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volume 6, issue 3, article 91, 2005.

Received 12 February, 2005;

accepted 17 June, 2005.

Communicated by:S.S. Dragomir

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Journal of Inequalities in Pure and Applied Mathematics

THE FIRST EIGENVALUE FOR THE p-LAPLACIAN OPERATOR

IDRISSA LY

Laboratoire de Mathématiques et Applications de Metz Ile du Saulcy, 57045 Metz Cedex 01, France.

EMail:idrissa@math.univ-metz.fr

c

2000Victoria University ISSN (electronic): 1443-5756 037-05

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The First Eigenvalue for the p-Laplacian Operator

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J. Ineq. Pure and Appl. Math. 6(3) Art. 91, 2005

http://jipam.vu.edu.au

Abstract

In this paper, using the Hausdorff topology in the space of open sets under some capacity constraints on geometrical domains we prove the strong continuity with respect to the moving domain of the solutions of ap-Laplacian Dirich- let problem. We are also interested in the minimization of the first eigenvalue of thep-Laplacian with Dirichlet boundary conditions among open sets and quasi open sets of given measure.

2000 Mathematics Subject Classification:35J70, 35P30, 35R35.

Key words:p-Laplacian, Nonlinear eigenvalue problems, Shape optimization.

1. Introduction

Let Ωbe an open subset of a fixed ball D inR^N, N ≥ 2 and1 < p < +∞.

Consider the Sobolev spaceW₀^1,p(Ω)which is the closure ofC^∞functions com- pactly supported inΩfor the norm

||u||^p_1,p = Z

Ω

|u(x)|^pdx+ Z

Ω

|∇u(x)|^pdx.

Thep-Laplacian is the operator defined by

∆_p :W₀^1,p(Ω)−→W^−1,q(Ω)

u7−→∆_pu=div(|∇u|^p−2∇u),

where W^−1,q(Ω) is the dual space of W₀^1,p(Ω) and we have 1 < p, q < ∞,

1

p + ¹_q = 1.

We are interested in the nonlinear eigenvalue problem (1.1)

( −∆_pu−λ|u|^p−2u = 0inΩ,

u = 0on∂Ω.

Let u be a function ofW₀^1,p(Ω), not identically 0. The function uis called an eigenfunction if

Z

Ω

|∇u(x)|^p−2∇u∇φdx =λ Z

Ω

|u(x)|^p−2uφdx

for allφ ∈ C₀^∞(Ω).The corresponding real numberλis called an eigenvalue.

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Contrary to the Laplace operator, the p-Laplacian spectrum has not been proved to be discrete. In [15], the first eigenvalue and the second eigenvalue are described.

LetD be a bounded domain inR^N andc > 0.Let us denote λ^p₁(Ω) as the first eigenvalue for the p-Laplacian operator. The aim of this paper is to study the isoperimetric inequality

min{λ^p₁(Ω),Ω⊆D and |Ω|=c}

and its continuous dependance with respect to the domain. We extend the Rayleigh-Faber-Khran inequality to thep-Laplacian operator and study the minimization of the first eigenvalue in two dimensions when Dis a box. By con- sidering a class of simply connected domains, we study the stability of the min- imizerΩp of the first eigenvalue with respect topthat is ifΩpis a minimizer of the first eigenvalue for thep-Laplacian Dirichlet, whenpgoes to2,Ω₂ is also a minimizer of the first eigenvalue of the Laplacian Dirichlet. Thus we will give a formal justification of the following conjecture: "Ωis a minimizer of given volume c,contained in a fixed boxDand ifDis too small to contain a ball of the same volume asΩ.Are the free parts of the boundary ofΩpieces of circle?"

Henrot and Oudet solved this question and proved by using the Hölmgren uniqueness theorem, that the free part of the boundary ofΩcannot be pieces of circle, see [10].

The structure of this paper is as follows: The first section is devoted to the definition of two eigenvalues. In the second section, we study the properties of geometric variations for the first eigenvalue. The third section is devoted to the minimization of the first eigenvalue among open (or, if specified, quasi open)

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sets of given volume. In the fourth part we discuss the minimization of the first eigenvalue in a box in two dimensions.

LetDbe a bounded open set inR^N which contains all the open (or, if specified, quasi open) subsets used.

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2. Definition of the First and Second Eigenvalues

The first eigenvalue is defined by the nonlinear Rayleigh quotient λ₁(Ω) = min

φ∈W₀^1,p(Ω),φ6=0

R

Ω|∇φ(x)|^pdx R

Ω|φ(x)|^p = R

Ω|∇u₁(x)|^pdx R

Ω|u₁(x)|^pdx ,

where the minimum is achieved by u₁ which is a weak solution of the Euler- Lagrange equation

(2.1)

−∆_pu−λ|u|^p−2u = 0 in Ω

u = 0 on ∂Ω.

The first eigenvalue has many special properties, it is strictly positive, simple in any bounded connected domain see [15]. Andu₁is the only positive eigenfunction for thep-Laplacian Dirichlet see also [15].

In [15], the second eigenvalue is defined by λ₂(Ω) = inf

C∈C2

maxC

R

Ω|∇φ(x)|^pdx R

Ω|φ(x)|^p , where

C₂ :={C ∈W₀^1,p(Ω) :C =−C such that genus(C)≥2}.

In [1], Anane and Tsouli proved that there does not exist any eigenvalue between the first and the second ones.

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3. Properties of the Geometric Variations

In this section we are interested in the continuity of the map Ω7−→λ1(Ω).

Then, we have to fix topology on the space of the open subsets of D. On the family of the open subsets ofD,we define the Hausdorff complementary topology, denotedH^c given by the metric

d_H^c(Ω^c₁,Ω^c₂) = sup

x∈R^N

|d(x,Ω^c₁)−d(x,Ω^c₂)|.

The H^c-topology has some good properties for example the space of the open subsets ofDis compact. Moreover ifΩ_n→^H^c Ω,then for any compactK ⊂⊂Ω we haveK ⊂⊂Ωn fornlarge enough.

However, perturbations in this topology may be very irregular and in general situations the continuity of the mappingΩ7−→λ₁(Ω)fails, see [4].

In order to obtain a compactness result we impose some additional constraints on the space of the open subsets of D which are expressed in terms of the Sobolev capacity. There are many ways to define the Sobolev capacity, we use the local capacity defined in the following way.

Definition 3.1. For a compact setKcontained in a ballB, cap(K, B) := inf

Z

B

|∇φ|^p, φ ∈ C₀^∞(B), φ ≥1 on K

.

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Definition 3.2.

1. It is said that a property holdsp-quasi everywhere (abbreviated as p-q.e) if it holds outside a set ofp-capacity zero.

2. A setΩ ⊂ R^N is said to be quasi open if for every > 0there exists an open setΩ such thatΩ⊆Ω,andcap(Ω\Ω)< .

3. A functionu : R^N −→ R is saidp-quasi continuous if for every > 0 there exists an open setΩsuch thatcap(Ω)< andu_|_R_\Ωis continuous inR\Ω.

It is well known that every Sobolev function u ∈ W^1,p(R^N) has ap-quasi continuous representative which we still denoteu.Therefore, level sets of Sobolev functions arep-quasi open sets; in particularΩv ={x∈D;|v(x)|>0}is quasi open subsets ofD.

Definition 3.3. We say that an open set Ωhas the p−(r, c) capacity density condition if

∀x∈∂Ω, ∀0< δ < r, cap(Ω^c∩B(x, δ), B(x,¯ 2δ)) cap( ¯B(x, δ), B(x,2δ)) ≥c whereB(x, δ)denotes the ball of raduisδ,centred atx.

Definition 3.4. We say that the sequence of the spacesW₀^1,p(Ω_n)converges in the sense of Mosco to the spaceW₀^1,p(Ω)if the following conditions hold

1. The first Mosco condition: For all φ ∈ W₀^1,p(Ω),there exists a sequence φn ∈W₀^1,p(Ωn)such thatφnconverges strongly inW₀^1,p(D)toφ.

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2. The second Mosco condition: For every sequenceφ_n_k ∈W₀^1,p(Ω_n_k)weakly convergent inW₀^1,p(D)to a functionφ,we haveφ∈W₀^1,p(Ω).

Definition 3.5. We say a sequence (Ωn)of open subsets of a fixed ballD γp- converges toΩif for anyf ∈W^−1,q(Ω) the solutions of the Dirichlet problem

−∆_pu_n =f in Ω_n, u_n∈W₀^1,p(Ω_n)

converge strongly inW₀^1,p(D),asn−→+∞, to the solution of the correspond- ing problem inΩ,see [7], [8].

By Op−(r,c)(D), we denote the family of all open subsets of D which satisfy the p −(r, c) capacity density condition. This family is compact in the H^c topology see [4]. In [2], D. Bucur and P. Trebeschi, using capacity constraints analogous to those introduce in [3] and [4] for the linear case, prove the γ_p-compactness result for the p-Laplacian. In the same way, they extend the continuity result of Šveràk [19] to thep-Laplacian forp∈(N −1, N], N ≥2.

The reason of the choice ofpis that inR^N the curves haveppositive capacity if p > N −1. The case p > N is trivial since all functions in W^1,p(R^N) are continuous.

Let us denote by

Ol(D) ={Ω⊆D, ]Ω^c ≤l}

where]denotes the number of the connected components. We have the following theorem.

Theorem 3.1 (Bucur-Trebeschi). LetN ≥p > N−1.Consider the sequence (Ω_n) ⊆ O_l(D) and assume that Ω_n converges in Hausdorff complementary topology toΩ.ThenΩ⊆ Ol(D)andΩn γp−converges toΩ.

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Proof of Theorem3.1. See [2].

ForN = 2andp = 2,Theorem3.1becomes the continuity result of Šveràk [19].

Back to the continuity result, we use the above results to prove the following theorem.

Theorem 3.2. Consider the sequence (Ω_n) ⊆ O_l(D). Assume that Ω_n con- verges in Hausdorff complementary topology toΩ.Thenλ₁(Ω_n) converges to λ₁(Ω).

Proof of Theorem3.2. Let us take

λ₁(Ω_n) = min

φn∈W₀^1,p(Ωn),φn6=0

R

Ωn|∇φ_n(x)|^pdx R

Ωn|φ_n(x)|^p = R

Ωn|∇u_n(x)|^pdx R

Ωn|u_n(x)|^p , where the minimum is attained by u_n,and

λ1(Ω) = min

φ∈W₀^1,p(Ω),φ6=0

R

Ω|∇φ(x)|^pdx R

Ω|φ(x)|^p = R

Ω|u₁(x)|^pdx , where the minimum is achieved byu₁.

By the Bucur and Trebeschi theorem,Ωn γp converges toΩ.This implies W₀^1,p(Ω_n)converges in the sense of Mosco toW₀^1,p(Ω).

If the sequence(u_n)is bounded inW₀^1,p(D),then there exists a subsequence still denotedu_nsuch thatu_nconverges weakly inW₀^1,p(D)to a functionu.The second condition of Mosco implies thatu∈W₀^1,p(Ω).

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Using the weak lower semicontinuity of theL^p−norm, we have the inequality

lim inf

n−→+∞

R

D|∇u_n(x)|^pdx R

D|u_n(x)|^p ≥ R

Ω|∇u(x)|^pdx R

Ω|u(x)|^p ≥ R

Ω|u₁(x)|^p , then

(3.1) lim inf

n→+∞λ₁(Ω_n)≥λ₁(Ω).

Using the first condition of Mosco, there exists a sequence (v_n) ∈ W₀^1,p(Ω_n) such thatv_nconverges strongly inW₀^1,p(D)to u₁.

We have

λ₁(Ω_n)≤ R

D|∇v_n(x)|^pdx R

D|v_n(x)|^p this implies that

lim sup

n−→+∞

λ₁(Ω_n)≤lim sup

n−→+∞

R

D|∇vn(x)|^pdx R

D|v_n(x)|^p

= lim

n−→+∞

R

D|∇v_n(x)|^pdx R

D|vn(x)|^p = R

Ω|u1(x)|^p then

(3.2) lim sup

n→+∞

λ₁(Ω_n)≤λ₁(Ω).

By the relations (3.1) and (3.2) we conclude that λ1(Ωn) converges toλ1(Ω).

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4. Shape Optimization Result

We extend the classical inequality of Faber-Krahn for the first eigenvalue of the Dirichlet Laplacian to the Dirichletp-Laplacian. We study this inequality when Ωis a quasi open subset ofD.

Definition 4.1. LetΩbe an open subset and bounded inR^N.We denote by B the ball centred at the origin with the same volume asΩ.Letube a non negative function inΩ,which vanishes on∂Ω.For allc >0,the set{x∈ Ω, u(x) > c}

is called the level set ofu.

The functionu^∗ which has the following level set

∀c >0, {x∈B, u^∗(x)> c}={x∈Ω, u(x)> c}^∗

is called the Schwarz rearrangement ofu.The level sets ofu^∗ are the balls that we obtain by rearranging the sets of the same volume ofu.

We have the following lemma.

Lemma 4.1. LetΩbe an open subset inR^N. Letψbe any continuous function on R^∗+,we have

1. R

Ωψ(u(x))dx =R

Ω^∗ψ(u^∗(x))dx u^∗ is equi-mesurable withu.

2. R

Ωu(x)v(x)dx≤R

Ω^∗u^∗(x)v^∗(x)dx.

3. Ifu∈W₀^1,p(Ω), p > 1 thenu^∗ ∈W₀^1,p(Ω^∗)and Z

Ω

|∇u(x)|^pdx≥ Z

Ω^∗

|∇u^∗(x)|^pdx Pòlya inequality.

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Proof of Lemma4.1. See [12].

The basic result for the minimization of eigenvalues is the conjecture of Lord Rayleigh: “The disk should minimize the first eigenvalue of the Laplacian Dirichlet among every open set of given measure”. We extend the Rayleigh- Faber-Krahn inequality to thep-Laplacian operator.

LetΩbe any open set inR^N with finite measure. We denote byλ1(Ω)the first eigenvalue for thep-Laplacian operator with Dirichlet boundary conditions.

We have the following theorem.

Theorem 4.2. LetB be the ball of the same volume asΩ,then λ₁(B) = min{λ₁(Ω),Ω open set of R^N,|Ω|=|B|}.

Proof of Theorem4.2. Let u₁ be the first eigenfunction of λ₁(Ω), it is strictly positive see [15]. By Lemma4.1, equi-mesurability of the function u₁ and its Schwarz rearrangementu^∗₁ gives

Z

Ω

|u₁(x)|^pdx= Z

B

|u^∗₁(x)|^pdx.

The Pòlya inequality implies that Z

Ω

|∇u₁(x)|^pdx≥ Z

B

|∇u^∗₁(x)|^pdx.

By the two conditions, it becomes R

B|∇u^∗₁(x)|^pdx R

B|u^∗₁(x)|^pdx ≤ R

Ω|u₁(x)|^pdx =λ₁(Ω).

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This implies that λ₁(B) = min

v∈W₀^1,p(B),v6=0

R

B|∇v(x)|^pdx R

B|v(x)|^pdx ≤ R

B|∇u^∗₁(x)|^pdx R

B|u^∗₁(x)|^pdx ≤λ₁(Ω).

Remark 1. The solution Ω must satisfy an optimality condition. We suppose that ΩC²−regular to compute the shape derivative. We deform the domain Ω with respect to an admissible vector fieldV to compute the shape derivative

dJ(Ω;V) = lim

t−→0

J(Id+tΩ)−J(Ω)

t .

We have the variation calculation

−div(|∇u|^p−2∇u) =λ|u|^p−2u

− Z

Ω

div(|∇u|^p−2∇u)φdx = Z

Ω

λ|u|^p−2uφdx, for all φ ∈ D(Ω) Z

Ω

|∇u|^p−2∇u∇φdx = Z

Ω

λ|u|^p−2uφdx, for all φ ∈ D(Ω) Let us takeJ(Ω) =R

Ω|∇u|^p−2∇u∇φdxandJ₁(Ω) =R

Ωλ|u|^p−2uφdx.We havedJ(Ω;V) =dJ₁(Ω;V).

We use the classical Hadamard formula to compute the Eulerian derivative of the functionalJ at the pointΩin the directionV.

dJ(Ω;V) = Z

Ω

(|∇u|^p−2∇u∇φ)⁰dx+ Z

Ω

div(|∇u|^p−2∇u∇φ.V(0))dx.

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We have Z

Ω

(|∇u|^p−2∇u∇φ)⁰dx= Z

Ω

(|∇u|^p−2)⁰∇u∇φdx +

Z

Ω

|∇u|^p−2∇u(∇φ)⁰dx+ Z

Ω

|∇u|^p−2(∇u)⁰∇φdx.

We have the expression

(|∇u|^p−2)⁰ =

(|∇u|²)^p−2² 0

= p−2

2 (|∇u|²)⁰(|∇u|²)^p−4² (|∇u|^p−2)⁰ = (p−2)∇u∇u⁰|∇u|^p−4. Then

dJ(Ω;V) = (p−2) Z

Ω

|∇u|^p−4|∇u|²∇u⁰∇φdx

− Z

Ω

div(|∇u|^p−2∇u)φ⁰dx− Z

Ω

div(|∇u|^p−2(∇u)⁰)φdx

dJ(Ω, V) = (p−2) Z

Ω

|∇u|^p−2∇u⁰∇φdx

− Z

Ω

div(|∇u|^p−2∇u)φ⁰dx− Z

Ω

div(|∇u|^p−2(∇u)⁰)φdx

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because Z

Ω

div(|∇u|^p−2∇u∇φ·V(0))dx= Z

∂Ω

|∇u|^p−2∇u∇φ·V(0)·νds= 0.

We obtain

dJ(Ω;V) =−(p−1) Z

Ω

div(|∇u|^p−2∇u⁰)φdx− Z

Ω

div(|∇u|^p−2∇u)φ⁰dx We have also

dJ₁(Ω;V) = Z

Ω

λ⁰|u|^p−2uφdx+ Z

Ω

λ|u|^p−2u⁰φdx +

Z

Ω

λ|u|^p−2uφ⁰dx+ (p−2) Z

Ω

λ|u|^p−2u⁰φdx, dJ₁(Ω;V) =

Z

Ω

λ|u|^p−2u⁰φdx dJ(Ω;V) =dJ₁(Ω, V)implies

−(p−1) Z

Ω

div(|∇u|^p−2∇u)φ⁰dx

= Z

Ω

λ|u|^p−2u⁰φdx.

By simplification we get

−(p−1) Z

Ω

div(|∇u|^p−2∇u)φ⁰dx

= Z

Ω

λ⁰|u|^p−2uφdx+ (p−1) Z

Ω

λ|u|^p−2u⁰φdx, for all φ∈ D(Ω).

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This implies that (4.1)

−(p−1)div(|∇u|^p−2∇u⁰) = λ⁰|u|^p−2u+ (p−1)λ|u|^p−2u⁰inD⁰(Ω) We multiply the equation (4.1) byuand by Green ’s formula we get

−(p−1) Z

Ω

div(|∇u|^p−2∇u)u⁰dx+ Z

∂Ω

|∇u|^p−2∇u·νu⁰ds

=λ⁰+ (p−1) Z

Ω

λ|u|^p−2uu⁰dx.

Finally we obtain the expression of

λ⁰(Ω;V) =−(p−1) Z

∂Ω

|∇u|^p∇u·νu⁰ds whereu⁰ satisfiesu⁰ =−^∂u_∂νV(0)·νon∂Ω.Then

λ⁰(Ω, V) =−(p−1) Z

∂Ω

|∇u|^pV ·νds.

We have a similar formula for the variation of the volumedJ₂(Ω, V) = R

∂ΩV · νds, where J₂(Ω) =R

Ωdx−c.

IfΩis an optimal domain then there exists a Lagrange multipliera <0such that

−(p−1) Z

∂Ω

|∇u|^pV ·νds =a Z

∂Ω

V ·νds.

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Then we obtain

|∇u|=

−a p−1

_p¹

on ∂Ω.

SinceΩisC²−regular andu= 0on∂Ω, then we get

−∂u

∂ν =

−a p−1

¹_p

on ∂Ω.

We are also interested the existence of a minimizer for the following problem min{λ₁(Ω),Ω∈ A,|Ω| ≤c},

whereAis a family of admissible domain defined by A={Ω⊆D,Ω is quasi open}

andλ₁(Ω)is defined by λ1(Ω) = min

φ∈W₀^1,p(Ω),φ6=0

R

Ω|∇φ(x)|^pdx R

Ω|φ(x)|^p = R

Ω|u₁(x)|^pdx .

The Sobolev space W₀^1,p(Ω) is seen as a closed subspace ofW₀^1,p(D) defined by

W₀^1,p(Ω) ={u∈W₀^1,p(D) :u= 0p−q.e on D\Ω}.

The problem is to look for weak topology constraints which would make the class Asequentially compact. This convergence is called weakγ_p-convergence for quasi open sets.

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Definition 4.2. We say that a sequence (Ω_n)ofAweaklyγ_p-converges toΩ∈ Aif the sequenceun converges weakly inW₀^1,p(D) to a functionu∈ W₀^1,p(D) (that we may take as quasi-continuous) such thatΩ ={u >0}.

We have the following theorem.

Theorem 4.3. The problem

(4.2) min{λ₁(Ω),Ω∈ A,|Ω| ≤c}

admits at least one solution.

Proof of Theorem4.3. Let us take λ₁(Ω_n) = min

φn∈W₀^1,p(Ωn),φn6=0

R

Ωn|∇φn(x)|^pdx R

Ωn|φ_n(x)|^pdx = R

Ωn|∇un(x)|^pdx R

Ωn|u_n(x)|^pdx . Suppose that (Ω_n)(n∈N) is a minimizing sequence of domain for the problem (4.2). We denote byu_na first eigenfunction onΩ_n,such thatR

Ωn|u_n(x)|^pdx = 1.

Sinceunis the first eigenfunction ofλ1(Ωn), un is strictly positive, cf [15], then the sequence(Ω_n)is defined byΩ_n ={u_n >0}.

If the sequence(u_n)is bounded inW₀^1,p(D),then there exists a subsequence still denoted byu_n such thatu_n converges weakly inW₀^1,p(D)to a functionu.

By compact injection, we have thatR

Ω|u(x)|^pdx= 1.

Let Ω be quasi open and defined byΩ = {u > 0}, this implies that u ∈ W₀^1,p(Ω).As the sequence(u_n)is bounded inW₀^1,p(D),then

lim inf

n→+∞

R

Ωn|∇u_n(x)|^pdx R

Ωn|u_n(x)|^pdx ≥ R

Ω|∇u(x)|^pdx R

Ω|u(x)|^pdx ≥ R

Ω|∇u1(x)|^pdx R

Ω|u₁(x)|^pdx =λ₁(Ω).

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Now we show that|Ω| ≤c.

We know that if the sequenceΩ_nweaklyγ_p- converges toΩand the Lebesgue measure is weakly γ_p-lower semicontinuous on the classA (see [5]), then we obtain|{u >0}| ≤lim inf

n→+∞ |{u_n >0}| ≤cthis implies that|Ω| ≤c.

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5. Domain in Box

Now let us take N = 2. We consider the class of admissible domains defined by

C ={Ω,Ω open subsets ofDand simply connected,|Ω|=c}.

• For p > 2, the p-capacity of a point is stricly positive and every W₀^1,p function has a continuous representative. For this reason, a property which holdsp−q.ewithp >2holds in fact everywhere. Forp >2,the domain Ω_p is a minimizer of the problemmin{λ^p₁(Ω_p),Ω_p ∈ C}.

Consider the sequence(Ω_p_n)⊆ C and assume thatΩ_p_nconverges in Haus- dorff complementary topology toΩ₂,whenp_ngoes to 2 andp_n>2.Then Ω₂ ⊆ C andΩ_p_n γ₂-converges toΩ₂.

By the Sobolev embedding theorem, we have W₀^1,pⁿ(Ω_p_n) ,→ H₀¹(Ω_p_n).

Theγ₂-convergence implies thatH₀¹(Ω_p_n)converges in the sense of Mosco toH₀¹(Ω2).Forpn>2,by the Hölder inequality we have

Z

|∇u_p_n|²dx ¹₂

≤ |Ω_p_n|¹²⁻^pn¹ Z

|∇u_p_n|^pⁿdx _pn¹

Z

|∇u_p_n|² ¹₂

dx≤c¹²⁻^pn¹ λ^p₁ⁿ(Ω_p_n).

Then the sequence(u_p_n)is uniformly bounded in H₀¹(Ω_p_n).There exists a subsequence still denotedu_p_n such thatu_p_n converges weakly inH₀¹(D) to a functionu.The second condition of Mosco implies thatu∈H₀¹(Ω2).

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Forp >2,we have the Sobolev embedding theoremW₀^1,p(D),→ C^0,α( ¯D).

Ascoli’s theorem implies thatu_p_n −→ u and ∇u_p_n −→ ∇ulocally uniformly inΩ₂,whenp_ngoes to 2 andp_n >2.

Now show that

pnlim−→2

Z

|u_p_n|²dx= 1 i.e.

Z

|u|²dx= 1.

For >0small, we havep_n>2−.Noting that Z

|∇u_p_n|²⁻dx ₂₋¹

≤ |Ω_p_n|²⁻¹ ⁻^pn¹ Z

|∇u_p_n|^pⁿdx _pn¹

=c²⁻¹ ⁻^pn¹ λ^p₁ⁿ(Ω_p_n),

this implies that the sequence upn is uniformly bounded inW₀^1,2−(Ωpn).

Then there exists a subsequence still denotedu_p_n such thatu_p_n is weakly convergent inH₀¹(D)tou.By the second condition of Mosco we getu∈ W₀^1,2−(Ω2).It follows that

Z

|u|²⁻dx= lim

pn−→2

Z

|u_p_n|²⁻dx

≤ lim

pn−→2|Ω_p_n|¹⁻²⁻^pn Z

|u_p_n|^pⁿdx ²⁻_pn

=c². Letting−→0,we obtain R

|u|²dx≤1.

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On the other hand, Lemma 4.2 of [14] implies that Z

|u|^pⁿdx≥ Z

|u_p_n|^pⁿdx+p_n Z

|u|^pⁿ⁻²dxu_p_n(u−u_p_n).

The second integral on the right-hand side approaches0aspn−→2.Thus we getR

|u|²dx≥1,and we conclude thatR

|u|²dx= 1.

In [11, Theorem 2.1 p. 3350],λ^p_kis continuous inpfork = 1,2,whereλ^p_k is thek−theigenvalue for thep-Laplacian operator.

We have (5.1)

Z

|∇upn|^pⁿ⁻²∇upn∇φdx

= Z

λ^p₁ⁿ|u_p_n|^pⁿ⁻²u_p_nφdx, for all φ∈ D(Ω₂).

Lettingp_ngo to2, p_n>2in (5.1), and noting thatu_p_nconverges uniformly touon the support ofφ,we obtain

Z

∇u∇φdx= Z

λ²₁uφdx, for all φ ∈ D(Ω₂), whence we have

−∆u = λ²₁u in D⁰(Ω2)

u = 0 on ∂Ω₂.

We conclude that whenp−→ 2andp > 2the free parts of the boundary ofΩpcannot be pieces of circle.

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• For p ≤ 2, we consider the sequence (Ω_p_n) ⊆ C and assume that Ω_p_n converges in Hausdorff complementary topology toΩ2,whenpngoes to 2 andp_n ≤2.Then by Theorem3.1, we getΩ₂ ⊆ C andΩ_p_nγ2−-converges toΩ₂.

In [16], the sequence (u_p_n) is bounded inW^1,2−(D), 0 < < 1 that is

∇u_p_n converges weakly inL²⁻(D)to ∇u andu_p_n converges strongly in L²⁻(D) tou.In [16], we get alsoR

|∇u|²dx≤β and R

|u|²dx <∞.

By Lemma 4.2 of [14], we have Z

|u|^pⁿdx≥ Z

|u_p_n|^pⁿdx+p_n Z

|u|^pⁿ⁻²dxu_p_n(u−u_p_n).

The second integral on the right-hand side approaches0aspn−→2.Thus we getR

|u|²dx≥1.This implies that

pnlim−→2

Z

|u_p_n|²⁻dx= Z

|u|²⁻dx= 1.

Letting−→0,we obtainR

|u|²dx= 1.

Theγ₂₋-convergence implies thatu_p_n converges strongly inW₀^1,2−(Ω₂) to u. According to P. Lindqvist see [16], we have u ∈ H¹(D), and we can deduce thatu ∈ H₀¹(Ω₂). As the first eigenvalue for thep-Laplacian operator is continuous inpcf [11], we have

(5.2) Z

|∇u_p_n|^pⁿ⁻²∇u_p_n∇φdx

= Z

λ^p₁ⁿ|u_p_n|^pⁿ⁻²u_p_nφdx, for all φ∈ D(Ω₂).

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Lettingp_ngo to2, p_n≤2in (5.2), and noting thatu_p_nconverges uniformly touon the support ofφ,we obtain

Z

∇u∇φdx= Z

λ²₁uφdx, for all φ ∈ D(Ω₂), whence we have

−∆u = λ²₁u in D⁰(Ω₂) u = 0 on ∂Ω₂.

We conclude that whenp −→ 2andp ≤ 2the free parts of the boundary ofΩpcannot be pieces of circle.

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References

[1] A. ANANE ANDN. TSOULI, The second eigenvalue of thep-Laplacian, Nonlinear Partial Differential Equations (Fès, 1994), Pitman Research Notes Math. Ser., 343, Longman, Harlow, (1996), 1–9.

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[3] D. BUCUR AND J.P. ZOLÉSIO, Wiener’s criterion and shape continuity for the Dirichlet problem, Boll. Un. Mat. Ital., B 11 (1997), 757–771.

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[8] G. DAL MASOANDA. DEFRANCESCHI, Limits of nonlinear Dirichlet problems in varying domains, Manuscripta Math., 61 (1988), 251–278.

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[9] G. FABER, Bewiw, dass unter allen homogenen Membranen von gleicher Fläche und gleicher Spannung die kreisförmige den tiefsten Grundton gibt, Sitz. Ber. Bayer. Akad. Wiss, (1923), 169–172.

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[13] E. KRAHN, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann., 94 (1924), 97–100.

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[18] J. SOKOLOWSKI AND J.P. ZOLESIO, Introduction to Shape Optimiza- tion. Shape Sensitivity Analysis. Springer series in Computational Mathe- matics, 16. Springer-Verlag, Berlin (1992).

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