The First Eigenvalue for the $P$-Laplacian Operator

Journal of Inequalities in Pure and Applied Mathematics http://jipam.vu.edu.au/

Volume 6, Issue 3, Article 91, 2005

THE FIRST EIGENVALUE FOR THE p-LAPLACIAN OPERATOR IDRISSA LY

LABORATOIRE DE MATHÉMATIQUES ET APPLICATIONS DE METZ ILEDU SAULCY, 57045 METZ CEDEX 01, FRANCE. [email protected]

Received 12 February, 2005; accepted 17 June, 2005 Communicated by S.S. Dragomir

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 a p-Laplacian Dirichlet problem. We are also interested in the minimization of the ﬁrst eigenvalue of the p-Laplacian with Dirichlet boundary conditions among open sets and quasi open sets of given measure.

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

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

1. INTRODUCTION Let Ω be an open subset of a ﬁxed ball D in RN ,N ≥ 2 and 1 < p < +∞. Consider the 1,p ∞ Sobolev space W0 (Ω) which is the closure of C functions compactly supported in Ω for the norm Z Z p p p ||u||1,p = |u(x)| dx + |∇u(x)| dx. Ω Ω The p-Laplacian is the operator deﬁned by

1,p −1,q ∆p : W0 (Ω) −→ W (Ω) p−2 u 7−→ ∆pu = div(|∇u| ∇u),

−1,q 1,p 1 1 where W (Ω) is the dual space of W0 (Ω) and we have 1 < p, q < ∞, p + q = 1. We are interested in the nonlinear eigenvalue problem ( p−2 −∆pu − λ|u| u = 0 in Ω, (1.1) u = 0 on ∂Ω.

1,p Let u be a function of W0 (Ω), not identically 0. The function u is called an eigenfunction if Z Z |∇u(x)|p−2∇u∇φdx = λ |u(x)|p−2uφdx Ω Ω ∞ for all φ ∈ C0 (Ω). The corresponding real number λ is called an eigenvalue. 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. N p Let D be a bounded domain in R and c > 0. Let us denote λ1(Ω) as the first eigenvalue for the p-Laplacian operator. The aim of this paper is to study the isoperimetric inequality p min{λ1(Ω), Ω ⊆ D and |Ω| = c} and its continuous dependance with respect to the domain. We extend the Rayleigh-Faber- Khran inequality to the p-Laplacian operator and study the minimization of the first eigenvalue in two dimensions when D is a box. By considering a class of simply connected domains, we study the stability of the minimizer Ωp of the first eigenvalue with respect to p that is if Ωp is a minimizer of the first eigenvalue for the p-Laplacian Dirichlet, when p goes to 2, Ω2 is also a minimizer of the first eigenvalue of the Laplacian Dirichlet. Thus we will give a formal justification of the following conjecture: "Ω is a minimizer of given volume c, contained in a fixed box D and if D is too small to contain a ball of the same volume as Ω. Are the free parts of the boundary of Ω pieces of circle?" Henrot and Oudet solved this question and proved by using the Hölmgren uniqueness theorem, that the free part of the boundary of Ω cannot 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) sets of given volume. In the fourth part we discuss the minimization of the first eigenvalue in a box in two dimensions. Let D be a bounded open set in RN which contains all the open (or, if specified, quasi open) subsets used.

2. DEFINITION OF THE FIRSTAND SECOND EIGENVALUES The ﬁrst eigenvalue is deﬁned by the nonlinear Rayleigh quotient R p R p Ω |∇φ(x)| dx Ω |∇u1(x)| dx λ1(Ω) = min = , 1,p R |φ(x)|p R |u (x)|pdx φ∈W0 (Ω),φ6=0 Ω Ω 1

where the minimum is achieved by u1 which is a weak solution of the Euler-Lagrange equation −∆ u − λ|u|p−2u = 0 in Ω (2.1) p u = 0 on ∂Ω. The ﬁrst eigenvalue has many special properties, it is strictly positive, simple in any bounded connected domain see [15]. And u1 is the only positive eigenfunction for the p-Laplacian Dirichlet see also [15]. In [15], the second eigenvalue is deﬁned by R p Ω |∇φ(x)| dx λ2(Ω) = inf max , C∈C C R p 2 Ω |φ(x)| where 1,p C2 := {C ∈ W0 (Ω) : C = −C such that genus(C) ≥ 2}.

J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005 http://jipam.vu.edu.au/ THE FIRST EIGENVALUE FOR THE p-LAPLACIAN OPERATOR 3

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

3. PROPERTIESOFTHE GEOMETRIC VARIATIONS In this section we are interested in the continuity of the map

Ω 7−→ λ1(Ω). Then, we have to fix topology on the space of the open subsets of D. On the family of the open subsets of D, we define the Hausdorff complementary topology, denoted Hc given by the metric c c c c dHc (Ω1, Ω2) = sup |d(x, Ω1) − d(x, Ω2)|. N x∈R The Hc-topology has some good properties for example the space of the open subsets of D is Hc compact. Moreover if Ωn → Ω, then for any compact K ⊂⊂ Ω we have K ⊂⊂ Ωn for n large enough. However, perturbations in this topology may be very irregular and in general situations the continuity of the mapping Ω 7−→ 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 set K contained in a ball B, Z p ∞ cap(K,B) := inf |∇φ| , φ ∈ C0 (B), φ ≥ 1 on K . B 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. N (2) A set Ω ⊂ R 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 : RN −→ 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(RN ) has a p-quasi continuous representative which we still denote u. Therefore, level sets of Sobolev functions are p-quasi open sets; in particular Ωv = {x ∈ D; |v(x)| > 0} is quasi open subsets of D. Definition 3.3. We say that an open set Ω has the p − (r, c) capacity density condition if cap(Ωc ∩ B¯(x, δ),B(x, 2δ)) ∀x ∈ ∂Ω, ∀0 < δ < r, ≥ c cap(B¯(x, δ),B(x, 2δ)) where B(x, δ) denotes the ball of raduis δ, centred at x.

1,p Deﬁnition 3.4. We say that the sequence of the spaces W0 (Ωn) converges in the sense of 1,p Mosco to the space W0 (Ω) if the following conditions hold 1,p 1,p (1) The ﬁrst Mosco condition: For all φ ∈ W0 (Ω),there exists a sequence φn ∈ W0 (Ωn) 1,p such that φn converges strongly in W0 (D) to φ. 1,p (2) The second Mosco condition: For every sequence φnk ∈ W0 (Ωnk ) weakly convergent 1,p 1,p in W0 (D) to a function φ, we have φ ∈ W0 (Ω).

J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005 http://jipam.vu.edu.au/ 4 IDRISSA LY

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 1,p −∆pun = f in Ωn, un ∈ W0 (Ωn) 1,p converge strongly in W0 (D), as n −→ +∞, to the solution of the corresponding problem in Ω, 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 Hc 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 γp-compactness result for the p-Laplacian. In the same way, they extend the continuity result of Šveràk [19] to the p-Laplacian for p ∈ (N − 1,N],N ≥ 2. The reason of the choice of p is that in RN the curves have p positive capacity if p > N − 1. The case p > N is trivial since all functions in W 1,p(RN ) are continuous. Let us denote by c Ol(D) = {Ω ⊆ D,]Ω ≤ l} where ] denotes the number of the connected components. We have the following theorem.

Theorem 3.1 (Bucur-Trebeschi). Let N ≥ p > N − 1. Consider the sequence (Ωn) ⊆ Ol(D) and assume that Ωn converges in Hausdorff complementary topology to Ω. Then Ω ⊆ Ol(D) and Ωn γp−converges to Ω. Proof of Theorem 3.1. See [2]. For N = 2 and p = 2, Theorem 3.1 becomes 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 (Ωn) ⊆ Ol(D). Assume that Ωn converges in Hausdorff complementary topology to Ω. Then λ1(Ωn) converges to λ1(Ω). Proof of Theorem 3.2. Let us take R p R p |∇φn(x)| dx |∇un(x)| dx λ (Ω ) = min Ωn = Ωn , 1 n R p R p φ ∈W 1,p(Ω ),φ 6=0 |φ (x)| |u (x)| n 0 n n Ωn n Ωn n

where the minimum is attained by un, and R p R p Ω |∇φ(x)| dx Ω |∇u1(x)| dx λ1(Ω) = min = , 1,p R |φ(x)|p R |u (x)|pdx φ∈W0 (Ω),φ6=0 Ω Ω 1

where the minimum is achieved by u1. 1,p By the Bucur and Trebeschi theorem, Ωn γp converges to Ω. This implies W0 (Ωn) con- 1,p verges in the sense of Mosco to W0 (Ω). 1,p If the sequence (un) is bounded in W0 (D), then there exists a subsequence still denoted 1,p un such that un converges weakly in W0 (D) to a function u. The second condition of Mosco 1,p implies that u ∈ W0 (Ω). Using the weak lower semicontinuity of the Lp−norm, we have the inequality R |∇u (x)|pdx R |∇u(x)|pdx R |∇u (x)|pdx lim inf D n ≥ Ω ≥ Ω 1 , n−→+∞ R p R p R p D |un(x)| Ω |u(x)| Ω |u1(x)| then

(3.1) lim inf λ1(Ωn) ≥ λ1(Ω). n→+∞

J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005 http://jipam.vu.edu.au/ THE FIRST EIGENVALUE FOR THE p-LAPLACIAN OPERATOR 5

1,p Using the ﬁrst condition of Mosco, there exists a sequence (vn) ∈ W0 (Ωn) such that vn 1,p converges strongly in W0 (D) to u1. We have R |∇v (x)|pdx λ (Ω ) ≤ D n 1 n R p D |vn(x)| this implies that R |∇v (x)|pdx lim sup λ (Ω ) ≤ lim sup D n 1 n R p n−→+∞ n−→+∞ D |vn(x)| R |∇v (x)|pdx = lim D n n−→+∞ R p D |vn(x)| R |∇u (x)|pdx = Ω 1 R p Ω |u1(x)| then

(3.2) lim sup λ1(Ωn) ≤ λ1(Ω). n→+∞

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

4. SHAPE OPTIMIZATION RESULT We extend the classical inequality of Faber-Krahn for the first eigenvalue of the Dirichlet Laplacian to the Dirichlet p-Laplacian. We study this inequality when Ω is a quasi open subset of D. Definition 4.1. Let Ω be an open subset and bounded in RN . 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 RN . ∗ Let ψ be any continuous function on R+, we have R R ∗ ∗ (1) Ω ψ(u(x))dx = Ω∗ ψ(u (x))dx u is equi-mesurable with u. R R ∗ ∗ (2) Ω u(x)v(x)dx ≤ Ω∗ u (x)v (x)dx. 1,p ∗ 1,p ∗ (3) If u ∈ W0 (Ω), p > 1 then u ∈ W0 (Ω ) and Z Z |∇u(x)|pdx ≥ |∇u∗(x)|pdx Pòlya inequality. Ω Ω∗ Proof of Lemma 4.1. 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 the p-Laplacian operator. N Let Ω be any open set in R with finite measure. We denote by λ1(Ω) the first eigenvalue for the p-Laplacian operator with Dirichlet boundary conditions. We have the following theorem.

J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005 http://jipam.vu.edu.au/ 6 IDRISSA LY

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

Proof of Theorem 4.2. Let u1 be the ﬁrst eigenfunction of λ1(Ω), it is strictly positive see [15]. ∗ By Lemma 4.1, equi-mesurability of the function u1 and its Schwarz rearrangement u1 gives Z Z p ∗ p |u1(x)| dx = |u1(x)| dx. Ω B The Pòlya inequality implies that Z Z p ∗ p |∇u1(x)| dx ≥ |∇u1(x)| dx. Ω B By the two conditions, it becomes R |∇u∗(x)|pdx R |∇u (x)|pdx B 1 ≤ Ω 1 = λ (Ω). R ∗ p R p 1 B |u1(x)| dx Ω |u1(x)| dx This implies that R p R ∗ p B |∇v(x)| dx B |∇u1(x)| dx λ1(B) = min ≤ ≤ λ1(Ω). 1,p R |v(x)|pdx R |u∗(x)|pdx v∈W0 (B),v6=0 B B 1 Remark 4.3. 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 ﬁeld V to compute the shape derivative J(Id + tΩ) − J(Ω) dJ(Ω; V ) = lim . t−→0 t We have the variation calculation −div(|∇u|p−2∇u) = λ|u|p−2u Z Z − div(|∇u|p−2∇u)φdx = λ|u|p−2uφdx, for all φ ∈ D(Ω) Ω Ω Z Z |∇u|p−2∇u∇φdx = λ|u|p−2uφdx, for all φ ∈ D(Ω) Ω Ω R p−2 R p−2 Let us take J(Ω) = Ω |∇u| ∇u∇φdx and J1(Ω) = Ω λ|u| uφdx. We have dJ(Ω; V ) = dJ1(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. Z Z dJ(Ω; V ) = (|∇u|p−2∇u∇φ)0dx + div(|∇u|p−2∇u∇φ.V (0))dx. Ω Ω We have Z (|∇u|p−2∇u∇φ)0dx Ω Z Z Z = (|∇u|p−2)0∇u∇φdx + |∇u|p−2∇u(∇φ)0dx + |∇u|p−2(∇u)0∇φdx. Ω Ω Ω

J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005 http://jipam.vu.edu.au/ THE FIRST EIGENVALUE FOR THE p-LAPLACIAN OPERATOR 7

We have the expression 0 p−2 0 2 p−2 (|∇u| ) = (|∇u| ) 2

p − 2 2 0 2 p−4 = (|∇u| ) (|∇u| ) 2 2 (|∇u|p−2)0 = (p − 2)∇u∇u0|∇u|p−4. Then Z dJ(Ω; V ) = (p − 2) |∇u|p−4|∇u|2∇u0∇φdx Ω Z Z − div(|∇u|p−2∇u)φ0dx − div(|∇u|p−2(∇u)0)φdx Ω Ω Z dJ(Ω,V ) = (p − 2) |∇u|p−2∇u0∇φdx Ω Z Z − div(|∇u|p−2∇u)φ0dx − div(|∇u|p−2(∇u)0)φdx Ω Ω because Z Z div(|∇u|p−2∇u∇φ · V (0))dx = |∇u|p−2∇u∇φ · V (0) · νds = 0. Ω ∂Ω We obtain Z Z dJ(Ω; V ) = −(p − 1) div(|∇u|p−2∇u0)φdx − div(|∇u|p−2∇u)φ0dx Ω Ω We have also Z Z 0 p−2 p−2 0 dJ1(Ω; V ) = λ |u| uφdx + λ|u| u φdx Ω Ω Z Z + λ|u|p−2uφ0dx + (p − 2) λ|u|p−2u0φdx, Ω Ω Z Z Z 0 p−2 p−2 0 p−2 0 dJ1(Ω; V ) = λ |u| uφdx + λ|u| uφ dx + (p − 1) λ|u| u φdx Ω Ω Ω dJ(Ω; V ) = dJ1(Ω,V ) implies Z Z − (p − 1) div(|∇u|p−2∇u0)φdx − div(|∇u|p−2∇u)φ0dx Ω Ω Z Z Z = λ0|u|p−2uφdx + λ|u|p−2uφ0dx + (p − 1) λ|u|p−2u0φdx. Ω Ω Ω By simpliﬁcation we get Z Z − (p − 1) div(|∇u|p−2∇u0)φdx − div(|∇u|p−2∇u)φ0dx Ω Ω Z Z = λ0|u|p−2uφdx + (p − 1) λ|u|p−2u0φdx, for all φ ∈ D(Ω). Ω Ω This implies that (4.1) −(p − 1)div(|∇u|p−2∇u0) = λ0|u|p−2u + (p − 1)λ|u|p−2u0 in D0(Ω)

J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005 http://jipam.vu.edu.au/ 8 IDRISSA LY

We multiply the equation (4.1) by u and by Green ’s formula we get Z Z Z −(p−1) div(|∇u|p−2∇u)u0dx + |∇u|p−2∇u · νu0ds = λ0 +(p−1) λ|u|p−2uu0dx. Ω ∂Ω Ω Finally we obtain the expression of Z λ0(Ω; V ) = −(p − 1) |∇u|p∇u · νu0ds ∂Ω 0 0 ∂u where u satisﬁes u = − ∂ν V (0) · ν on ∂Ω. Then Z λ0(Ω,V ) = −(p − 1) |∇u|pV · νds. ∂Ω R We have a similar formula for the variation of the volume dJ2(Ω,V ) = V · νds, where R ∂Ω J2(Ω) = Ω dx − c. If Ω is an optimal domain then there exists a Lagrange multiplier a < 0 such that Z Z −(p − 1) |∇u|pV · νds = a V · νds. ∂Ω ∂Ω Then we obtain 1 −a p |∇u| = on ∂Ω. p − 1 Since Ω is C2−regular and u = 0 on ∂Ω, then we get 1 ∂u −a p − = on ∂Ω. ∂ν p − 1 We are also interested the existence of a minimizer for the following problem

min{λ1(Ω), Ω ∈ A, |Ω| ≤ c}, where A is a family of admissible domain deﬁned by A = {Ω ⊆ D, Ω is quasi open}

and λ1(Ω) is deﬁned by R p R p Ω |∇φ(x)| dx Ω |∇u1(x)| dx λ1(Ω) = min = . 1,p R |φ(x)|p R |u (x)|pdx φ∈W0 (Ω),φ6=0 Ω Ω 1 1,p 1,p The Sobolev space W0 (Ω) is seen as a closed subspace of W0 (D) deﬁned by 1,p 1,p W0 (Ω) = {u ∈ W0 (D): u = 0p − q.e on D\Ω}. The problem is to look for weak topology constraints which would make the class A sequen- tially compact. This convergence is called weak γp-convergence for quasi open sets.

Deﬁnition 4.2. We say that a sequence (Ωn) of A weakly γp-converges to Ω ∈ A if the sequence 1,p 1,p un converges weakly in W0 (D) to a function u ∈ W0 (D) (that we may take as quasi- continuous) such that Ω = {u > 0}. We have the following theorem. Theorem 4.4. The problem

(4.2) min{λ1(Ω), Ω ∈ A, |Ω| ≤ c} admits at least one solution.

J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005 http://jipam.vu.edu.au/ THE FIRST EIGENVALUE FOR THE p-LAPLACIAN OPERATOR 9

Proof of Theorem 4.4. Let us take R p R p |∇φn(x)| dx |∇un(x)| dx λ (Ω ) = min Ωn = Ωn . 1 n R p R p φ ∈W 1,p(Ω ),φ 6=0 |φ (x)| dx |u (x)| dx n 0 n n Ωn n Ωn n

Suppose that (Ωn)(n∈N) is a minimizing sequence of domain for the problem (4.2). We denote by u a first eigenfunction on Ω , such that R |u (x)|pdx = 1. n n Ωn n Since un is the first eigenfunction of λ1(Ωn), un is strictly positive, cf [15], then the sequence (Ωn) is defined by Ωn = {un > 0}. 1,p If the sequence (un) is bounded in W0 (D), then there exists a subsequence still denoted by 1,p un such that un converges weakly in W0 (D) to a function u. By compact injection, we have R p that Ω |u(x)| dx = 1. 1,p Let Ω be quasi open and defined by Ω = {u > 0}, this implies that u ∈ W0 (Ω). As the 1,p sequence (un) is bounded in W0 (D), then R p R p R p |∇un(x)| dx |∇u(x)| dx |∇u (x)| dx Ωn Ω Ω 1 lim inf ≥ ≥ = λ1(Ω). n→+∞ R |u (x)|pdx R |u(x)|pdx R |u (x)|pdx Ωn n Ω Ω 1 Now we show that |Ω| ≤ c. We know that if the sequence Ωn weakly γp- converges to Ω and the Lebesgue measure is weakly γp-lower semicontinuous on the class A (see [5]), then we obtain |{u > 0}| ≤ lim inf |{un > 0}| ≤ c this implies that |Ω| ≤ c. n→+∞

5. DOMAININ BOX Now let us take N = 2. We consider the class of admissible domains deﬁned by C = {Ω, Ω open subsets of D and simply connected, |Ω| = c}. 1,p • For p > 2, the p-capacity of a point is stricly positive and every W0 function has a continuous representative. For this reason, a property which holds p − q.e with p > 2 holds in fact everywhere. For p > 2, the domain Ωp is a minimizer of the problem p min{λ1(Ωp), Ωp ∈ C}.

Consider the sequence (Ωpn ) ⊆ C and assume that Ωpn converges in Hausdorff com-

plementary topology to Ω2, when pn goes to 2 and pn > 2. Then Ω2 ⊆ C and Ωpn γ2-converges to Ω2. 1,pn 1 By the Sobolev embedding theorem, we have W0 (Ωpn ) ,→ H0 (Ωpn ). The γ2 - 1 1 convergence implies that H0 (Ωpn ) converges in the sense of Mosco to H0 (Ω2). For pn > 2, by the Hölder inequality we have 1 1 Z 2 Z pn 2 1 − 1 p 2 pn n |∇upn | dx ≤ |Ωpn | |∇upn | dx

1 Z 2 1 1 2 − pn 2 pn |∇upn | dx ≤ c λ1 (Ωpn ).

1 Then the sequence (upn ) is uniformly bounded in H0 (Ωpn ). There exists a subsequence 1 still denoted upn such that upn converges weakly in H0 (D) to a function u. The second 1 condition of Mosco implies that u ∈ H0 (Ω2). 1,p 0,α ¯ For p > 2, we have the Sobolev embedding theorem W0 (D) ,→ C (D).

Ascoli’s theorem implies that upn −→ u and ∇upn −→ ∇u locally uniformly in Ω2, when pn goes to 2 and pn > 2.

J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005 http://jipam.vu.edu.au/ 10 IDRISSA LY

Now show that Z Z 2 2 lim |upn | dx = 1 i.e. |u| dx = 1. pn−→2

For > 0 small, we have pn > 2 − . Noting that

1 1 Z 2− 1 1 Z pn 1 1 2− − p − pn 2− pn n 2− pn |∇upn | dx ≤ |Ωpn | |∇upn | dx = c λ1 (Ωpn ),

1,2− this implies that the sequence upn is uniformly bounded in W0 (Ωpn ). Then there 1 exists a subsequence still denoted upn such that upn is weakly convergent in H0 (D) to 1,2− u. By the second condition of Mosco we get u ∈ W0 (Ω2). It follows that

2− Z Z Z pn 2− 2− 1− 2− p pn n 2 |u| dx = lim |upn | dx ≤ lim |Ωpn | |upn | dx = c . pn−→2 pn−→2

Letting −→ 0, we obtain R |u|2dx ≤ 1. On the other hand, Lemma 4.2 of [14] implies that Z Z Z pn pn pn−2 |u| dx ≥ |upn | dx + pn |u| dxupn (u − upn ).

The second integral on the right-hand side approaches 0 as pn −→ 2. Thus we get R |u|2dx ≥ 1, and we conclude that R |u|2dx = 1. p p In [11, Theorem 2.1 p. 3350], λk is continuous in p for k = 1, 2, where λk is the k − th eigenvalue for the p-Laplacian operator. We have Z Z pn−2 pn pn−2 (5.1) |∇upn | ∇upn ∇φdx = λ1 |upn | upn φdx, for all φ ∈ D(Ω2).

Letting pn go to 2, pn > 2 in (5.1), and noting that upn converges uniformly to u on the support of φ, we obtain Z Z 2 ∇u∇φdx = λ1uφdx, for all φ ∈ D(Ω2),

whence we have 2 0 −∆u = λ1u in D (Ω2) u = 0 on ∂Ω2.

We conclude that when p −→ 2 and p > 2 the free parts of the boundary of Ωp cannot be pieces of circle.

• For p ≤ 2, we consider the sequence (Ωpn ) ⊆ C and assume that Ωpn converges in Hausdorff complementary topology to Ω2, when pn goes to 2 and pn ≤ 2. Then by

Theorem 3.1, we get Ω2 ⊆ C and Ωpn γ2−-converges to Ω2. 1,2− In [16], the sequence (upn ) is bounded in W (D), 0 < < 1 that is ∇upn con- 2− 2− verges weakly in L (D) to ∇u and upn converges strongly in L (D) to u. In [16], we get also R |∇u|2dx ≤ β and R |u|2dx < ∞. By Lemma 4.2 of [14], we have Z Z Z pn pn pn−2 |u| dx ≥ |upn | dx + pn |u| dxupn (u − upn ).

J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005 http://jipam.vu.edu.au/ THE FIRST EIGENVALUE FOR THE p-LAPLACIAN OPERATOR 11

The second integral on the right-hand side approaches 0 as pn −→ 2. Thus we get R |u|2dx ≥ 1. This implies that Z Z 2− 2− lim |upn | dx = |u| dx = 1. pn−→2 Letting −→ 0, we obtain R |u|2dx = 1. 1,2− The γ2−-convergence implies that upn converges strongly in W0 (Ω2) to u. Ac- cording to P. Lindqvist see [16], we have u ∈ H1(D), and we can deduce that u ∈ 1 H0 (Ω2). As the ﬁrst eigenvalue for the p-Laplacian operator is continuous in p cf [11], we have Z Z pn−2 pn pn−2 (5.2) |∇upn | ∇upn ∇φdx = λ1 |upn | upn φdx, for all φ ∈ D(Ω2).

Letting pn go to 2, pn ≤ 2 in (5.2), and noting that upn converges uniformly to u on the support of φ, we obtain Z Z 2 ∇u∇φdx = λ1uφdx, for all φ ∈ D(Ω2), whence we have 2 0 −∆u = λ1u in D (Ω2) u = 0 on ∂Ω2.

We conclude that when p −→ 2 and p ≤ 2 the free parts of the boundary of Ωp cannot be pieces of circle.

REFERENCES

[1] A. ANANE AND N. TSOULI, The second eigenvalue of the p-Laplacian, Nonlinear Partial Differ- ential Equations (Fès, 1994), Pitman Research Notes Math. Ser., 343, Longman, Harlow, (1996), 1–9.

[2] D. BUCUR AND P. TREBESCHI, Shape optimization problem governed by nonlinear sate equation, Proc. Roy. Edinburgh, 128 A(1998), 945–963.

[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.

[4] D. BUCUR AND J.P. ZOLÉSIO, N-Dimensional shape Optimization under Capacity Constraints, J. Differential Equations, 123(2) (1995), 504–522.

[5] D. BUCUR AND G. BUTTAZZO, Variational Methods in Shape Optimization Problems, SNS Pisa 2002.

[6] G. BUTTAZZO AND G. DAL MASO, An existence result for a class of shape optimization problems, Arch. Rational. Mech. Anal., 122 (1993), 183–195. [7] G. DAL MASO, Γ-convergence and µ capacities, Ann. Scuola Norm. Sup. Pisa,14 (1988), 423– 464.

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

[10] A. HENROT AND E. OUDET, Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions, Arch. Ration. Mech. Anal., 169(1) (2003), 73–87.

J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005 http://jipam.vu.edu.au/ 12 IDRISSA LY

[11] Y.X. HUANG, On the eigenvalue of the p-Laplacian with varying p, Proc. Amer. Math. Soc., 125(11) (1997), 3347–3354. [12] B. KAWOHL, Rearrangements and Convexity of Level Sets in PDE, (Lecture Notes In Mathematics 1150), Springer-Verlag, Heidelberg. [13] E. KRAHN, Über eine von Rayleigh formulierte Minimaleigenschaft des Kreises, Math. Ann., 94 (1924), 97–100. [14] P. LINDQVIST, On the equation div(|∇u|p−2∇u)+λ|u|p−2u = 0, Proc. Amer. Math. Soc., 109(1) (1990), 157–164. Addendum, ibiden, 116(2) (1992), 583–584. [15] P. LINDQVIST, On a nonlinear eigenvalue problem, Berichte Univ. Jyvaskyla Math. Inst., 68 (1995), 33–54. [16] P. LINDQVIST, Stability for the solutions of div(|∇u|p−2∇u) = f with varying p, J. Math. Anal. Appl., 127(1) (1987), 93–102. [17] J. SIMON, Differential with respect to the domain in boundary value problems, Numer. Funct. Anal. and Optimiz., 2(7-8) (1980), 649–687.

[18] J. SOKOLOWSKI AND J.P. ZOLESIO, Introduction to Shape Optimization. Shape Sensitivity Anal- ysis. Springer series in Computational Mathematics, 16. Springer-Verlag, Berlin (1992). [19] V. ŠVERÀK, On optimal shape design, J. Math. Pures Appl., 72 (1993), 537–551.

J. Inequal. Pure and Appl. Math., 6(3) Art. 91, 2005 http://jipam.vu.edu.au/