pu + b(x)uq = 0 with p > 2 and ∂tu −∆(um) + b(x)uq = 0 with m > 1where 0 ≤ q < 1. Some criteria involving the asymptotics as h → 0 of thefirst eigenvalue of −∆p + b(x)

hp are given which imply such a phenomenon.Applying these criteria we deduce that the time vanishing property occurswhen 1/b belongs to Ls(Ω) for some s depending on q, and p or m.

1991 Mathematics Subject Classification. 35K55, 35P15.Key words. evolution equations, p-Laplacian, porous-medium, strong absorption, regularizing effects, semi-

classical limits

1 Introduction

Let Ω be a smooth bounded domain of IRN (N ≥ 1), b a non-negative Borel measurablefunction and q ∈ [0, 1). The aim of this paper is to investigate the time compactproperty (shortly the TCS-property) of weak solutions of the degenerate parabolicequations appearing in the theory of nonlinear filtration such as :

ut − div(|∇u|p−2∇u

)+ b(x)|u|q−1u = 0 in Ω× (0,∞),

∂νu = 0 on ∂Ω,u(x, 0) = u0(x) in Ω,

(1.1)

1

with p > 2, andut −∆ (um) + b(x)|u|q−1u = 0 in Ω× (0,∞),

∂νu = 0 on ∂Ω,u(x, 0) = u0(x) in Ω,

(1.2)

with m > 1.

Definition 1.1 A solution u satisfies the TCS-property if there exists a time T suchthat ∀t ≥ T, ∀x ∈ Ω, u(x, t) = 0.

The method we shall develop below originated in a paper by Kondratiev and Veron [7]where only the semilinear case of equations (1.1)-(1.2) (p = 2,m = 1) was considered.Later on, Belaud, Helffer and Veron [1] put a new light on the proof by linking thisproperty to the growth of the first eigenvalue of some associated Schrdinger operator,using semi-classical methods. More precisely, in [7], it is shown that

µ(M) = inf∫

Ω

(|∇v|2 +M b(x) |v|2

)dx : v ∈W 1,2(Ω),

∫Ω|v|2 dx = 1

, (1.3)

where M is a non-negative real number, and that if+∞∑n=0

lnµ(2n)µ(2n)

<∞, then the TCS-

property holds. In [1], this criterion is enlightened by the use of the semi-classicalanalysis of Schrdinger operators, and integral expressions are given which imply theTCS-property. Cases where this property does not hold are also studied.

Consider first the merely semilinear equation :ut −∆u+ b(x)uq = 0 in Ω× (0,∞),

∂νu = 0 on ∂Ω,u(x, 0) = 1 in Ω.

(1.4)

1) Suppose there exists a real number γ such as b(x) ≥ γ > 0 a.e. in Ω. Fromthe maximum principle, u(x, t) ≤ (1− γ(1− q)t)

11−q in Ω × (0,∞). The nonlinear

absorption is stronger than the diffusion and the TCS-property holds. Notice thatµ(2n) ≥ γ2n, so the above series converges.

2) If we assume that there exists a connected open set ω such as b(x) = 0 a.e. in ω (noabsorption in ω), then usually, u does not have the compact support property. Indeed,if we denote by λ(ω) the first eigenvalue of −∆ in W 1,2

0 (ω) and ζ the first eigenfunctionwith ||ζ||L∞(ω) = 1 and ζ ≥ 0, then from the maximum principle, u(x, t) ≥ ζ(x) e−λ(ω)t

for all x in ω and for all t ≥ 0.The criterion in [1] does not apply, but the above series diverges.Up to some minor changes, the previous examples are also valid if u satisfies (1.1) and

2

(1.2).

The compact support property is related to x : b(x) = 0 and the behaviour of thefunction b in a neighbourhood of this set.

In this paper, our aim is to find a sufficient condition on b implying the TCS-propertyfor a much more general class of quasilinear equations. More precisely, we study theweak solutions of an equation of the following type

ut − div(A(x, t, u,∇u)) + f(x, t, u) = 0 in Ω× (0,∞),∂νu = 0 on ∂Ω,

u(x, 0) = u0(x) in Ω.(1.5)

Besides the standard Caratheodory assumptions on A and f , we assume the followingminimal conditions in Section 2 :

|A(x, t, ρ, s)| ≤ C|s|p−1,A(x, t, ρ, s).s ≥ C ′|s|p, (1.6)

for some positive constants C and C ′. As for the function f , we shall suppose thatthere exists a nonnegative, bounded and measurable function b and a real number q in[0, 1) such that

f(x, t, ρ)ρ ≥ b(x)|ρ|q+1. (1.7)

Furthermore, it is always possible to assume that b satisfies

ess inf bΩ

= 0 and∫Ωb(x) dx > 0, (1.8)

since the case ess inf bΩ

> 0 has already been treated in the second example.

Before starting, we recall Veron’s result [10], [11] on the regularizing effects associatedto this type of equation.

Theorem A Suppose A satisfies (1.6) for some C, C ′ > 0 and p > 1, and let u be aweak solution of

ut − div (A(x, t, u,∇u)) +B(x, t, u) = 0 in Ω× (0,∞),∂νu = 0 on ∂Ω,

u(x, 0) = u0(x) ∈ Lr(Ω),(1.9)

where B is a Caratheodory function which satisfies B(x, t, ρ)ρ ≥ 0 a.e. in Ω× (0,∞).If u0 ∈ Lr(Ω) with r ≥ 1, r > N(2/p− 1) then

||u(., t)||L∞(Ω) ≤ C

(1 +

1t

)δ(r)||u(., 0)||σ(r)

Lr(Ω), (1.10)

3

with C = C(Ω, p) , δ(r) =N

rp+N(p− 2)and σ(r) =

rp

rp+N(p− 2).

Notice that if p > 2 , σ(r) < 1 and σ(r) → 1 when r → +∞.

Since any solution of (1.9) becomes bounded for positive f whenever p ≥ 2, we shallalways consider bounded initial data.

We introduce the following quantity :

µ(α, p)=inf∫

Ω

(|∇ψ|p + αq−(p−1)b(x)|ψ|p

)dx : ψ ∈W 1,p(Ω),

∫Ω|ψ|pdx = 1

. (1.11)

We first prove

Theorem 2.1 Let 0 ≤ q < 1, p > 2 and assume that (1.6), (1.7) and (1.8) hold.Assume that there exist two sequences of positive real numbers (αn)n∈IN and (rn) suchthat (αn)n∈IN is decreasing and

∞∑n=0

rp−1n

αp−2n+1µ(αn, p)σ(rn)

< +∞. (1.12)

Then any solution of (1.5) with initial bounded data satisfies the TCS-property.

The next criterion is close to the expression obtained in [1].

Corollary 2.1 Under the same assumptions on q, p, A and f , if there exists a de-creasing sequence of positive real numbers (αn)n∈IN such that

∞∑n=0

(lnµ(αn, p))p−1

αp−2n+1µ(αn, p)

< +∞, (1.13)

then any solution of (1.5) satisfies the TCS-property.

It is clear that conditions of the corollary 2.1 are not easy to apply. It is more convenientto replace them by an integral condition which does not depend on (αn). So we derivedthe following theorem.

Theorem 2.2 (Integral criterion) Let 0 ≤ q < 1, p > 2 and assume that (1.6), (1.7)and (1.8) hold. If ∫ 1

0

(lnµ(t, p))p−1

tp−1µ(t, p)dt < +∞, (1.14)

then all solutions of (1.5) satisfy the TCS-property.

4

In [1], they also give an integral criterion which is formally the previous one by takingp = 2.

As in [1], the main difficulty to transform the above criteria into practical ones is to giveprecise estimates on µ(α, p) when α tends to 0. In the semilinear case, these estimatesare provided by means of the semi-classical expansion of eigenvalues of Schrdingeroperators. Clearly, this method does not work in the case p > 2. If we denote by λ1(h)(h > 0) the first eigenvalue of the operator

u 7→ −div(|∇u|p−2∇u

)+ h−pb(x)up−1, (1.15)

in W 1,p(Ω), with the Neumann boundary condition, we prove that under assumption(1.8),

λ1(h)meas(b(x) ≤ hpλ1(h))1γ ≥ C, (1.16)

for some C > 0 independent of h and γ = γ(N, p) > 0. From this inequality, we derivea sharp enough estimate on the µ(α, p) and we establish

Theorem 2.3 Let 0 ≤ q < 1, p > 2 and assume that (1.6), (1.7) and (1.8) hold. If(1b

)s∈ L1(Ω) with

s >

p− 21− q

N

pfor p ≤ N,

s >p− 21− q

for p > N,

(1.17)

then any solution of (1.5) has the TCS-property.

Notice that in the case p = 2, the previous result does not apply, and actually thecriterion for the TCS-property is drastically different since it is

ln1b∈ Ls(Ω) (1.18)

with s >N

2(see [1]).

As an application of the previous result, we have

Corollary 2.2 Suppose that 0 ∈ Ω and b(x) = |x|β.

1) If p ≤ N and β <p(1− q)p− 2

then u holds the TCS-property.

5

2) One can draw the same conclusion if p > N and β <N(1− q)p− 2

.

The Section 3 is dedicated to another type of operators. More precisely, u satisfies(1.5), A and f satisfy the standart Caratheodory assumptions and

|A(x, t, ρ, s)| ≤ C|s|,A(x, t, ρ, s).s ≥ C ′mρm−1|s|2, (1.19)

for m > 1 and for some positive constants C and C ′. In a same way, f satisfies (1.7), b(1.8), and the initial data is also bounded. The same method yields to similar theoremsin spite of differences between the two operators. In this spirit, we also define

µ′(α,m)=inf∫

Ω

(|∇ψ|2 + αq−mb(x)|ψ|2

)dx : ψ ∈W 1,2(Ω),

∫Ω|ψ|2dx = 1

. (1.20)

Porous media equations are also subject to regularizing effects : ([10] , [11])

Theorem B Suppose A satisfies (1.19) for some C, C ′ > 0 and m > 1, and let u bea weak solution of

ut − div (A(x, t, u,∇u)) +B(x, t, u) = 0 in Ω× (0,∞),∂νu = 0 on ∂Ω,

u(x, 0) = u0(x) ∈ Lr(Ω),(1.21)

where B is a Caratheodory function which satisfies B(x, t, ρ)ρ ≥ 0 a.e. in Ω× (0,∞).If u0 ∈ Lr(Ω) with r ≥ 1, r > N(1−m)/2 then

||u(., t)||L∞(Ω) ≤ C

(1 +

1t

)δ(r)||u(., 0)||σ(r)

Lr(Ω), (1.22)

with C = C(Ω,m) , δ(r) =N

2r +N(m− 1)and σ(r) =

2r2r +N(m− 1)

.

When m > 1 , σ(r) < 1 and σ(r) → 1 when r → +∞.

Formally, changing p− 1 into m provides the following theorems :

Theorem 3.1 Let 0 ≤ q < 1, m > 1 and assumptions (1.19)-(1.7)-(1.8) hold. Assumethat there exist two sequences of positive real numbers (αn)n∈IN and (rn) such that(αn)n∈IN is decreasing and

∞∑n=0

rmnαm−1n+1 µ

′(αn,m)σ(rn)< +∞. (1.23)

6


and

Corollary 3.1 Under the same assumptions on q, m, A and f , if there exists a de-creasing sequence of positive real numbers (αn)n∈IN such that

∞∑n=0

(lnµ′(αn,m))m

αm−1n+1 µ

′(αn,m)< +∞, (1.24)


The integral version of corollary (3.1) is

Theorem 3.2 (Integral criterion) Let 0 ≤ q < 1, m > 1 and assumptions (1.19)-(1.7)-(1.8) hold. If ∫ 1

0

(lnµ′(t,m))m

tmµ′(t,m)dt < +∞, (1.25)


Then we establish that

Theorem 3.3 Let 0 ≤ q < 1, m > 1 and assumptions (1.19)-(1.7)-(1.8) hold. If(1b

)s∈ L1(Ω) with

s >

m− 11− q

N

2for N ≥ 2,

s >m− 11− q

for N = 1,

(1.26)



1) If N ≥ 2 and β <2(1− q)m− 1


2) One can draw the same conclusion if N = 1 and β <(1− q)m− 1

.

The last section is dedicated to semi-classical analysis. It can be read independentlyfrom the others parts. It is some sort of ”technical” support for applications. It givesa lower bound for the first eigenvalue for

u 7→ −div(|∇u|p−2∇u

)+ h−pb(x)up−1,

7

under weak assumptions.Contrary to the well-known formula of Cwickel-Lieb-Rosenblyum, the theory of pseudo-differential operators is not available for p-Laplacian when p 6= 2.We first tackle with the case p < N and Ω = IRN , in which case the Gagliardo-Nirenberg-Sobolev inequality holds

||∇ψ||pLp(IRN )

≥ C(p,N)||ψ||pL

pNN−p (IRN )

, (1.27)

for all ψ in W 1,p(IRN ).Assume that V is a measurable function defined in IRN and

W 1,p,V (IRN ) = ψ ∈W 1,p(IRN ) : V (x)ψp ∈ L1(IRN ), (1.28)

is not empty and putting

FV (ψ) =∫Ω|∇ψ|p + V (x)|ψ|p dx, (1.29)

for ψ ∈W 1,p(IRN ), and

λ1 = inf∫

IRN|∇ψ|p + V (x)|ψ|p dx : ψ ∈W 1,p(IRN ),

∫IRN

|ψ|p dx = 1, (1.30)

we obtain :

Theorem 4.1 Suppose N > p > 1. Then either λ1 = −∞ or(∫V (x)≤λ1

(λ1 − V (x))Np dx

) pN

≥ C(p,N), (1.31)

where C = C(p,N) > 0 is the positive constant of the Sobolev inequality.In addition, if there exists a minimizer in W 1,p,V (IRN ),(∫

V (x)<λ1


) pN

≥ C(p,N). (1.32)

For applications to the TCS-property, we assume that

V ∈ L∞(Ω), ess inf VΩ

= 0 and∫ΩV (x) dx > 0. (1.33)

We put λ1(h) = λ1(h−pV ) andγ = N

p for 1 N,

(1.34)

8

withγ

γ − 1= +∞ if γ = 1.

Theorem 4.2 Assume that Ω is a C1 bounded domain and (1.33) holds. Then for hsmall enough, (∫

V (x)<hpλ1(h)

(λ1(h)−

V (x)hp

)γdx

) 1γ

≥ C, (1.35)

where C = C(p,N, γ,Ω, V ) is a positive real constant.

The following corollary is used for theorems 2.3 and 3.3 :

Corollary 4.2 If (1.33) holds then for h small enough

λ1(h) (meas(V (x) < hpλ1(h)))1γ ≥ C, (1.36)

where C = C(p,N, γ,Ω, V ) is a positive constant.

When p ≥ N or Ω 6= IRN , a more restrictive version of theorem 4.1 is obtained. We set

λ1(h) = inf∫

Ω|∇ψ|p + h−pV (x)|ψ|p dx : ψ ∈W 1,p,V (Ω),

∫Ω|ψ|p dx = 1

, (1.37)

where Ω is a connected open set (bounded or not, regular or not) of IRN and h apositive real number.

Definition 4.1 We say that V has a well in U if U is a C1 bounded, connected,non-empty open set of Ω and if there exists ψ0 ∈ W 1,p,V (Ω) with ||ψ0||Lp(Ω) = 1 suchthat ∫

ΩV (x)|ψ0|p dx < a = ess inf V

Ω\U, (1.38)

with meas(Ω\U) > 0.

With it, we establish the following estimates :

Theorem 4.3 If V has a well in U and for h small enough, either λ1(h) = −∞ or(∫V (x)≤hpλ1(h)

(λ1(h)− h−pV (x)

)γdx

) 1γ

≥ C(p,N,U, γ,Ω, θ), (1.39)

where C(p,N,U, γ,Ω, θ) is a positive constant.In addition, if there exists a minimizer in W 1,p,V (Ω),(∫

V (x)<hpλ1(h)

(λ1(h)− h−pV (x)

)γdx

) 1γ

≥ C(p,N,U, γ,Ω, θ). (1.40)

9

and a simplified version when σ = ess inf VΩ

> −∞ :

Corollary 4.3 (λ1(h)−

σ

hp

)meas(x : V (x) ≤ hpλ1(h))

1γ ≥ C. (1.41)

In addition, if there exists a minimizer in W 1,p,V (Ω),(λ1(h)−

σ

hp

)meas(x : V (x) < hpλ1(h))

1γ ≥ C. (1.42)

Our paper is organized as follow :1-Introduction.2-The p-Laplacian diffusion equation.3-The Porous Media diffusion equation.4-Lower bound for the first eigenvalue.5-References

AcknowledgementsI want to thank L. Veron for his help and advice during the preparation of this article.

2 The p-Laplacian diffusion equation (p > 2)

In this section, Ω is a bounded domain of IRN (N ≥ 1) with a Lipschitz-continuousboundary, A : (x, t, r, s) 7→ A(x, t, r, s) and f : (x, t, r) 7→ f(x, t, r) are measurablefunctions from respectively Ω×IR+×IR×IRN and Ω×IR+×IR with values respectivelyin IRN and IR. We assume that A and f are continuous in the variables (r, s) ∈ IR×IRNand satisfy

|A(x, t, r, s)| ≤ C|s|p−1,A(x, t, r, s).s ≥ C ′|s|p, (2.1)

f(x, r)r ≥ b(x)|r|q+1, (2.2)

for all (x, t, r, s) ∈ Ω× IR+× IR× IRN where C and C ′ are some positive real constants.In the sequel, b is a bounded, nonnegative and measurable function and q ∈ [0, 1).

Definition 2.1 We say that u belonging to C ([0,∞);Lp(Ω)) ∩ Lploc([0,∞);W 1,p(Ω)

)and such that f(., ., u) ∈ Lploc (Ω× [0,∞)) is a weak solution (resp. subsolution) of theproblem


u(x, 0) = u0(x) in Ω,(2.3)

10

with u0 ∈ L1(Ω) if, for t > 0 there holds∫ t

0

∫Ω

(−uζt +A(x, t, u,∇u).∇ζ + f(x, t, u)ζ) dxdτ = (2.4)

∫Ωζ(x, 0)u0(x) dx−

∫Ωζ(x, t)u(x, t) dx,

for any ζ ∈ Lploc([0,∞);W 1,p(Ω)

)∩W 1,p

loc ([0,∞);Lp(Ω)).(resp. ∫ t

0

∫Ω

(−uζt +A(x, t, u,∇u).∇ζ + f(x, t, u)ζ) dxdτ ≤ (2.5)

∫Ωζ(x, 0)u0(x) dx−


for any ζ ∈ Lploc([0,∞);W 1,p(Ω)

)∩W 1,p

loc ([0,∞);Lp(Ω)), ζ ≥ 0).

It is fundamental to notice that assumptions (2.1)-(2.2) allow us to use the energy-method in the space W 1,p(Ω).

The sketch of the method

Given a decreasing sequence (αn), the aim is to construct an increasing sequence (tn)such that ||u(., tn)||L∞(Ω) ≤ αn. If (tn) is bounded and lim

n→+∞αn = 0 then u vanishes

identically on Ω for t ≥ sup tn. At the beginning, we suppose that ||u0||L∞(Ω) ≤ α0.The first step is to get a Lr decay at time t

2 . For this purpose, we use u|u|r−2 as test-function for r large enough, (1.6), (1.7), the definition of µ(α, p) and Hlder’s inequality.Finally, the regularizing effect transforms a Lr decay at time t

2 into a L∞ decay attime t (see [10], [11]). Consequently, as α1 < α0, we compute a first time t1 such that∀t ≥ t1, ||u(., t)||L∞(Ω) ≤ α1. This process is iterated and gives an upper bound for theseries

∑(tn+1 − tn) and therefore for (tn).

2.1 Iterative method

Let us denote by r a fixed real constant in [2,+∞) then

1(r − 1)

(p+ r − 2

p

)p≥ 1. (2.6)

Lemma 2.1 We suppose there exists a measurable function u in Ω×IR+ which satisfiesweakly


u(x, 0) = u0(x) in Ω.(2.7)

11

with ||u0||L∞(Ω) ≤ α and that (1.6) and (1.7) hold. Then

||u(., t)||Lr(Ω) ≤

1||u(., 0)||2−pLr(Ω) + C1µ(α, p) t

1p−2

, (2.8)

where C1 = C1(Ω, r, p) is a positive real constant, and there exist two positive realnumbers C = C(Ω, p) and C2 = C2(r, p) such that

||u(., t)||L∞(Ω) ≤ min

C (1 +2t

)δ(r) 1||u(., 0)||2−pL∞(Ω) + C2µ(α, p)t

σ(r)p−2

, 1

. (2.9)

(δ(r) and σ(r) are defined in Theorem A)

Proof: Taking u|u|r−2 as a test-function (justified by Theorem A) gives

1r

d

dt

∫Ω|u|r dx+ (r − 1)

∫ΩA(x, t, u,∇u)∇u|u|r−2 +

∫Ωf(x, t, u)u|u|r−2 dx = 0,

since u∇(|u|) = |u|∇(u). By (1.6) and (1.7), we have

1r

d

dt

∫Ω|u|r dx+ (r − 1)

∫Ω|∇u|p|u|r−2 dx+

∫Ωb(x)|u|q+r−1 dx ≤ 0.

Using ||u(., t)||L∞(Ω) ≤ α, and replacing |u|q+r−1 by αq+r−1 leads to

1r

d

dt

∫Ω|u|r dx+ (r − 1)

∫Ω|∇u|p|u|r−2 dx+ αq−(p−1)

∫Ωb(x)|u|p+r−2 dx ≤ 0.

But |∇u|p|u|r−2 =(

p

p+ r − 2

)p ∣∣∣∣∇(|u| p+r−2p

)∣∣∣∣p so we have

1r

d

dt

∫Ω|u|r dx+ (r − 1)

(p

p+ r − 2

)p ∫Ω

∣∣∣∣∇(|u| p+r−2p

)∣∣∣∣p dx +

αq−(p−1)∫Ωb(x)

(|u|

p+r−2p

)pdx ≤ 0.

Thus,1r

d

dt

∫Ω|u|r dx+ (r − 1)

(p

p+ r − 2

)p[∫

Ω

∣∣∣∣∇(|u| p+r−2p

)∣∣∣∣p dx+(p+ r − 2

p

)p αq−(p−1)

(r − 1)

∫Ωb(x)

(|u|

p+r−2p

)pdx

]≤ 0.

12

By (2.6) and |u|p+r−2

p as test-function for µ(α, p) gives

1r

d

dt

∫Ω|u|r dx+ (r − 1)

(p

p+ r − 2

)pµ(α, p)

∫Ω|u|p+r−2 dx ≤ 0.

Hlder’s inequality leads to∫Ω|u|r dx ≤

(∫Ω|u|p+r−2 dx

) rp+r−2

meas(Ω)p−2

p+r−2 .

As a consequence,

d

dt

∫Ω|u|r dx+ (r − 1)

(p

p+ r − 2

)p rµ(α, p)

meas(Ω)p−2

r

(∫Ω|u|r dx

) p+r−2r

≤ 0.

Set y =∫Ω |u|r dx. We obtain a Bernoulli differential inequality. Thus,

y(t) ≤

1

y(0)2−p

r +(p−2)(r−1)

(p

p+r−2

)p

meas(Ω)p−2

r

µ(α, p) t

r

p−2

,

which yields to (2.8) with

C1 =(p− 2)(r − 1)

(p

p+r−2

)pmeas(Ω)

p−2r

.

From theorem A, there exists a constant C ′ = C ′(Ω, p) such as

||u(., t)||L∞(Ω) ≤ C ′(

1 +1τ

)δ(r)||u(., t− τ)||σ(r)

Lr(Ω),

with 0 < τ < t. One should keep in mind that δ(r) = Nrp+N(p−2) and σ(r) =

rprp+N(p−2) < 1.

Applying (2.8) at time t− τ leads to

||u(., t− τ)||Lr(Ω) ≤

1

||u(., 0)||2−pL∞(Ω)meas(Ω)2−p

r + C1µ(α, p) (t− τ)

1p−2

.

Remark that 2− p < 0 and so ||u(., 0)||2−pLr(Ω) ≥ ||u(., 0)||2−pL∞(Ω)meas(Ω)2−p

r .Consequently,

||u(., t)||L∞(Ω) ≤ C ′(

1 +1τ

)δ(r)13

1

||u(., 0)||2−pL∞(Ω)meas(Ω)2−p

r + C1µ(α, p) (t− τ)

σ(r)p−2

.

r is large enough so C ′ meas(Ω)σ(r)

r ≤ C ′ meas(Ω)12 = C. As a consequence,

||u(., t)||L∞(Ω) ≤ C

(1 +

1τ

)δ(r) 1||u(., 0)||2−pL∞(Ω) + 2C2µ(α, p) (t− τ)

σp−2

,

whereC2 =

12meas(Ω)

p−2r C1 =

12(p− 2)(r − 1)

(p

p+ r − 2

)p.

It follows at once that taking τ = t2 completes the proof of (2.9). \

Remark 2.1 If u is only a subsolution of (2.7), then we replace the test-functionu|u|r−2 by (u+)r−1, and (2.8)-(2.9) hold but for u+ instead of u.

For the next theorem, it is very important to notice that

C2 = C2(r) ∼12

(p− 2) pp r1−p (2.10)

when r tends to infinity.

Theorem 2.1 Let 0 ≤ q < 1, p > 2 and assume that (1.6), (1.7), and (1.8) hold.Assume that there exist two sequences of positive real numbers (αn)n∈IN and (rn) suchthat (αn)n∈IN is decreasing and

∞∑n=0

rp−1n


< +∞. (2.11)


Proof: First, notice that limn→+∞

αn = 0 since the series converges.

In the previous Lemma, r is a free parameter. The proof of the theorem is based on aniterative method, so at each step, we can choose a different r. As a consequence, wehave labeled these numbers by a sequence (rn)n∈IN .The value of α0 can be taken equal to 1. So we can apply Lemma 2.1 with α = α0 = 1and r = r0. Thus,

||u(., t)||L∞(Ω) ≤ C

(1 +

2t

)δ(r0) ( 11 + C2(r0)µ(1, p) t

)σ(r0)

p−2

.

14

Let t1 be such that C(

1 +2t1

)δ(r0) ( 11 + C2(r0)µ(1, p) t1

)σ(r0)

p−2

= α1 .

Consequently, ||u(., t)||L∞(Ω) ≤ α1 if t > t1. Moreover, applying Lemma 2.1 withα = α1 and r = r1,

||u(., t)||L∞(Ω) ≤ C

(1 +

2t− t1

)δ(r1)(

1α2−p

1 + C2(r1)µ(α1, p) (t− t1)

)σ(r1)

p−2

.

Thus, there exists t2 such that

C

(1 +

2t2 − t1

)δ(r1)(

1α2−p

1 + C2(r1)µ(α1, p) (t2 − t1)

)σ(r1)

p−2

= α2.

We iterate the process and we construct an increasing sequence (tn) such as

αn+1 = C

(1 +

2tn+1 − tn

)δ(rn)(

1α2−pn + C2(rn)µ(αn, p) (tn+1 − tn)

)σ(rn)p−2

.

Then (tn+1 − tn

2 + tn+1 − tn

)δ(rn) (α2−pn + C2(rn)µ(αn, p) (tn+1 − tn)

)σ(rn)p−2 =

C

αn+1,

and finally,

(tn+1 − tn)δ(rn)

(1 + C2(rn)αp−2

n µ(αn, p) (tn+1 − tn))σ(rn)

p−2 =

Cασ(rn)n

αn+1(2 + tn+1 − tn)

δ(rn) .

We shall prove successively that the sequence (tn+1 − tn) tends to zero and that theseries

∑(tn+1 − tn) converges. First, it is easy to check that σ(r) + (p− 2)δ(r) = 1 for

all r ≥ 2. Consequently,

(tn+1 − tn)1

p−2 C2(rn)σ(rn)p−2 µ(αn, p)

σ(rn)p−2 ≤ C

1αn+1

(2 + tn+1 − tn)δ(rn) ,

and

(tn+1 − tn)C2(rn)σ(rn)µ(αn, p)σ(rn) ≤ C1

αp−2n+1

(2 + tn+1 − tn)(p−2)δ(rn) .

Thus,

(tn+1 − tn) ≤ CC2(rn)(p−2)δ(rn)

C2(rn)1


(2 + tn+1 − tn)(p−2)δ(rn) .

15

r 7→ C2(r)δ(r) is bounded when r is large enough :

C2(r)δ(r) =

(12(p− 2)pp

(1− 1

r

)(1

1 + p−2r

)p)δ(r)exp

((1− p)

N ln rpr +N(p− 2)

). (2.12)

Once again, we use the relation between σ(r) and δ(r) :

tn+1 − tn2 + tn+1 − tn

≤ K1

(2 + tn+1 − tn)σ(rn)

r(p−1)n


≤ Kr(p−1)n


,

where K is a positive real which does not depend on n.The series in the right-hand side converges, so its general term tends to zero and(tn+1 − tn) too. We deduce also that

∑n

(tn+1 − tn) converges and as a consequence

the sequence (tn) has a limit which we call T . For t > T ≥ tn, ||u(., t)||L∞(Ω) ≤ αn. Apassage to the limit leads to the conclusion. \

By taking rn = lnµ(αn, p) in the assumption of theorem 2.1, then

µ(αn, p)δ(rn) = exp(

N lnµ(αn, p)p lnµ(αn, p) +N(p− 2)

),

is bounded. Since σ(rn) = 1− (p− 2)δ(rn), we derive the following result.

Corollary 2.1 Under the same assumptions on q, p, A and f , if there exists a de-creasing sequence of positive real numbers (αn)n∈IN such that

∞∑n=0

(lnµ(αn, p))p−1

αp−2n+1µ(αn, p)

< +∞, (2.13)


Remark 2.2 The choice of rn and so the sufficient condition can be slightly improved.But it involves tedious calculations. A good choice is µ(αn, p)(p−2)δ(rn) = rp−1

n whichgives terms like ln(lnµ(αn, p)) at the denominator of the fraction.

2.2 Integral conditions

We change the previous sum into an integral.

Proposition 2.1 Let f be a non-negative and a non-increasing function defined on(0, 1]. Assume that

0 < lim inf f(αn)(αn − αn+1)(

lnµ(αn, p)lnµ(αn+1, p)

)p−1

(2.14)

16

≤ lim sup f(αn+1)(αn − αn+1)(

lnµ(αn+1, p)lnµ(αn, p)

)p−1

< +∞.

Then∞∑n=0

(lnµ(αn, p))p−1

αp−2n µ(αn, p)

is finite if and only if∫ 1

0

f(t) (lnµ(t, p))p−1

tp−2µ(t, p)dt is finite.

Proof: Since f is non-increasing and t 7→ µ(t, p) is a decreasing function, the followinginequalities hold ∫ αn

αn+1


tp−2µ(t, p)dt ≤

f(αn+1)(αn − αn+1)(αnαn+1

)p−2 ( lnµ(αn+1, p)lnµ(αn, p)

)p−1 (lnµ(αn, p))p−1

αp−2n µ(αn, p)

,

and ∫ αn

αn+1


tp−2µ(t, p)dt ≥

f(αn)(αn − αn+1)(αn+1

αn

)p−2 ( lnµ(αn, p)lnµ(αn+1, p)

)p−1 (lnµ(αn+1, p))p−1

αp−2n+1µ(αn+1, p)

,

which complete the proof. \

We have to find a ”good” function f . In our case, we take αn = 2−n so the sequences(lnµ(αn+1, p)lnµ(αn, p)

)and

(lnµ(αn, p)

lnµ(αn+1, p)

)are bounded. This allows us to take f(t) =

1t.

Theorem 2.2 (Integral criterion) Let 0 ≤ q < 1, p > 2 and assume that (1.6), (1.7)and (1.8) hold. If ∫ 1

0

(lnµ(t, p))p−1

tp−1µ(t, p)dt < +∞, (2.15)


Remark 2.3 In [1], it is proved that if∞∑n=1

1µ(αn)

lnαnαn+1

is finite then∫ 1

0

dt

tµ(t)is finite too. When p → 2,

∫ 1

0

(lnµ(t, p))p−1

tp−1µ(t, p)dt →

∫ 1

0

lnµ(t, 2)tµ(t, 2)

dt which is

exactly the same expression since from the definitions, µ(t, 2) = µ(t).

On the other hand, in [1], it is also proved that if∫ 1

0

lnµ(t)tµ(t)

dt converges, solutions hold

the compact support property. Theorem 2.2 is therefore valid for p = 2.

17

From now on, in this paragraph, we use the semi-classical notations. We recall that

µ(α, p)=inf∫

Ω

(|∇ψ|p + αq−(p−1)b(x)ψp

)dx : ψ ∈W 1,p(Ω),

∫Ω|ψ|pdx = 1

, (2.16)

for α > 0 and

λ1(h) = inf∫

Ω|∇ψ|p + h−pb(x)|ψ|p dx : ψ ∈W 1,p(Ω),

∫Ω|ψ|p dx = 1

, (2.17)

for h > 0. Thenµ(α, p) = λ1(α

(p−1)−qp ),

and the integral criterion becomes

∫ 1

0

(lnλ1

(t

p−1−qp

))p−1

tp−1λ1

(t

p−1−qp

) dt < +∞.

By a change of variables h = tp−1−q

p , we have∫ 1

0

(lnλ1(h))p−1

hp(p−1)−(1+q)

p−(1+q) λ1(h)dh < +∞. (2.18)

The convergence depends on the growth of λ1(h). If we have an estimate of the followingtype

λ1(h) ≥ C1hθ, (2.19)

where C and θ are two positive real numbers then (2.18) holds provided

−θ +p(p− 1)− (1 + q)

p− (1 + q)< 1 ⇐⇒ θ >

p(p− 2)p− (1 + q)

. (2.20)

We notice thatp− 2

p− (1 + q)< 1. Such θ exists. Indeed, by taking some test-function

in the definition of λ1(h), we see that λ1(h) ≤ Ch−p for h small enough, and somepositive real number C. The growth of the first eigenvalue is less than h−p.The point is to find functions b for which estimate (2.19) holds. From (4.32), functionslike |x|β match the condition for some β > 0. The next theorem is more accurate.

18

Theorem 2.3 Let 0 ≤ q < 1, p > 2 and assume that (1.6), (1.7) and (1.8) hold. If(1b

)s∈ L1(Ω) with

s >

p− 21− q

N

pfor p ≤ N,

s >p− 21− q

for p > N,

(2.21)


Proof: We recall that γ = N

p for 1 N.

(2.22)

Since b satisfies (1.8), by corollary 4.2, we have

λ1(h)γ meas (x : b(x) ≤ hpλ1(h)) ≥ C,

where C is a positive constant which does not depend on h. Clearly,

meas(x : b(x) ≤ hpλ1(h)) = meas(

x :1b(x)

≥ 1hpλ1(h)

),

andmeas

(x :

1b(x)

≥ 1hpλ1(h)

)≤ (hpλ1(h))

s∫Ω

(1b(x)

)sdx.

Therefore, we derive an estimate

λ1(h) ≥C ′

hps

γ+s

,

for some C ′ > 0. From (2.20), we have

s

γ + s>

p− 2p− (1 + q)

.

As a consequence,s(1− q) > γ(p− 2),

which completes the proof for p 6= N .For p = N , we have taken the limit of γ which is 1. \

When p > N , estimates of a lower bound for first eigenvalues are not sharp. Anyimprovement in this field will generate better sufficient conditions.One ends with some applications :

19


1) If p ≤ N and β <p(1− q)p− 2

then u satisfies the TCS-property.

2) One can draw the same conclusion if p > N and β <N(1− q)p− 2

.

Proof: We assume that p ≤ N . Let s ∈(

0,N

β

). β <

p(1− q)p− 2

implies that s >

p− 21− q

N

2. Furthermore, βs < N leads to

∫ 1

0

1rβs

rN−1 dr < +∞. Thus1bs∈ L1(Ω).

The previous theorem ends the proof.The case p > N is treated the same way. \

Remark 2.4 For p ≤ N , that is, when estimates are sharp, the condition does not

depend on N . If p→ 2,p(1− q)p− 2

tends to infinity. It can be compared to the condition

found in [1].

3 The Porous Media diffusion equation (m > 1)

We assume that Ω is a bounded domain of IRN (N ≥ 1) with a Lipschitz-continuousboundary, A : (x, t, r, s) 7→ A(x, t, r, s) and f : (x, t, r) 7→ f(x, t, r) are measurablefunctions from respectively Ω×IR+×IR×IRN and Ω×IR+×IR with values respectivelyin IRN and IR. We assume also that A and f are continuous in the variables (r, s) ∈IR× IRN and satisfy

|A(x, t, r, s)| ≤ C|s|rm−1,A(x, t, r, s).s ≥ C ′mrm−1|s|2, (3.1)

f(x, r)r ≥ b(x)|r|q+1, (3.2)

for all (x, t, r, s) ∈ Ω× IR+× IR× IRN , where C and C ′ are some positive real numbers.In the sequel, b is a bounded, nonnegative and measurable function and q ∈ [0, 1).

Definition 3.1 We say that u belonging to C([0,∞);L2(Ω)

)∩ L2

loc

([0,∞);W 1,2(Ω)

)and such that f(., ., u) ∈ L2

loc (Ω× [0,∞)) is a weak solution (resp. subsolution) of theproblem


u(x, 0) = u0(x) in Ω,(3.3)

20

with u0 ∈ L2(Ω), if for t > 0 there holds∫ t

0

∫Ω

(−uζt +A(x, t, u,∇u).∇ζ + f(x, t, u)ζ) dxdτ = (3.4)∫Ωζ(x, 0)u0(x) dx−


for any ζ ∈ L2loc

([0,∞);W 1,2(Ω)

)∩W 1,2

loc

([0,∞);L2(Ω)

).

(resp. ∫ t

0

∫Ω

(−uζt +A(x, t, u,∇u).∇ζ + f(x, t, u)ζ) dxdτ ≤ (3.5)∫Ωζ(x, 0)u0(x) dx−


for any ζ ∈ L2loc

([0,∞);W 1,2(Ω)

)∩W 1,2

loc

([0,∞);L2(Ω)

), ζ ≥ 0).

Since (1.19) holds, an energy-method is available in W 1,2(Ω) and we define

µ′(α,m)=inf∫

Ω

(|∇ψ|2 + αm−qb(x)|ψ|2

)dx : ψ ∈W 1,2(Ω),

∫Ω|ψ|2dx = 1

. (3.6)

3.1 Iterative method

In this paragraph, r is a fixed real constant in [m+ 1,+∞).Following 2.1, we first establish lemma 3.1 :

Lemma 3.1 We suppose there exists a measurable function u in Ω×IR+ which satisfiesweakly


u(x, 0) = u0(x) in Ω.(3.7)

with ||u0||L∞(Ω) ≤ α, and that (1.19) and (1.7) hold. Then

||u(., t)||Lr(Ω) ≤

1||u(., 0)||1−mLr(Ω) + C1µ′(α,m) t

1m−1

, (3.8)

with C1 = C1(Ω, r,m), and there exist two positive constants C = C(Ω,m) and C2 =C2(r,m) such that

||u(., t)||L∞(Ω) ≤ min

C (1 +2t

)δ(r) 1||u(., 0)||1−mL∞(Ω) + C2µ′(α,m)t

σ(r)m−1

, 1

. (3.9)

(δ(r) and σ(r) are defined in Theorem B)

21

Proof: As in Lemma (2.1), we have

1r

d

dt

∫Ω|u|r dx+ (r − 1)

∫ΩA(x, t, u,∇u)∇u|u|r−2 +

∫Ωf(x, t, u)u|u|r−2 dx = 0,

with the same test-function u|u|r−2. Repeating the computation, we obtain

1r

d

dt

∫Ω|u|r dx+m(r − 1)

∫Ω|∇|u|2um+r−3 dx+

∫Ωαq−m b(x)|u|m+r−1 dx ≤ 0.

Thus,

1r

d

dt

∫Ω|u|r dx+

4m(r − 1)(m+ r − 1)2

∫Ω|∇ (|u|)

m+r−12 |2 dx+

∫Ωαq−m b(x)|u|m+r−1 dx ≤ 0.

It is clear that

(m+ r − 1)2

4m(r − 1)≥ 1. (3.10)

Both the preceding argument and the choice of |u|m+r−1

2 as a test-function in (3.6) yield

1r

d

dt

∫Ω|u|r dx+

4m(r − 1)(m+ r − 1)2

µ′(α,m)∫Ω|u|m+r−1 dx ≤ 0.

By Hlder’s inequality,∫Ω|u|r dx ≤

(∫Ω|u|m+r−1 dx

) rm+r−1

meas(Ω)m−1

m+r−1 ,

so we have a differential inequality

d

dt

∫Ω|u|r dx+

4m(r − 1)(m+ r − 1)2

rµ′(α,m)

meas(Ω)m−1

r

(∫Ω|u|r dx

)m+r−1r

≤ 0.

A similar computation leads to

∫Ω|u|r dx ≤

1

(∫Ω |u0|r)

1−mr +

(m−1)4m(r−1)

(m+r−1)2

meas(Ω)m−1

rµ′(α,m) t

r

m−1

,

and (3.8) holds for C1 =(m− 1) 4m(r−1)

(m+r−1)2

meas(Ω)m−1

r

.

22

For (3.9), by theorem B , there exists a constant C ′ = C ′(Ω,m) such as

||u(., t)||L∞(Ω) ≤ C ′(

1 +1τ

)δ(r)||u(., t− τ)||σ(r)

Lr(Ω),

with δ(r) = N2r+N(m−1) , σ(r) = 2r

2r+N(m−1) < 1 and 0 < τ < t.

From (3.8), we have

||u(., t− τ)||Lr(Ω) ≤

1

||u(., 0)||1−mL∞(Ω)meas(Ω)1−m

r + C1µ′(α,m) (t− τ)

1m−1

.

Notice 1−m < 0 implies that ||u(., 0)||1−mLr(Ω) ≥ ||u(., 0)||1−mL∞(Ω) (meas(Ω))m−1

r . Thus,

||u(., t)||L∞(Ω) ≤ C ′(

1 +1τ

)δ(r) 1

||u(., 0)||1−mL∞(Ω)meas(Ω)1−m

r + C1µ′(α,m) (t− τ)

σ(r)m−1

;

r large enough implies that C ′meas(Ω)σ(r)

r ≤ C ′meas(Ω)12 . Then,

||u(., t)||L∞(Ω) ≤ C

(1 +

1τ

)δ(r) 1||u(., 0)||1−mL∞(Ω) + 2C2µ′(α,m) (t− τ)

σ(r)m−1

,

where C2 =12C1 (meas(Ω))

m−1r =

12(m − 1)

4m(r − 1)(m+ r − 1)2

. Inequality (3.9) follows by

taking τ = t2 . \

Theorem 3.1 Let 0 ≤ q < 1, m > 1 and assumptions (1.19)-(1.7)-(1.8) hold. Assumethat there exist two sequences of positive real numbers (αn)n∈IN and (rn) such that(αn)n∈IN is decreasing and

∞∑n=0

rmnαm−1n+1 µ

′(αn,m)σ(rn)< +∞. (3.11)


Proof: Similar arguments hold for the sequence (αn). Therefore, limn→+∞

αn = 0 and we

set α0 = 1. In a same way, r is taken large enough.We apply Lemma 3.1 with α = α0 = 1 and r = r0.

||u(., t)||L∞(Ω) ≤ C

(1 +

2t

)δ(r0) ( 11 + C2µ′(1,m) t

)σ(r0)

m−1

.

23

We define a time t1 by

α1 = C

(1 +

2t1

)δ(r0) ( 11 + C2µ′(1,m) t1

)σ(r0)

m−1

.

This implies that for t > t1,

||u(., t)||L∞(Ω) ≤ C

(1 +

2t− t1

)δ(r1)(

1α1−m

1 + C2µ′(α1,m) (t− t1)

)σ(r1)

m−1

.

Therefore, it is easy to define a new time t2 by

α2 = C

(1 +

2t2 − t1

)δ(r1)(

1α1−m

1 + C2µ′(α1,m) (t2 − t1)

)σ(r1)

m−1

.

Continuing this process, one constructs an increasing sequence (tn) hence

αn+1 = C

(1 +

2tn+1 − tn

)δ(rn)(

1α1−mn + C2µ′(αn,m) (tn+1 − tn)

)σ(rn)m−1

.

Similar arguments complete the proof. \We deduce also

Corollary 3.1 Under the same assumptions on q, m, A and f , if there exists a de-creasing sequence of positive real numbers (αn)n∈IN such that

∞∑n=0

(lnµ′(αn,m))m

αm−1n+1 µ

′(αn,m)< +∞, (3.12)


3.2 Integral condition

Similarly,

Proposition 3.1 Let f be a non-negative and a non-increasing function defined on(0, 1]. Assume that

0 < lim inf f(αn)(αn − αn+1)(

lnµ′(αn,m)lnµ′(αn+1,m)

)m(3.13)

≤ lim sup f(αn+1)(αn − αn+1)(

lnµ′(αn+1,m)lnµ′(αn,m)

)m< +∞.

Then∞∑n=0

(lnµ′(αn,m))m

αm−1n µ′(αn,m)

is finite if and only if∫ 1

0

f(t) (lnµ′(t,m))m

tm−1µ′(t,m)dt is finite.

24

As a consequence,

Theorem 3.2 (Integral criterion) Let 0 ≤ q < 1, m > 1 and assumptions (1.19)-(1.7)-(1.8) hold. If ∫ 1

0

(lnµ′(t,m))m

tmµ′(t,m)dt < +∞, (3.14)


Remark 3.1 Notice that Remark 2.3 is also valid by changing p − 2 into m − 1 andµ(α, p) into µ′(α,m).

What remains now is to find a condition like theorem 2.3 on the function b. First, wetransform the previous integral by a change of variables.

µ′(α,m) = λ1(αm−q

2 ) so∫ 1

0

(lnµ′(t,m))m

tmµ′(t,m)dt converges if and only if

∫ 1

0

(lnλ1(t

m−q2 ))m

tmλ1(tm−q

2 )dt =

2m− q

∫ 1

0

(lnλ1(h))m

h2(m−1)+m−q

m−q λ1(h)dh converges.

This happens if λ1(h) ≥C

hθwith

−θ(1− ε) +2(m− 1) +m− q

m− q< 1.

Letting ε go to 0 gives

θ >2(m− 1)m− q

. (3.15)

Theorem 3.3 Let 0 ≤ q < 1, m > 1 and assumptions (1.19)-(1.7)-(1.8) hold. If(1b

)s∈ L1(Ω) with

s >

m− 11− q

N

2for N ≥ 2,

s >m− 11− q

for N = 1,

(3.16)


25

Proof: The main point is

C ≤ λ1(h)γmeas(x : b(x) ≤ h2λ1(h)

)≤ λ1(h)γ

(h2λ1(h)

)s ∫Ω

(1b(x)

)sdx,

with C a positive constant and h small enough. Thus, we have

λ1(h) ≥C ′

h2s

s+γ

,

so by (3.15), 0 ≤ q < 1 and

s > γm− 11− q

.

The definition of γ ends the proof. \.

The section ends with an application.


1) If N ≥ 2 and β <2(1− q)m− 1


2) One can draw the same conclusion if N = 1 and β <(1− q)m− 1

.

4 Lower bound for the first eigenvalue

The aim of this section is to provide a p-Laplace analogy with the semi-classical ex-pansion of the first eigenvalue of a Schrdinger operator. Our results are stated for thep-Laplace operator plus a potential V

u 7→ −∆pu+ V (x)|u|p−2u. (4.1)

Our estimates are obtained under very weak assumptions on the domain and the po-tential, without assuming the existence of a minimizer. Let us consider a non-emptyconnected open subset Ω ⊂ IRN and a measurable function V defined in Ω. We set

W 1,p,V (Ω) = ψ ∈W 1,p(Ω) : V (x)|ψp| ∈ L1(Ω). (4.2)

If W 1,p,V (Ω) 6= 0 and ψ ∈W 1,p,V (Ω), we set

FV (ψ) =∫Ω|∇ψ|p + V (x)|ψ|p dx, (4.3)

and define

λ1 = infFV (ψ) : ψ ∈W 1,p,V (Ω),

∫Ω|ψ|p dx = 1

, (4.4)

26

and for h > 0,

λ1(h) = infFh−pV (ψ) : ψ ∈W 1,p,V (Ω),

∫Ω|ψ|p dx = 1

, (4.5)

Thus λ1(h) is the first eigenvalue of the operator

u 7→ −∆pu+ h−pV (x)|u|p−2u,

in W 1,p,V (Ω) with Neumann boundary condition if the infimum is achieved by a regularenough element of W 1,p,V (Ω) and ∂Ω C1.Before starting, we define a well for a measurable function V .

Definition 4.1 We say that V has a well in U if U is a C1 bounded, connected, non-empty open set of Ω and if there exists ψ0 ∈ W 1,p,V (Ω) with ||ψ0||Lp(Ω) = 1 suchthat ∫

ΩV (x)|ψ0|p dx < a = ess inf V

Ω\U, (4.6)

with meas(Ω\U) > 0.

The notion of a well generalizes the definition in [6]. For example, if x : V (x) < a isa C1 bounded, connected, open non-empty subset of Ω, then V has a well in it.

Lower bounds are based on the Sobolev and Morrey inequalities (See [3], pages 168,169).The reason of the choice of IRN with p < N in a first part is that it gives prototypes ofarguments used in more general situations. A comparison with the well-known formulaof Cwickel-Lieb-Rosenblyum is provided.For applications to the TCS-property, we assume that V is bounded, ess inf V

Ω= 0,∫

ΩV (x) dx > 0 and Ω is a C1 bounded domain. These conditions seem restrictive but

they are very often assumed.In more general situations, we must use the assumption of a well for V : it avoidsthe vanishing of the gradient by concentrating the minimizing sequences in a boundedregion.

4.1 A simple case : minimization in IRN for 1 p > 1. Then, either λ1 = −∞ or(∫V (x)≤λ1


) pN

≥ C(p,N), (4.7)

27

where C = C(p,N) > 0 is the positive constant of the Sobolev inequality.In addition, if there exists a minimizer in W 1,p,V (IRN ),(∫

V (x)<λ1


) pN

≥ C(p,N). (4.8)

Proof: Suppose that λ1 > −∞. Let ψ be in W 1,p,V (IRN ) with ||ψ||Lp(IRN ) = 1 then∫IRN

|∇ψ|p dx+∫IRN

V (x)|ψ|p dx = FV (ψ) = FV (ψ)∫IRN

|ψ|p dx.

The integral involving V is split in two parts :∫IRN

|∇ψ|p dx =∫V (x)<FV (ψ)

(FV (ψ)− V (x))|ψ|p dx (4.9)

+∫V (x)≥FV (ψ)

(FV (ψ)− V (x))|ψ|p dx.

Note that V (x)ψp belongs to L1(IRN ). Therefore,∫IRN

|∇ψ|p dx ≤∫V (x)<FV (ψ)

(FV (ψ)− V (x))|ψ|p dx. (4.10)

Applying Hlder’s inequality leads to∫IRN

|∇ψ|p dx ≤(∫

V (x)<FV (ψ)(FV (ψ)− V (x))

Np dx

) pN(∫

V (x)<FV (ψ)|ψ|p∗ dx

)1− pN

.

Obviously,∫IRN

|∇ψ|p dx ≤(∫

V (x)<FV (ψ)(FV (ψ)− V (x))

Np dx

) pN (∫

IRN|ψ|p∗ dx

)1− pN

. (4.11)

Then we use Sobolev inequality. Indeed, non zero constants do not belong to

W 1,p,V (IRN ) and so all functions ψ satisfy∫IRN

|∇ψ|p dx > 0.

The Beppo-Levi theorem ends the proof. \

Notice that p∗ =pN

N − pand 1− p

N=

p

p∗.

Remark 4.1 If Ω is any open domain of IRN , we define

W 1,p,V0 (Ω) = ψ ∈W 1,p

0 (Ω) : V (x)|ψp| ∈ L1(Ω), (4.12)

and if W 1,p,V0 (Ω) 6= 0,

λ1 = infFV (ψ) : ψ ∈W 1,p,V

0 (Ω),∫Ω|ψ|p dx = 1

, (4.13)

then the estimates in theorem 4.1 hold for λ1.

28

Remark 4.2 If limz→−∞

∫V (x)<z

(z − V (x))Np dx = 0 then λ1 > −∞. It always happens

if ess inf VIRN

= σ > −∞.

Remark 4.3 (4.8) generalizes the well-known Cwickel-Lieb-Rozenblyum formula forthe energy of the ground state when p = 2. But, the inequality is slightly more precisethan the previous one because the function x 7→ (λ1 − V (x))

Np is integrated over the

set x : V (x) < λ1 instead of x : V (x) ≤ λ1.

Remark 4.4 The quantity in the left-hand side may be +∞. Hence, this theoremis not efficient for some classes of potential. But in many applications, when thereexists z0 > λ1 such that (z0 − V (x))

Np ∈ L1(x : V (x) < z0), this inequality is very

practical.

It is sometimes convenient to express a lower bound in a simple way.

Corollary 4.1 If ess inf VIRN

= σ > −∞ and meas(x : V (x) ≤ λ1) <∞ then

(λ1 − σ)meas(x : V (x) ≤ λ1)pN ≥ C(p,N). (4.14)

In addition, if there exists a minimizer,

(λ1 − σ)meas(x : V (x) < λ1)pN ≥ C(p,N). (4.15)

Proof: We have∫V (x)<λ1

(λ1 − V (x))Np dx ≤ (λ1 − σ)

Np meas(x : V (x) < λ1),

which gives (4.15) and ends the proof. \

Remark 4.5 Directly from the theorem, if u is a minimizer, then λ1 > σ = ess inf VΩ

.

From corollary 4.1, if λ1 = σ > −∞ then there is no minimizer andmeas(x : V (x) < z) = +∞ for all z > λ1.

4.2 The case ess inf(V)=0

In this section, we assume that Ω is a bounded domain with a C1 boundary and thatV is a bounded measurable function such that

ess inf VΩ

= 0 and∫ΩV (x) dx > 0. (4.16)

29

It is well-known that the infimum λ1(h) is attained for a function of W 1,p(Ω). We calleduh the nonnegative function of W 1,p(Ω) such that Fh(uh) = λ1(h) and ||uh||Lp(Ω) = 1.From now on, let us denote by γ a real number which satisfies :

γ = Np for 1 N,

(4.17)

withγ

γ − 1= +∞ if γ = 1.

Note that W 1,p(Ω) imbeds Lq(Ω) continuously with q =pγ

γ − 1, that is,

||∇ψ||pLp(Ω) + ||ψ||pLp(Ω) ≥ C(p,N,Ω, γ) ||ψ||pLq(Ω) (4.18)

for some positive constant C(p,N,Ω, γ) and for all ψ in W 1,p(Ω).The major change between the previous case and the others cases lies in the imbeddinginequality of W 1,p(Ω) into Lp(Ω).

Theorem 4.2 Assume that (4.16) holds. Then for h small enough,(∫V (x)<hpλ1(h)

(λ1(h)−

V (x)hp

)γdx

) 1γ

≥ C, (4.19)

where C = C(p,N, γ,Ω, V ) is a positive real constant.

Proof: The beginning of the proof is similar. Instead of IRN , ψ and V , we take Ω, uhand V

hp . The Hlder’s inequality gives

∫Ω|∇uh|p dx ≤

(∫V (x)<hpλ1(h)

(λ1(h)−

V (x)hp

)γdx

) 1γ (∫

Ω|uh|q dx

) pq

. (4.20)

Thus, by (4.18),(∫V (x)<hpλ1(h)

(λ1(h)−

V (x)hp

)γdx

) 1γ

≥ C||∇uh||pLp(Ω)

1 + ||∇uh||pLp(Ω)

,

with C = C(p,N,Ω, γ) a positive real number. The main point is to prove that

lim infh→0

||∇uh||Lp(Ω) > 0.

Suppose that there exists a sequence (hn) of positive real numbers converging to zerosuch that

limn→+∞

||∇uhn ||Lp(Ω) = 0.

30

(uhn) is bounded in W 1,p(Ω) so there exists a function u0 in W 1,p(Ω) such that, up toa subsequence, uhn u0 weakly in W 1,p(Ω). Obviously, ||∇u0||Lp(Ω) = 0. Therefore,u0 = C, where C is a real number. Thanks to the Rellich-Kondrachov theorem, up

to a subsequence, uhn → C strongly in Lp(Ω), so C =(

1meas(Ω)

) 1p

. We deduce that

limn→+∞

hpnλ1(hn) =∫Ω V (x) dxmeas(Ω)

.

Now, consider the set E =x : V (x) ≤

∫Ω V (x) dx4meas(Ω)

. The set E has a positive measure,

so one can find a sequence (fn) ofW 1,p(Ω) such that ||fn||Lp(Ω) = 1 and fn →χE

meas(E)1p

for the norm Lp(Ω), V is bounded, and therefore∫ΩV (x)fpn dx→

1meas(E)

∫ΩV (x)χE dx =

∫E V (x) dxmeas(E)

≤∫Ω V (x) dx4meas(Ω)

.

One can find an integer N1 such that∫ΩV (x)fpN1

dx ≤∫Ω V (x) dx2meas(Ω)

. We use fN1 as a

test-function for λ1(hn), which leads to

hpnλ1(hn) ≤ hpn

∫Ω|∇fN1 |p dx+

∫ΩV (x)fpN1

dx.

Finally,

lim supn→+∞

hpnλ1(hn) ≤∫Ω V (x) dx2meas(Ω)

,

which gives a contradiction. \

Remark 4.6 The condition V ∈ L∞(Ω) can be weakened, see [1], lemma 3.2, whenp = 2. More precisely, by changing the set E, it is easy to prove that lim

h→0hpλ1(h) = 0.

Similarly, one has

Corollary 4.2 If (4.16) holds then for h small enough

λ1(h) (meas(V (x) < hpλ1(h)))1γ ≥ C, (4.21)

where C = C(p,N, γ,Ω, V ) is a positive constant.

4.3 The case when V has a well.

We suppose that Ω is an open connected set of IRN and V is a measurable function on

Ω which has a well in U . ψ0 is associated to the well in U , that is,∫ΩV (x)|ψ0|p dx <

a = ess inf VΩ\U

and we fix θ > 0 small enough such that

∫ΩV (x)|ψ0|p dx < a− θ. (4.22)

31

One can prove easily the next proposition.

Proposition 4.1 If V has a well in U then for h small enough, hpλ1(h) < a− θ, andx : h−pV (x) ≤ λ1(h)

is a subset of U .

Proof: Clearly, hpλ1(h) ≤ hp∫Ω|∇ψ0|p +

∫ΩV (x)|ψ0|p dx. Therefore, for h small

enough, hpλ1(h) < a− θ andx : h−pV (x) ≤ λ1(h)

⊂ x : V (x) < a ⊂ U.

\

We recall that γ has been defined in (4.17). For the sake of simplicity, FV is denotedby F and Fh−pV by Fh.

Theorem 4.3 If V has a well in U and h is small enough, either λ1(h) = −∞ or

(∫V (x)≤hpλ1(h)

(λ1(h)− h−pV (x)

)γdx

) 1γ

≥ C(p,N,U, γ,Ω, θ), (4.23)

where C(p,N,U, γ,Ω, θ) is a positive constant.In addition, if there exists a minimizer in W 1,p,V (Ω),

(∫V (x)<hpλ1(h)

(λ1(h)− h−pV (x)

)γdx

) 1γ

≥ C(p,N,U, γ,Ω, θ). (4.24)

Proof: Suppose that λ1(h) > −∞ for h small enough. From proposition 4.1, we canfind h ≤ 1 small enough such that hpλ1(h) < a−θ. Thus, one can choose ψ ∈W 1,p,V (Ω)which satisfies ||ψ||pLp(Ω) = 1 and hpFh(ψ) ≤ a− θ. We have

Fh(ψ) =∫Ω|∇ψ|p dx+

∫Ωh−pV (x)|ψ|p dx.

Once again, we obtain∫Ω|∇ψ|p dx+

∫V (x)≥hpFh(ψ)

(h−pV (x)− Fh(ψ)

)|ψ|p dx

=∫V (x)<hpFh(ψ)

(Fh(ψ)− h−pV (x)

)|ψ|p dx.

Since x : V (x) < a ⊂ U , Ω\U ⊂ x : V (x) ≥ a which yields∫V (x)<hpFh(ψ)


)|ψ|p dx

32

≥∫Ω|∇ψ|p dx+

∫Ω\U

(h−pV (x)− Fh(ψ)

)|ψ|p dx,


(a

hp− Fh(ψ)

)∫Ω\U

|ψ|p dx,


θ

hp

∫Ω\U

|ψ|p dx,

≥∫Ω|∇ψ|p dx+ θ

∫Ω\U

|ψ|p dx,

since h ≤ 1. Using Hlder’s inequality yields to∫V (x)<hpFh(ψ)


)|ψ|p dx

≤(∫

V (x)<hpFh(ψ)

(F (ψ)− h−pV (x)

)γdx

) 1γ

||ψ||pLq([V (x)<hpF (ψ)]),

≤(∫

V (x)<hpFh(ψ)

(F (ψ)− h−pV (x)

)γdx

) 1γ

||ψ||pLq(U).

By the continuous imbedding of W 1,p(U) into Lq(U), it follows that,

(∫V (x)<hpFh(ψ)


)γdx

) 1γ

≥ C

||∇ψ||pLp(Ω) + θ

∫Ω\U

|ψ|p dx

||∇ψ||pLp(U) + ||ψ||pLp(U)

,

where C = C(p,N,U, γ) is a positive constant. Finally,

(∫V (x)<hpFh(ψ)


)γdx

) 1γ

≥ C

||∇ψ||pLp(Ω) + θ

∫Ω\U

|ψ|p dx

||∇ψ||pLp(U) + 1,

since ||ψ||pLp(U) ≤ ||ψ||pLp(Ω) = 1. Let (ψn) be a minimizing sequence of λ1(h). For nlarge enough hpFh(ψn) ≤ a− θ. It suffices to prove that

lim inf

||∇ψn||pLp(Ω) + θ

∫Ω\U

|ψn|p dx

||∇ψn||pLp(U) + 1

> 0,

independently of h. To do so, we suppose that there exists a sequence (φn) in W 1,p(Ω)such that

limn→+∞

||∇φn||pLp(Ω) + θ

∫Ω\U

|φn|p dx

||∇φn||pLp(U) + 1= 0.

a) If meas(Ω) is infinite, the contradiction comes from the following lemma :

33

Lemma 4.1 Let (φn) be a sequence of W 1,p(Ω) such that ||φn||Lp(Ω) = 1 and||∇φn||Lp(Ω) goes to zero. Then

limn→+∞

∫U|φn|p dx = 0. (4.25)

proof of the lemma 4.1: (φn) is bounded in W 1,p(Ω), there exists a function φ inW 1,p(Ω) such that φn φ weakly in W 1,p(Ω). Obviously, ||∇φ||Lp(Ω) = 0. Therefore,φ = C where C is a real number. If C is not equal to zero, φ /∈ Lp(Ω) (meas(Ω) = ∞). As a consequence, φn 0 weakly in W 1,p(Ω). Thanks to the Rellich-Kondrachovtheorem over U , up to a subsequence, φn → 0 strongly in Lp(U) which finishes theproof of the lemma. \

b) If meas(Ω) is finite, in a same way, up to a subsequence, φn C weakly in Lp(Ω)

and φn → C strongly in Lp(U). Clearly, we have 0 ≤ C ≤(

1meas(Ω)

) 1p

. Thus,

0 ≤ lim inf∫U|φn|p dx ≤ lim sup

∫U|φn|p dx ≤

meas(U)meas(Ω)

. It follows that

lim inf∫Ω\U

|φn|p dx ≥ 1− meas(U)meas(Ω)

> 0 due to the definition of a well, which yields a

contradiction.

The theorem of Beppo-Levi ends the proof. \

Remark 4.7 It has be shown that

inf||∇φn||pLp(Ω) +∫Ω\U

|φn|p dx : φ ∈W 1,p(Ω) ,∫Ω|φn|p dx = 1 > 0. (4.26)

It can be compared to the proof of theorem 3.1 in [2], page 167.

The next estimate is more applicable.

Corollary 4.3 If ess inf VIRN

= σ > −∞ then(λ1(h)−

σ

hp

)meas(x : V (x) ≤ hpλ1(h))

1γ ≥ C. (4.27)

In addition, if there exists a minimizer in W 1,p,V (Ω),(λ1(h)−

σ

hp

)meas(x : V (x) < hpλ1(h))

1γ ≥ C. (4.28)

Many remarks made in the previous paragraph remains valid. It is important to noticethat C = C(p,N,U, γ,Ω, θ) tends to zero when γ → 1 for p = N .

34

4.4 Estimate of the first eigenvalue for radial functions

In this paragraph, we shall apply the inequality (4.27) in the following case :There exists a point x0 in Ω and a positive real r0 such that

V has a well in Br0(x0) and ∀x ∈ Br0(x0), V (x) ≥ f(|x− x0|), (4.29)

where

f is a continuous and increasing function on [0, r0] with f(0) = 0. (4.30)

This simple example provides enlightments for the compact support property. Othersassumptions on V are more technical, for instance if V vanishes on a manifold or onmany isolated points. We do not use (4.28) since this formula is more suitable when Vis not continuous in a neighbourhood of the set x : V (x) = 0.

Proposition 4.2 Under (4.29) and (4.30), for h small enough,

λ1(h)(f−1 (hpλ1(h))

)Nγ ≥ C, (4.31)

where C is a positive constant which does not depend on h.

Proof: We first observe that x : V (x) ≤ hpλ1(h) is a subset of x : f(|x − x0|) ≤hpλ1(h) for h small enough. From (4.27), we have

λ1(h)meas(x : f(|x− x0|) ≤ hpλ1(h))1γ =

λ1(h)meas(x : |x− x0| ≤ f−1 (hpλ1(h))

)

1γ ≥ C,

which completes the proof. \.

Let us recall that for 1 0 :

λ1(h) ≥ C h−pN

βγ+N , (4.32)

and if 1 0 :

λ1(h) ≥ C (− lnh)Nβγ , (4.34)

and if 1 < p < N ,λ1(h) ≥ C (− lnh)

pβ . (4.35)

For 2), a simple asymptotic calculus is needed. All the results can be compared to thosein [1] when p = 2. We have seen that for p-Laplacian and porous media equation, theTCS-property appears for power function. But functions like 2) are very importantwhen A is linear (see [2]).

35

References

[1] Y. Belaud, B. Helffer, L. Veron, Long-time vanishing properties of solutions ofsublinear parabolic equations and semi-classical limit of Schrdinger operator, Ann.Inst. Henri Poincarre Anal. nonlinear 18, 1 (2001), 43-68

[2] F.A. Berezin, M.A. Shubin, The Schrdinger Equation, Kluwer Academic Pub-lishers, 1991.

[3] H. Brezis, Analyse fonctionelle. Theorie et applications, CollectionMathematiques appliquees pour la matrise, Masson, 1986.

[4] M. Cwikel, Weak type estimates for singular value and the number of bound statesof Schrdinger operator, Ann. Math. 106 (1977), 93-100.

[5] D. Gilbarg, N. Trudinger, Elliptic Partial Differential Equations of Second Order,Springer Verlag, 1977.

[6] B. Helffer, Semi-classical analysis for the Schrdinger operator and applications,Lecture Notes in Math. 1336, Springer-Verlag, 1989.

[7] V.A. Kondratiev and L. Veron, Asymptotic behaviour of solutions of some non-linear parabolic or elliptic equations, Asymptotic Analysis 14 (1997), 117-156.

[8] E.H. Lieb, W. Thirring, Inequalities for the moments of the eigenvalues of theSchrdinger Hamiltonian and their relations to Sobolev Inequalities, In Studies inMath. Phys., essay in honour of V. Bargmann, Princeton Univ. Press, 1976.

[9] G. V. Rosenblyum, Distribution of the discrete spectrum of singular differentialoperators, Doklady Akad. Nauk USSR 202 (1972), 1012-1015.

[10] L. Veron, Effets regularisants de semi-groupes non lineaires dans des espaces deBanach, Annales faculte des Sciences Toulouse 1 (1979), 171-200.

[11] L. Veron, Coercivite et proprietes regularisantes des semi-groupes non lineairesdans les espaces de Banach, Publication de l’Universite Francois Rabelais - Tours(1976).

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