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

**The First Eigenvalue for the**
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**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 conti- nuity 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.

**Contents**

1 Introduction. . . 3

2 Definition of the First and Second Eigenvalues. . . 6

3 Properties of the Geometric Variations. . . 7

4 Shape Optimization Result. . . 12

5 Domain in Box . . . 21 References

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**1.** **Introduction**

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

Consider the Sobolev spaceW_{0}^{1,p}(Ω)which is the closure ofC^{∞}functions com-
pactly supported inΩfor the norm

||u||^{p}_{1,p} =
Z

Ω

|u(x)|^{p}dx+
Z

Ω

|∇u(x)|^{p}dx.

Thep-Laplacian is the operator defined by

∆_{p} :W_{0}^{1,p}(Ω)−→W^{−1,q}(Ω)

u7−→∆_{p}u=div(|∇u|^{p−2}∇u),

where W^{−1,q}(Ω) is the dual space of W_{0}^{1,p}(Ω) and we have 1 < p, q < ∞,

1

p + ^{1}_{q} = 1.

We are interested in the nonlinear eigenvalue problem (1.1)

( −∆_{p}u−λ|u|^{p−2}u = 0inΩ,

u = 0on∂Ω.

Let u be a function ofW_{0}^{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−2}uφdx

for allφ ∈ C_{0}^{∞}(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^{p}_{1}(Ω) as the
first eigenvalue for the p-Laplacian operator. The aim of this paper is to study
the isoperimetric inequality

min{λ^{p}_{1}(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 min-
imization 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,Ω_{2} 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 spec-
ified, 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
λ_{1}(Ω) = min

φ∈W_{0}^{1,p}(Ω),φ6=0

R

Ω|∇φ(x)|^{p}dx
R

Ω|φ(x)|^{p} =
R

Ω|∇u_{1}(x)|^{p}dx
R

Ω|u_{1}(x)|^{p}dx ,

where the minimum is achieved by u_{1} which is a weak solution of the Euler-
Lagrange equation

(2.1)

−∆_{p}u−λ|u|^{p−2}u = 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]. Andu_{1}is the only positive eigenfunc-
tion for thep-Laplacian Dirichlet see also [15].

In [15], the second eigenvalue is defined by
λ_{2}(Ω) = inf

C∈C2

maxC

R

Ω|∇φ(x)|^{p}dx
R

Ω|φ(x)|^{p} ,
where

C_{2} :={C ∈W_{0}^{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 topol-
ogy, denotedH^{c} given by the metric

d_{H}^{c}(Ω^{c}_{1},Ω^{c}_{2}) = sup

x∈R^{N}

|d(x,Ω^{c}_{1})−d(x,Ω^{c}_{2})|.

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−→λ_{1}(Ω)fails, see [4].

In order to obtain a compactness result we impose some additional con- straints 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 set*K

*contained in a ball*B, cap(K, B) := inf

Z

B

|∇φ|^{p}, φ ∈ C_{0}^{∞}(B), φ ≥1 *on* K

.

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**Definition 3.2.**

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

*2. A set*Ω ⊂ R^{N} *is said to be quasi open if for every* > 0*there exists an*
*open set*Ω_{} *such that*Ω⊆Ω_{},*and*cap(Ω_{}\Ω)< .

*3. A function*u : R^{N} −→ R *is said*p-quasi continuous if for every > 0
*there exists an open set*Ω_{}*such that*cap(Ω_{})< *and*u_{|}_{R}_{\Ω}_{}*is continuous*
*in*R\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* Ω

*has 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
*where*B(x, δ)*denotes the ball of raduis*δ,*centred at*x.

* Definition 3.4. We say that the sequence of the spaces*W

_{0}

^{1,p}(Ω

_{n})

*converges in*

*the sense of Mosco to the space*W

_{0}

^{1,p}(Ω)

*if the following conditions hold*

*1. The first Mosco condition: For all* φ ∈ W_{0}^{1,p}(Ω),there exists a sequence
φn ∈W_{0}^{1,p}(Ωn)*such that*φn*converges strongly in*W_{0}^{1,p}(D)*to*φ.

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*2. The second Mosco condition: For every sequence*φ_{n}_{k} ∈W_{0}^{1,p}(Ω_{n}_{k})*weakly*
*convergent in*W_{0}^{1,p}(D)*to a function*φ,*we have*φ∈W_{0}^{1,p}(Ω).

* Definition 3.5. We say a sequence* (Ωn)

*of open subsets of a fixed ball*D γp

*-*

*converges to*Ω

*if for any*f ∈W

^{−1,q}(Ω)

*the solutions of the Dirichlet problem*

−∆_{p}u_{n} =f *in* Ω_{n}, u_{n}∈W_{0}^{1,p}(Ω_{n})

*converge strongly in*W_{0}^{1,p}(D),*as*n−→+∞, *to the solution of the correspond-*
*ing problem in*Ω,*see [7], [8].*

By Op−(r,c)(D), we denote the family of all open subsets of D which sat-
isfy 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 con-
straints 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 follow- ing theorem.

* Theorem 3.1 (Bucur-Trebeschi). Let*N ≥p > N−1.

*Consider the sequence*(Ω

_{n}) ⊆ O

_{l}(D)

*and assume that*Ω

_{n}

*converges in Hausdorff complementary*

*topology to*Ω.

*Then*Ω⊆ Ol(D)

*and*Ωn 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 to*Ω.

*Then*λ

_{1}(Ω

_{n})

*converges to*λ

_{1}(Ω).

*Proof of Theorem3.2. Let us take*

λ_{1}(Ω_{n}) = min

φn∈W_{0}^{1,p}(Ωn),φn6=0

R

Ωn|∇φ_{n}(x)|^{p}dx
R

Ωn|φ_{n}(x)|^{p} =
R

Ωn|∇u_{n}(x)|^{p}dx
R

Ωn|u_{n}(x)|^{p} ,
where the minimum is attained by u_{n},and

λ1(Ω) = min

φ∈W_{0}^{1,p}(Ω),φ6=0

R

Ω|∇φ(x)|^{p}dx
R

Ω|φ(x)|^{p} =
R

Ω|∇u_{1}(x)|^{p}dx
R

Ω|u_{1}(x)|^{p}dx ,
where the minimum is achieved byu_{1}.

By the Bucur and Trebeschi theorem,Ωn γp converges toΩ.This implies
W_{0}^{1,p}(Ω_{n})converges in the sense of Mosco toW_{0}^{1,p}(Ω).

If the sequence(u_{n})is bounded inW_{0}^{1,p}(D),then there exists a subsequence
still denotedu_{n}such thatu_{n}converges weakly inW_{0}^{1,p}(D)to a functionu.The
second condition of Mosco implies thatu∈W_{0}^{1,p}(Ω).

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Using the weak lower semicontinuity of theL^{p}−norm, we have the inequal-
ity

lim inf

n−→+∞

R

D|∇u_{n}(x)|^{p}dx
R

D|u_{n}(x)|^{p} ≥
R

Ω|∇u(x)|^{p}dx
R

Ω|u(x)|^{p} ≥
R

Ω|∇u_{1}(x)|^{p}dx
R

Ω|u_{1}(x)|^{p} ,
then

(3.1) lim inf

n→+∞λ_{1}(Ω_{n})≥λ_{1}(Ω).

Using the first condition of Mosco, there exists a sequence (v_{n}) ∈ W_{0}^{1,p}(Ω_{n})
such thatv_{n}converges strongly inW_{0}^{1,p}(D)to u_{1}.

We have

λ_{1}(Ω_{n})≤
R

D|∇v_{n}(x)|^{p}dx
R

D|v_{n}(x)|^{p}
this implies that

lim sup

n−→+∞

λ_{1}(Ω_{n})≤lim sup

n−→+∞

R

D|∇vn(x)|^{p}dx
R

D|v_{n}(x)|^{p}

= lim

n−→+∞

R

D|∇v_{n}(x)|^{p}dx
R

D|vn(x)|^{p} =
R

Ω|∇u_{1}(x)|^{p}dx
R

Ω|u1(x)|^{p}
then

(3.2) lim sup

n→+∞

λ_{1}(Ω_{n})≤λ_{1}(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. Let*Ω

*be an open subset and bounded in*R

^{N}.

*We denote by*B

*the ball centred at the origin with the same volume as*Ω.

*Let*u

*be a non negative*

*function in*Ω,

*which vanishes on*∂Ω.

*For all*c >0,

*the set*{x∈ Ω, u(x) > c}

*is called the level set of*u.

*The function*u^{∗} *which has the following level set*

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

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

We have the following lemma.

* Lemma 4.1. Let*Ω

*be an open subset in*R

^{N}

*.*

*Let*ψ

*be any continuous function on*R

^{∗}+,

*we have*

*1.* R

Ωψ(u(x))dx =R

Ω^{∗}ψ(u^{∗}(x))dx u^{∗} *is equi-mesurable with*u.

*2.* R

Ωu(x)v(x)dx≤R

Ω^{∗}u^{∗}(x)v^{∗}(x)dx.

*3. If*u∈W_{0}^{1,p}(Ω), p > 1 *then*u^{∗} ∈W_{0}^{1,p}(Ω^{∗})*and*
Z

Ω

|∇u(x)|^{p}dx≥
Z

Ω^{∗}

|∇u^{∗}(x)|^{p}dx *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. Let*B

*be the ball of the same volume as*Ω,

*then*λ

_{1}(B) = min{λ

_{1}(Ω),Ω

*open set of*R

^{N},|Ω|=|B|}.

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

Z

Ω

|u_{1}(x)|^{p}dx=
Z

B

|u^{∗}_{1}(x)|^{p}dx.

The Pòlya inequality implies that Z

Ω

|∇u_{1}(x)|^{p}dx≥
Z

B

|∇u^{∗}_{1}(x)|^{p}dx.

By the two conditions, it becomes R

B|∇u^{∗}_{1}(x)|^{p}dx
R

B|u^{∗}_{1}(x)|^{p}dx ≤
R

Ω|∇u_{1}(x)|^{p}dx
R

Ω|u_{1}(x)|^{p}dx =λ_{1}(Ω).

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This implies that
λ_{1}(B) = min

v∈W_{0}^{1,p}(B),v6=0

R

B|∇v(x)|^{p}dx
R

B|v(x)|^{p}dx ≤
R

B|∇u^{∗}_{1}(x)|^{p}dx
R

B|u^{∗}_{1}(x)|^{p}dx ≤λ_{1}(Ω).

* Remark 1. The solution* Ω

*must satisfy an optimality condition. We suppose*

*that*ΩC

^{2}−

*regular to compute the shape derivative. We deform the domain*Ω

*with respect to an admissible vector field*V

*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−2}u

− Z

Ω

div(|∇u|^{p−2}∇u)φdx =
Z

Ω

λ|u|^{p−2}uφdx, *for all* φ ∈ D(Ω)
Z

Ω

|∇u|^{p−2}∇u∇φdx =
Z

Ω

λ|u|^{p−2}uφdx, *for all* φ ∈ D(Ω)
*Let us take*J(Ω) =R

Ω|∇u|^{p−2}∇u∇φdx*and*J_{1}(Ω) =R

Ωλ|u|^{p−2}uφdx.*We*
*have*dJ(Ω;V) =dJ_{1}(Ω;V).

*We use the classical Hadamard formula to compute the Eulerian derivative*
*of the functional*J *at the point*Ω*in the direction*V.

dJ(Ω;V) = Z

Ω

(|∇u|^{p−2}∇u∇φ)^{0}dx+
Z

Ω

div(|∇u|^{p−2}∇u∇φ.V(0))dx.

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*We have*
Z

Ω

(|∇u|^{p−2}∇u∇φ)^{0}dx=
Z

Ω

(|∇u|^{p−2})^{0}∇u∇φdx
+

Z

Ω

|∇u|^{p−2}∇u(∇φ)^{0}dx+
Z

Ω

|∇u|^{p−2}(∇u)^{0}∇φdx.

*We have the expression*

(|∇u|^{p−2})^{0} =

(|∇u|^{2})^{p−2}^{2} 0

= p−2

2 (|∇u|^{2})^{0}(|∇u|^{2})^{p−4}^{2}
(|∇u|^{p−2})^{0} = (p−2)∇u∇u^{0}|∇u|^{p−4}.
*Then*

dJ(Ω;V) = (p−2) Z

Ω

|∇u|^{p−4}|∇u|^{2}∇u^{0}∇φdx

− Z

Ω

div(|∇u|^{p−2}∇u)φ^{0}dx−
Z

Ω

div(|∇u|^{p−2}(∇u)^{0})φdx

dJ(Ω, V) = (p−2) Z

Ω

|∇u|^{p−2}∇u^{0}∇φdx

− Z

Ω

div(|∇u|^{p−2}∇u)φ^{0}dx−
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}∇u^{0})φdx−
Z

Ω

div(|∇u|^{p−2}∇u)φ^{0}dx
*We have also*

dJ_{1}(Ω;V) =
Z

Ω

λ^{0}|u|^{p−2}uφdx+
Z

Ω

λ|u|^{p−2}u^{0}φdx
+

Z

Ω

λ|u|^{p−2}uφ^{0}dx+ (p−2)
Z

Ω

λ|u|^{p−2}u^{0}φdx,
dJ_{1}(Ω;V) =

Z

Ω

λ^{0}|u|^{p−2}uφdx+
Z

Ω

λ|u|^{p−2}uφ^{0}dx+ (p−1)
Z

Ω

λ|u|^{p−2}u^{0}φdx
dJ(Ω;V) =dJ_{1}(Ω, V)*implies*

−(p−1) Z

Ω

div(|∇u|^{p−2}∇u^{0})φdx−
Z

Ω

div(|∇u|^{p−2}∇u)φ^{0}dx

= Z

Ω

λ^{0}|u|^{p−2}uφdx+
Z

Ω

λ|u|^{p−2}uφ^{0}dx+ (p−1)
Z

Ω

λ|u|^{p−2}u^{0}φdx.

*By simplification we get*

−(p−1) Z

Ω

div(|∇u|^{p−2}∇u^{0})φdx−
Z

Ω

div(|∇u|^{p−2}∇u)φ^{0}dx

= Z

Ω

λ^{0}|u|^{p−2}uφdx+ (p−1)
Z

Ω

λ|u|^{p−2}u^{0}φdx, *for all* φ∈ D(Ω).

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*This implies that*
(4.1)

−(p−1)div(|∇u|^{p−2}∇u^{0}) = λ^{0}|u|^{p−2}u+ (p−1)λ|u|^{p−2}u^{0}*in*D^{0}(Ω)
*We multiply the equation (4.1) by*u*and by Green ’s formula we get*

−(p−1) Z

Ω

div(|∇u|^{p−2}∇u)u^{0}dx+
Z

∂Ω

|∇u|^{p−2}∇u·νu^{0}ds

=λ^{0}+ (p−1)
Z

Ω

λ|u|^{p−2}uu^{0}dx.

*Finally we obtain the expression of*

λ^{0}(Ω;V) =−(p−1)
Z

∂Ω

|∇u|^{p}∇u·νu^{0}ds
*where*u^{0} *satisfies*u^{0} =−^{∂u}_{∂ν}V(0)·ν*on*∂Ω.*Then*

λ^{0}(Ω, V) =−(p−1)
Z

∂Ω

|∇u|^{p}V ·νds.

*We have a similar formula for the variation of the volume*dJ_{2}(Ω, V) = R

∂ΩV ·
νds, *where* J_{2}(Ω) =R

Ωdx−c.

*If*Ω*is an optimal domain then there exists a Lagrange multiplier*a <0*such*
*that*

−(p−1) Z

∂Ω

|∇u|^{p}V ·νds =a
Z

∂Ω

V ·νds.

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*Then we obtain*

|∇u|=

−a p−1

_{p}^{1}

*on* ∂Ω.

*Since*Ω*is*C^{2}−regular andu= 0*on*∂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{λ_{1}(Ω),Ω∈ A,|Ω| ≤c},

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

andλ_{1}(Ω)is defined by
λ1(Ω) = min

φ∈W_{0}^{1,p}(Ω),φ6=0

R

Ω|∇φ(x)|^{p}dx
R

Ω|φ(x)|^{p} =
R

Ω|∇u_{1}(x)|^{p}dx
R

Ω|u_{1}(x)|^{p}dx .

The Sobolev space W_{0}^{1,p}(Ω) is seen as a closed subspace ofW_{0}^{1,p}(D) defined
by

W_{0}^{1,p}(Ω) ={u∈W_{0}^{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})

*of*A

*weakly*γ

_{p}

*-converges to*Ω∈ A

*if the sequence*un

*converges weakly in*W

_{0}

^{1,p}(D)

*to a function*u∈ W

_{0}

^{1,p}(D)

*(that we may take as quasi-continuous) such that*Ω ={u >0}.

We have the following theorem.

**Theorem 4.3. The problem**

(4.2) min{λ_{1}(Ω),Ω∈ A,|Ω| ≤c}

*admits at least one solution.*

*Proof of Theorem4.3. Let us take*
λ_{1}(Ω_{n}) = min

φn∈W_{0}^{1,p}(Ωn),φn6=0

R

Ωn|∇φn(x)|^{p}dx
R

Ωn|φ_{n}(x)|^{p}dx =
R

Ωn|∇un(x)|^{p}dx
R

Ωn|u_{n}(x)|^{p}dx .
Suppose that (Ω_{n})(n∈N) is a minimizing sequence of domain for the problem
(4.2). We denote byu_{n}a first eigenfunction onΩ_{n},such thatR

Ωn|u_{n}(x)|^{p}dx =
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 inW_{0}^{1,p}(D),then there exists a subsequence
still denoted byu_{n} such thatu_{n} converges weakly inW_{0}^{1,p}(D)to a functionu.

By compact injection, we have thatR

Ω|u(x)|^{p}dx= 1.

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

lim inf

n→+∞

R

Ωn|∇u_{n}(x)|^{p}dx
R

Ωn|u_{n}(x)|^{p}dx ≥
R

Ω|∇u(x)|^{p}dx
R

Ω|u(x)|^{p}dx ≥
R

Ω|∇u1(x)|^{p}dx
R

Ω|u_{1}(x)|^{p}dx =λ_{1}(Ω).

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Now we show that|Ω| ≤c.

We know that if the sequenceΩ_{n}weaklyγ_{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 W_{0}^{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^{p}_{1}(Ω_{p}),Ω_{p} ∈ C}.

Consider the sequence(Ω_{p}_{n})⊆ C and assume thatΩ_{p}_{n}converges in Haus-
dorff complementary topology toΩ_{2},whenp_{n}goes to 2 andp_{n}>2.Then
Ω_{2} ⊆ C andΩ_{p}_{n} γ_{2}-converges toΩ_{2}.

By the Sobolev embedding theorem, we have W_{0}^{1,p}^{n}(Ω_{p}_{n}) ,→ H_{0}^{1}(Ω_{p}_{n}).

Theγ_{2}-convergence implies thatH_{0}^{1}(Ω_{p}_{n})converges in the sense of Mosco
toH_{0}^{1}(Ω2).Forpn>2,by the Hölder inequality we have

Z

|∇u_{p}_{n}|^{2}dx
^{1}_{2}

≤ |Ω_{p}_{n}|^{1}^{2}^{−}^{pn}^{1}
Z

|∇u_{p}_{n}|^{p}^{n}dx
_{pn}^{1}

Z

|∇u_{p}_{n}|^{2}
^{1}_{2}

dx≤c^{1}^{2}^{−}^{pn}^{1} λ^{p}_{1}^{n}(Ω_{p}_{n}).

Then the sequence(u_{p}_{n})is uniformly bounded in H_{0}^{1}(Ω_{p}_{n}).There exists
a subsequence still denotedu_{p}_{n} such thatu_{p}_{n} converges weakly inH_{0}^{1}(D)
to a functionu.The second condition of Mosco implies thatu∈H_{0}^{1}(Ω2).

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

Ascoli’s theorem implies thatu_{p}_{n} −→ u and ∇u_{p}_{n} −→ ∇ulocally uni-
formly inΩ_{2},whenp_{n}goes to 2 andp_{n} >2.

Now show that

pnlim−→2

Z

|u_{p}_{n}|^{2}dx= 1 i.e.

Z

|u|^{2}dx= 1.

For >0small, we havep_{n}>2−.Noting that
Z

|∇u_{p}_{n}|^{2−}dx
_{2−}^{1}

≤ |Ω_{p}_{n}|^{2−}^{1} ^{−}^{pn}^{1}
Z

|∇u_{p}_{n}|^{p}^{n}dx
_{pn}^{1}

=c^{2−}^{1} ^{−}^{pn}^{1} λ^{p}_{1}^{n}(Ω_{p}_{n}),

this implies that the sequence upn is uniformly bounded inW_{0}^{1,2−}(Ωpn).

Then there exists a subsequence still denotedu_{p}_{n} such thatu_{p}_{n} is weakly
convergent inH_{0}^{1}(D)tou.By the second condition of Mosco we getu∈
W_{0}^{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}|^{p}^{n}dx
^{2−}_{pn}

=c^{}^{2}.
Letting−→0,we obtain R

|u|^{2}dx≤1.

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

|u|^{p}^{n}dx≥
Z

|u_{p}_{n}|^{p}^{n}dx+p_{n}
Z

|u|^{p}^{n}^{−2}dxu_{p}_{n}(u−u_{p}_{n}).

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

|u|^{2}dx≥1,and we conclude thatR

|u|^{2}dx= 1.

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

We have (5.1)

Z

|∇upn|^{p}^{n}^{−2}∇upn∇φdx

= Z

λ^{p}_{1}^{n}|u_{p}_{n}|^{p}^{n}^{−2}u_{p}_{n}φdx, for all φ∈ D(Ω_{2}).

Lettingp_{n}go to2, p_{n}>2in (5.1), and noting thatu_{p}_{n}converges uniformly
touon the support ofφ,we obtain

Z

∇u∇φdx= Z

λ^{2}_{1}uφdx, for all φ ∈ D(Ω_{2}),
whence we have

−∆u = λ^{2}_{1}u in D^{0}(Ω2)

u = 0 on ∂Ω_{2}.

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 getΩ_{2} ⊆ C andΩ_{p}_{n}γ2−-converges
toΩ_{2}.

In [16], the sequence (u_{p}_{n}) is bounded inW^{1,2−}(D), 0 < < 1 that is

∇u_{p}_{n} converges weakly inL^{2−}(D)to ∇u andu_{p}_{n} converges strongly in
L^{2−}(D) tou.In [16], we get alsoR

|∇u|^{2}dx≤β and R

|u|^{2}dx <∞.

By Lemma 4.2 of [14], we have Z

|u|^{p}^{n}dx≥
Z

|u_{p}_{n}|^{p}^{n}dx+p_{n}
Z

|u|^{p}^{n}^{−2}dxu_{p}_{n}(u−u_{p}_{n}).

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

|u|^{2}dx≥1.This implies that

pnlim−→2

Z

|u_{p}_{n}|^{2−}dx=
Z

|u|^{2−}dx= 1.

Letting−→0,we obtainR

|u|^{2}dx= 1.

Theγ_{2−}-convergence implies thatu_{p}_{n} converges strongly inW_{0}^{1,2−}(Ω_{2})
to u. According to P. Lindqvist see [16], we have u ∈ H^{1}(D), and we
can deduce thatu ∈ H_{0}^{1}(Ω_{2}). As the first eigenvalue for thep-Laplacian
operator is continuous inpcf [11], we have

(5.2) Z

|∇u_{p}_{n}|^{p}^{n}^{−2}∇u_{p}_{n}∇φdx

= Z

λ^{p}_{1}^{n}|u_{p}_{n}|^{p}^{n}^{−2}u_{p}_{n}φdx, for all φ∈ D(Ω_{2}).

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Lettingp_{n}go to2, p_{n}≤2in (5.2), and noting thatu_{p}_{n}converges uniformly
touon the support ofφ,we obtain

Z

∇u∇φdx= Z

λ^{2}_{1}uφdx, for all φ ∈ D(Ω_{2}),
whence we have

−∆u = λ^{2}_{1}u in D^{0}(Ω_{2})
u = 0 on ∂Ω_{2}.

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

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[1] A. ANANE ANDN. TSOULI, The second eigenvalue of thep-Laplacian,
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[9] G. FABER, Bewiw, dass unter allen homogenen Membranen von gleicher
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