The eigenvalue problem with conductivity boundary conditions

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The eigenvalue problem with conductivity boundary conditions

Dorin Bucur, Ioana Durus, Edouard Oudet

To cite this version:

Dorin Bucur, Ioana Durus, Edouard Oudet. The eigenvalue problem with conductivity boundary

conditions. Control and Cybernetics, Polish Academy of Sciences, 2008. �hal-00385139�

The conductivity eigenvalue problem

Dorin Bucur

^∗

, Ioana Durus

^†

, Edouard Oudet

^‡

Abstract

In this paper we study isoperimetric inequalities for the eigenvalues of the Laplace operator with constant and locally constant boundary conditions. Existence and sta- bility results are presented in the frame of the gamma and weak gamma convergencies, together with identification of optimal sets by analytical and numerical methods.

1 Introduction

The main purpose of this paper is to study the eigenvalues associated to the Laplace operator with (locally) constant boundary conditions in relationship with the geometric domain. On the one hand we consider globally constant boundary conditions (simply denoted const. b.c.) and on the other hand locally constant boundary conditions, which we call conductivity boundary conditions (denoted cond. b.c.).

The isoperimetric inequality for the first non zero eigenvalue for the const. b.c. eigenval- ues was discussed by Greco and Lucia in [10]. There is proved that the union of two equal balls minimizes the first non-zero eigenvalue among all domains of prescribed measure. Sev- eral more properties are also obtained in [10], among which relationships with the Dirichlet eigenvalues of the Laplacian and with the twisted eigenvalues of the Laplacian (see also [8]).

The first purpose of the paper is to investigate the shape stability of the spectrum of const. b.c Laplacian in relationship with the γ-convergence (see [3]). The γ-convergence is, roughly speaking, the geometric convergence of shapes for which the solution of the Dirichlet- Laplacian is stable. We prove that the spectrum of the const. b.c. Laplacian is stable for the γ-convergence if and only if the Lebesque measure of the moving domains is preserved at the γ-limit (which in general is not the case).

As a second purpose, we discuss the isoperimetric inequality for a general shape functional depending on the const. b.c. Laplacian eigenvalues, of the form F (Λ

1

(Ω), ..., Λ

k

(Ω)). We prove that for every bounded design region D the problem

Ω⊆D,|Ω|=c

min F (Λ

1

(Ω), ..., Λ

k

(Ω))

∗Laboratoire de Math´ematiques, CNRS UMR 5127 Universit´e de Savoie, Campus Scientifique, 73376 Le-Bourget-Du-Lac, France, dorin.bucur@univ-savoie.fr

†LMAM CNRS UMR 7122, Universit´e de Metz, Ile du Saulcy, 57045 Metz, France, ioana.durus@univ- metz.fr

‡Laboratoire de Math´ematiques, CNRS UMR 5127 Universit´e de Savoie, Campus Scientifique, 73376 Le-Bourget-Du-Lac, France, edouard.oudet@univ-savoie.fr

1

has a solution in the family of quasi-open sets, provided mild monotonicity and continuity assumptions on F . A similar result was obtained by Buttazzo and Dal Maso in [5] for the Dirichlet eigenvalues of the Laplacian, based on γ continuity of the eigenvalues. In our problem, the mapping

Ω → Λ

k

(Ω)

is not γ-continuous so that the main theorem of Buttazzo and Dal Maso does not apply. Our argument is based on the study of a modified version of the weak gamma convergence (see [3]) which preserves the measure. We also refer the reader to [10], where existence solutions for isoperimetric inequalities for the const. b.c. Laplacian eigenvalues were studied in the class of convex sets.

The third purpose of the paper is to study the isoperimetric inequalities associated to the conductivity eigenvalues of the Laplacian. This study is motivated by the analysis of the defect identification of a material by electrostatic boundary measurements. Friedman and Vogelius introduced in [9] a dual problem to the defect identification problem, where the perfectly insulating defects become perfectly conductive regions. In this last model, the locally constant boundary conditions are naturally used (see also Alessandrini [1] and [4, 2]).

In [2], the following space is introduced for the study of the conductivity problem in an arbitrary two dimensional bounded domain. For a bounded open set of R

ⁿ

we introduce

H

_cond¹

(Ω) = cl

H¹(Ω)

{ u ∈ H

_loc¹

( R

ⁿ

) | ∃ ε > 0 ∇ u = 0 a.e.(Ω

^c

)

^ε

} where K

^ε

is the Minkowski sum

K ⊕ B(0, ε) = [

x∈K

B (x, ε).

It turns out that if Ω

^c

is connected, every function in H

_cond¹

(Ω) is constant on the boundary, in the sense that there exists c ∈ R such that u − c ∈ H

₀¹

(Ω). Consequently, in this case the space H

_cond¹

(Ω) coincides with the space considered previously by Greco and Lucia (up to the zero mean value). If Ω

^c

is not connected, on the boundary of the smooth connected components of Ω

^c

, a function u ∈ H

_cond¹

(Ω) is locally constant (the constant may differ from one to another connected component). These regions can be seen as perfectly conductive regions in the frame of the dual defect identification problems of Friedeman and Vogelius.

The eigenvalue problem for the cond. b.c Laplacian is well posed, and the same questions (see [10]), as for the const. b.c. Laplacian can be raised. Unfortunately, we are not able to perform a complete shape analysis of the problem because of major new difficulties which are related to the (open) problem of the relaxation of the Neumann Laplacian.

The isoperimetric inequality of the first non-zero cond. b.c eigenvalue can be treated by a two steps rearrangement. As in [10], by a Schwarz rearrangement of two level sets of a test function u, we reduce the search of the optimum to the family of the unions of two sets of concentric annuli. In a second step we prove by a second rearrangement procedure that the annuli can be streched to balls and finally, that two equal disjoint balls minimize the first non zero eigenvalue among domains of constant volume. We refer the reader to [11] for a recent survey concerning isoperimetric inequalities for eigenvalues.

Shape stability is achieved only in two dimensions provided that the number of connected components of the moving complements is uniformly bounded and the Lebesque measure is

2

stable. Some numerical computations for the solutions of the isoperimetric inequalities for the first three cond. b.c eigenvalues are also provided.

2 The Laplacian with constant boundary conditions

Let D ⊆ R

^N

, N ≥ 2 be a bounded open set. We denote by Ω an open or quasi-open subset of D (we refer the reader to [3] for a detailed introduction to the shape analysis of quasi-open sets).

We introduce the following space:

H

_const¹

(Ω) = { u ∈ H

¹

(Ω) | ∃ c ∈ R , u − c ∈ H

₀¹

(Ω) } and consider the closed subspace:

U

^const

(Ω) = { u ∈ H

_const¹

(Ω) | Z

Ω

udx = 0 } .

It is easy to observe that U

^const

(Ω) can be equivalently endowed with the scalar product (u, v)

_Uconst(Ω)

= ( ∇ u, ∇ v )

L²

.

Indeed, the following Poincar´e inequality holds. There exists a constant M > 0, depending only on the measure of Ω such that for every u ∈ U

^const

(Ω)

k u k

^L2(Ω)

≤ M k∇ u k

^L2(Ω)

.

To prove this assertion, one can apply the Poincar´e inequality to u − c ∈ H

₀¹

(Ω) and obtain k u − c k

^L2(Ω)

≤ M k∇ (u − c) k

^L2(Ω)

. Since ∇ u = ∇ (u − c) and since R

Ω

udx = 0, we immediately get

k u k

^L2(Ω)

≤ k u − c k

^L2(Ω)

≤ M k∇ u k

^L2(Ω)

.

We consider the Laplace operator acting on the space U

^const

(Ω). More precisely we define T

Ω

: L

²₀

(Ω) −→ L

²₀

(Ω), T

Ω

f = R

Ω

f − 1

| Ω | Z

Ω

R

Ω

f dx.

Here L

²₀

(Ω) = { f ∈ L

²

(Ω) | R

Ω

f dx = 0 } and R

_Ω

: L

²

(Ω) −→ L

²

(Ω) is the resolvent of the Dirichlet Laplacian on Ω, i.e. R

Ω

f = u, where u solves

− ∆u = f in Ω

u ∈ H

₀¹

(Ω). (2.1)

in the weak variational sense. The eigenvalues of T

_Ω⁻¹

are the inverses of the eigenvalues of the const. b.c. Laplacian.

Alternatively, they are defined with the Rayleigh quotient λ

^ct_k

(Ω) = min

V^k⊂Uconst(Ω)

max

v∈V^k\{0}

R

Ω

|∇ v |

²

dx R

Ω

v

²

dx . (2.2)

3

where V

^k

stands for an arbitrary subspace of dimension k in U

^const

(Ω).

We notice at this point that we can also define the Laplacian on H

_const¹

(Ω), in which case the first eigenvalue is zero. For this reason we refer to λ

^ct₁

(Ω) as the first non zero eigenvalue.

If λ

^ct_k

(Ω) is an eigenvalue defined by (2.2), there exists a non zero function u ∈ U

^const

(Ω)

such that formally

− ∆u = λ

^ct_k

(Ω)u

u ∈ U

^const

(Ω). (2.3)

This formal relation is understood in the following sense: u ∈ U

^const

(Ω) and

∀ φ ∈ U

^const

(Ω) Z

Ω

∇ u · ∇ φdx = λ

^ct_k

(Ω) Z

Ω

uφdx.

Notice that the relationship

− ∆u = λ

^ct_k

(Ω)u (2.4)

holds also in the sense of distributions. Indeed, ∀ φ ∈ D (Ω), and a suitable constant c such that φ − c ∈ U

^const

(Ω) we have

Z

Ω

∇ u · ∇ (φ − c)dx = λ

^ct_k

(Ω) Z

Ω

u(φ − c)dx.

Since R

Ω

udx = 0 one can eliminate c on both sides above.

We recall from [10] the following properties of λ

^ct_k

(Ω):

1. if Ω

1

⊆ Ω

2

then λ

^ct_k

(Ω

2

) ≤ λ

^ct_k

(Ω

1

), 2. λ

^ct_k

(tΩ) = t

⁻²

λ

^ct_k

(Ω),

3. λ

k

(Ω) ≤ λ

^ct_k

(Ω) ≤ λ

^t_k

(Ω) ≤ λ

_k+1

(Ω),

where λ

k

(Ω) stands for the k-th Dirichlet eigenvalue of the Laplacian and λ

^t_k

(Ω) for the k-twisted eigenvalue (see [8]).

Greco and Lucia studied in [10] the isoperimetric inequality for λ

^ct₁

. There is proved that among all domains of given volume (say m), the union of two disjoint balls of volume

^m₂

minimizes λ

^ct₁

. By a Schwarz rearrangement of the level sets (u − c)

⁺

, (u − c)

⁻

of a test function u ∈ U

^const

(Ω) which is such that u − c ∈ H

₀¹

(Ω), the optimization problem is reduced to the family of unions of two balls.

In order to study more general isoperimetric inequalities associated to λ

^ct_k

(Ω) we recall some results for the Dirichlet Laplacian. For the convenience of the reader, we recall the definition of the γ-convergence.

Definition 2.1 Let Ω

n

, Ω ⊆ D be open (or quasi open) sets. It is said that Ω

n

γ-converges to Ω if R

_Ωn

−→ R

_Ω

in L (L

²

(D)).

For a detailed study of the γ-convergence, we refer the reader to [3].

From the following general inequality (see for instance [7])

| λ

k

(R

_Ωn

) − λ

k

(R

_Ω

) | ≤ k R

_Ωn

− R

_Ω

k

L(L²(D))

4

and from the fact that

λ

k

(Ω

n

) = 1 λ

k

(R

Ωn

) we get that

Ω

n

−→

γ

Ω = ⇒ λ

k

(Ω

n

) −→ λ

k

(Ω).

We also notice that if Ω

n

−→

γ

Ω, we have 1

Ω

(x) ≤ lim inf

n−→∞

1

Ωn

(x) a.e. x ∈ D and consequently

| Ω | ≤ lim inf

n−→∞

| Ω

n

| .

If we assume both γ convergence and convergence of measures, then we get L

¹

convergence of the characteristic functions 1

Ωn

−→ 1

Ω

.

The following general existence result is due to Buttazzo and Dal Maso [5]. Let us denote A (D) the family of quasi open subsets of D.

Theorem 2.2 Let F : A (D) −→ R be a γ-lower semi continuous functional which is non increasing to set inclusions (up to zero capacity). Then

min { F (Ω) | Ω ∈ A (D), | Ω | = m } has a solution.

If F (Ω) = Φ(λ

1

(Ω), ..., λ

k

(Ω)) where Φ : R

^k

−→ R is l.s.c. and non decreasing in each variable, then Theorem 2.2 applies for

F (Ω) = Φ(λ

1

(Ω), ..., λ

k

(Ω)).

In particular, the isoperimetric problem

Ω∈A(D),|Ω|=m

min λ

k

(Ω) has at least one solution.

In order to answer a similar question for shape functionals depending on λ

^ct₁

(Ω), ..., λ

^ct_k

(Ω) one can not use Theorem 2.2, since the mappings Ω −→ λ

^ct_k

(Ω) are not in general γ- continuous. This lack of γ-continuity is due to the possible ”loss” of measure in the γ-limit process and of the failure of convergence of the resolvent operators.

We prefer to (re)define the operators T

Ω

on the fixed space associated to the design region D ⊆ R

ⁿ

. In this way, all operators on the moving sets are defined on the same functional space, so that the comparison of their eigenvalues is easier to handle. For every (quasi) open set Ω ⊆ D, we introduce T e

Ω

: L

²₀

(D) −→ L

²₀

(D) by

T e

Ω

(f )(x) =

T

Ω

(f

|Ω

−

_|Ω|¹

R

Ω

f dx)(x) if x ∈ Ω, 0 if x ∈ D \ Ω.

We state the following proposition which has a standard proof.

5

Proposition 2.3 For every (quasi) open set Ω ⊆ D, T e

Ω

is positive, self adjoint and compact.

In the sequel we establish the relationship between the eigenvalues of T

Ω

and T e

Ω

.

Proposition 2.4 Let Ω be a (quasi) open set. The operators T

Ω

and T e

Ω

have the same set of eigenvalues, and there exists a natural bijection between eigenspaces.

Proof Let λ be an eigenvalue of T

_Ω

. There exists u ∈ L

²₀

(Ω) such that T

_Ω

u = λu. One can prove that there exist u

^∗

∈ L

²₀

(D) such that T e

Ω

u

^∗

= λu

^∗

. Indeed, let u

^∗

=

u if x ∈ Ω 0 if x ∈ D \ Ω.

Since R

D

u

^∗

dx = 0 then u

^∗

∈ L

²₀

(D). We obtain T e

Ω

u

^∗

= T

Ω

(u

^∗

− 1

| Ω | Z

Ω

u

^∗

dx) = T

Ω

u

^∗

= λu

^∗

. since R

Ω

u

^∗

dx = R

Ω

udx = 0. Consequently λ is an eigenvalue for T e

_Ω

. Conversely, let λ be an eigenvalue for T e

Ω

. There exists u ∈ L

²₀

(Ω) such that T e

Ω

u = λu. Since u ∈ Im T e

Ω

we get that u = 0 in D \ Ω and R

Ω

udx = 0. Consequently we have T e

Ω

u = T

Ω

(u −

_|Ω|¹

R

Ω

udx) = T

Ω

(u −

|Ω|¹

R

Ω

udx) = T

Ω

u = λu.

2 Here is the shape stability theorem for the eigenvalues of T e

Ω

operator.

Theorem 2.5 Let Ω

n

, Ω ⊆ D be non empty (quasi) open sets such that Ω

n

−→

γ

Ω. Then T e

Ωn

−→ T e

Ω

in L (L

²₀

(D)) if and only if | Ω

n

| −→ | Ω | .

Proof Assume that Ω

n

−→

γ

Ω and | Ω

n

| −→ | Ω | . There exists f

n

in the unit ball of L

²₀

(D) which realizes the supremum in

sup

kfk_L2

0(D)≤1

k T e

Ωn

f − T e

Ω

f k

L²₀(D)

= k T e

Ωn

(f

n

) − T e

Ω

(f

n

) k

L²₀(D)

. (2.5) We may assume, without restricting the generality that (f

n

)

n

converges weakly to f. We have

k T e

Ωn

(f

n

) − T e

Ω

(f

n

) k

^L2₀(D)

= k T e

Ωn

(f

n

) − T e

Ω

f + T e

Ω

f − T e

Ω

(f

n

) k

^L2₀(D)

≤ k T e

Ωn

(f

n

) − T e

Ω

f k

L²₀(D)

+ k T e

Ω

f − T e

Ω

(f

n

) k

L²₀(D)

.

Let φ ∈ L

²₀

(D). The convergence of T e

_Ωn

φ to T e

_Ω

φ is strong in L

²₀

(D), since Ω

n

−→

γ

Ω.

Indeed, T e

Ωn

φ = T

Ωn

(φ −

_|Ω¹_n|

R

Ωn

φdx) = T

Ωn

g

n

, and g

n

converge to g, where g

n

= φ

|Ωn

−

1

|Ωn|

R

Ωn

φdx and g = φ

_|Ω

−

_|Ω|¹

R

Ω

φdx. Using the continuity of R

Ω

and the hypothesis of γ -convergence, we obtain

T

Ωn

g

n

= R

Ωn

g

n

− 1

| Ω

n

| Z

Ωn

R

Ωn

g

n

dx −→ R

Ω

g − 1

| Ω | Z

Ω

R

Ω

gdx = T

Ω

g.

6

Consequently, we have in L

²₀

(D)

< T e

Ωn

f

n

, φ >=< f

n

, T e

Ωn

φ > −→ < f, T e

Ω

φ >=< T e

Ω

f, φ > .

The weak L

²₀

(D) convergence of T e

Ωn

(f

n

) to T e

Ω

(f ) implies the strong L

²₀

(D) convergence of a subsequence (still denoted using the same index).

To conclude, it remains to discuss k T e

Ω

f − T e

Ω

f

n

k

^L2₀(D)

. This term converges to zero as a direct consequence of the continuity/compactness of T e

Ω

and the weak L

²₀

(D) convergence of f

n

⇀ f .

Conversely, we assume that Ω

n

−→

γ

Ω and T e

_Ωn

−→ T e

_Ω

. Let φ ∈ L

²₀

(D). The convergence T e

Ωn

φ −→ T e

Ω

φ reads

R

Ωn

φ − 1

| Ω

n

| Z

Ωn

R

Ωn

φdx − 1

| Ω

n

| Z

Ωn

φdx

R

Ωn

1 + 1

| Ω

n

|

²

Z

Ωn

φdx Z

Ωn

R

Ωn

1dx

−→ R

Ω

φ − 1

| Ω | Z

Ω

R

Ω

φdx − 1

| Ω | Z

Ω

φdx

R

Ω

1 + 1

| Ω |

²

Z

Ω

φdx Z

Ω

R

Ω

1dx.

We denote by g

n

= R

Ωn

1, f

n

= R

Ωn

φ. The hypothesis on the γ convergence yields that g

n

and f

n

converge strongly H

₀¹

(D) to g = R

Ω

1, and f = R

Ω

φ, respectively, in H

₀¹

(D). We obtain

− 1

| Ω

n

| Z

Ωn

φdxg

n

− 1

| Ω

n

| Z

D

f

n

dx + 1

| Ω

n

|

²

Z

Ωn

φdx Z

Ωn

g

n

dx

−→ − 1

| Ω | Z

Ω

φdxg − 1

| Ω | Z

D

f dx + 1

| Ω |

²

Z

Ω

φdx Z

Ω

gdx.

We can rewrite simply a

n

g

n

( · ) + b

n

−→ ag( · ) + b, where a

n

= −

_|Ω¹_n|

R

Ωn

φdx, a = −

_|Ω|¹

R

Ω

φdx, b

n

= −

_|Ω¹_n|

R

D

f

n

dx +

_|Ω¹

n|²

R

Ωn

φdx R

Ωn

g

n

dx and b = −

_|Ω|¹

R

D

f dx +

_|Ω|¹2

R

Ω

φdx R

Ω

gdx. Since the function g is non vanishing, we get a

n

−→ a.

Assume | Ω

n

| −→ α > | Ω | and define φ by φ =

( c in Ω

−

_|D\Ω|^c|Ω|

in D \ Ω.

where c is a positive constant. The convergence a

n

−→ a implies that R

Ωn

φdx −→

_|Ω|^α

R

Ω

φdx.

On the other hand Z

Ωn

φdx = c | Ω | − c | Ω | | Ω

n

\ Ω |

| D \ Ω | .

We obtain 1 −

^|Ω_|D\Ω|^n\Ω|

−→

_|Ω|^α

> 1, which is absurd, since

^|Ω_|D\Ω|^n\Ω|

> 0. Consequently α = | Ω | . 2 A direct consequence of Theorem 2.5 is the following.

Corollary 2.6 Let Ω

n

, Ω ⊆ D be such that Ω

n

−→

γ

Ω and | Ω

n

| −→ | Ω | . Then λ

^ct_k

(Ω

n

) −→

λ

^ct_k

(Ω).

7

We recall the definition of the weak gamma convergence of quasi open sets. It is said that Ω

n

weakly gamma converges to Ω if R

Ωn

1 converges weakly H

₀¹

(D) to a function w, and Ω = { w > 0 } . We write Ω

n

−→

wγ

Ω. The main property of the weak gamma convergence is the following: assume that φ

n

∈ H

₀¹

(Ω

n

) is weakly convergent in H

₀¹

(D) to a function φ.

Then φ ∈ H

₀¹

(Ω).

The following result is the key of the optimization problem.

Theorem 2.7 Assume Ω

n

−→

wγ

Ω and Ω ⊆ Ω

^∗

, such that | Ω

^∗

| = lim inf

n→∞

| Ω

n

| . Then λ

^ct_k

(Ω

^∗

) ≤ lim

n−→∞

λ

^ct_k

(Ω

n

)

Proof Let ǫ > 0 and S

_n^k

⊆ U

^const

(Ω

n

)be a k-dimensional space such that:

λ

^ct_k

(Ω

n

) + ε ≥ max

v∈S^k_n\{0}

R

Ωn

|∇ v |

²

dx R

Ωn

v

²

dx , and let v

n

be a maximizer function such that R

Ωn

v

_n²

dx = 1. Let v

_iⁿ

, i = 1, .., k be a L

²₀

- orthonormal basis for S

_n^k

.

We assume that lim inf

n→∞

λ

^ct_k

(Ω

n

) < ∞ , otherwise the conclusion is obvious. Since R

Ωn

| v

_iⁿ

|

²

dx = 1 we get that R

Ωn

|∇ v

_iⁿ

|

²

dx is bounded. Consequently R

Ωn

|∇ (v

_iⁿ

− c

ⁿ_i

) |

²

dx is bounded (here v

ⁿ_i

− c

ⁿ_i

∈ H

₀¹

(Ω

n

)). Up to a subsequence we may assume that v

_iⁿ

− c

ⁿ_i

converges weakly in H

₀¹

(D) to w

i

, so the convergence Ω

n

wγ

→ Ω implies w

i

∈ H

₀¹

(Ω). Let Ω

^∗

⊂ D be such that | Ω

^∗

| = α, where α is such that | Ω

n

| −→ α. Since H

₀¹

(Ω) ⊂ H

₀¹

(Ω

^∗

) we obtain w

i

∈ H

₀¹

(Ω

^∗

).

Assume that | Ω

n

| → α and c

ⁿ_i

−→ c

i

. Then Z

D

(v

_iⁿ

− c

ⁿ_i

)dx → Z

D

(w

i

)dx ⇒ Z

Ωn

v

_iⁿ

dx − Z

Ωn

c

ⁿ_i

dx → Z

Ω^∗

w

i

dx ⇒ − c

ⁿ_i

| Ω

n

| → Z

Ω^∗

w

i

dx, and we get R

Ω^∗

w

i

dx = − c

i

α. Let v

i

= w

i

+ c

i

∈ U

^const

(Ω

^∗

). Indeed, we have R

Ω^∗

v

i

dx = R

Ω^∗

(w

i

+ c

i

)dx = − c

i

α + c

i

| Ω

^∗

| = 0.

We introduce the space S = span(v

1

, ..., v

k

). One can verify that (v

i

, v

j

)

L²

= δ

ij

so that S is a k dimensional subspace of U

^const

(Ω

^∗

). The construction of S and the convergence Ω

n

−→

wγ

Ω implies that for all v ∈ S, v = Σ

^k_i=1

α

i

v

i

the sequence h

n

∈ S

n

defined by h

n

= Σ

^k_i=1

α

i

v

_iⁿ

converges weakly to v in H

₀¹

(D). Moreover,

lim inf

n−→∞

Z

Ωn

|∇ h

n

|

²

dx ≥ Z

Ω^∗

|∇ v |

²

dx. (2.6)

To prove that

lim inf

n−→∞

R

Ωn

|∇ h

n

|

²

dx R

Ωn

h

²_n

dx ≥ R

Ω^∗

|∇ v |

²

dx R

Ω^∗

v

²

dx (2.7)

it is enough to prove that

lim sup

n→∞

Z

Ωn

h

²_n

dx ≤ Z

Ω^∗

v

²

dx. (2.8)

8

By the definition of v

i

we obtain Z

Ω^∗

v

²

dx = Z

Ω^∗

(Σ

^k_i=1

α

i

v

i

)

²

dx = Z

Ω^∗

(Σ

^k_i=1

α

i

(w

i

+ c

i

))

²

dx

= Z

Ω^∗

(Σ

^k_i=1

α

i

w

i

+ Σ

^k_i=1

α

i

c

i

)

²

dx = Z

Ω^∗

(Σ

^k_i=1

α

i

w

i

)

²

dx − 2C

²

α + C

²

α

= Z

Ω^∗

(Σ

^k_i=1

α

i

w

i

)

²

dx − C

²

α, where C = Σ

^k_i=1

α

i

c

i

. On the other hand we get

Z

Ωn

h

²_n

dx = Z

Ωn

(Σ

^k_i=1

α

i

v

ⁿ_i

)

²

dx = Z

Ωn

(Σ

^k_i=1

α

i

(v

_iⁿ

− c

ⁿ_i

) + Σ

^k_i=1

α

i

c

ⁿ_i

)

²

dx

= Z

Ωn

(Σ

^k_i=1

α

i

(v

_iⁿ

− c

ⁿ_i

))

²

dx + 2Σ

^k_i=1

α

i

c

ⁿ_i

Z

Ωn

Σ

^k_i=1

α

i

(v

_iⁿ

− c

ⁿ_i

)dx + (Σ

^k_i=1

α

i

c

ⁿ_i

)

²

| Ω

n

|

= Z

Ωn

(Σ

^k_i=1

α

i

(v

_iⁿ

− c

ⁿ_i

))

²

dx − 2C

²

| Ω

n

| + C

²

α.

Inequality (2.8) is equivalent to lim sup

n→∞

( Z

Ωn

(Σ

^k_i=1

α

i

(v

_iⁿ

− c

ⁿ_i

))

²

dx − 2C

²

| Ω

n

| + C

²

α) ≤ Z

Ω^∗

(Σ

^k_i=1

α

i

w

i

)

²

dx − C

²

α (2.9) Passing to the limit gives

− lim inf

n→∞

2C

²

| Ω

n

| + C

²

α ≤ C

²

α.

Finally,

lim inf

n→∞

λ

^ct_k

(Ω

n

) + ε ≥ lim inf

n→∞

max

v∈S_n^k\{0}

R

Ωn

|∇ v |

²

dx R

Ωn

v

²

dx ≥ lim inf

n→∞

R

Ωn

|∇ h

n

|

²

dx R

Ωn

h

²_n

dx ≥ max

v∈S

R

Ω^∗

|∇ v |

²

dx R

Ω^∗

v

²

dx ≥ λ

^ct_k

(Ω

^∗

).

so lim inf

n→∞

λ

^ct_k

(Ω

n

) + ε ≥ λ

^ct_k

(Ω

^∗

). Making ε → 0, we conclude the proof. 2 Theorem 2.8 Let Φ : R

^k

−→ R ∪{ + ∞} be a lower semi continuous function, nondecreasing in each variable. Then

Ω∈A(D),|Ω|=m

min Φ(λ

^ct₁

(Ω), ..., λ

^ct_k

(Ω)) (2.10) has at least one solution.

Proof Let us denote λ

^ct

(Ω) = (λ

^ct₁

(Ω), ..., λ

^ct_k

(Ω)). From the monotonicity property of λ

^ct_k

, the function Φ(λ

^ct

(Ω)) is non-increasing with respect to set inclusions. We assume that Φ(λ

^ct

(Ω)) is not identically + ∞ , otherwise every open set of measure m is a solution. We notice that Φ(λ

^ct

(Ω)) is bounded from below by Φ(λ

^ct

(D)).

9

Let Ω

n

be a minimizing sequence for the problem (2.10). From the compactness of the wγ convergence, we may assume that Ω

n

−→

wγ

Ω, and we get | Ω | ≤ m. If | Ω | = m than Ω is solution from Theorem 2.7. Otherwise, we find Ω

^∗

such that | Ω |

^∗

= m and Ω ⊆ Ω

^∗

. Then, from the monotonicity and the lower semicontinuity of Φ we get Φ(λ

^ct

(Ω

^∗

)) ≤

lim inf Φ(λ

^ct

(Ω

n

)), consequently Ω

^∗

is a solution. 2

3 The conductivity eigenvalue problem

Let D be a bounded, connected open subset of R

^N

such that R

^N

\ D is connected. For every open set Ω ⊆ D we introduce the conductivity space

H

_cond¹

(Ω) = cl

H¹(Ω)

{ u ∈ H

_loc¹

( R

ⁿ

) | ∃ ǫ > 0, ∇ u = 0 a.e. on ( R

ⁿ

\ Ω)

^ǫ

} . Here, for a set K ⊆ R

ⁿ

and for ǫ > 0 we denote

K

^ǫ

= [

x∈K

B (x, ǫ).

Let u ∈ H

_cond¹

(Ω). Clearly, there exists a constant c such that u − c ∈ H

₀¹

(D). Indeed, by definition there exists a sequence of functions u

n

∈ H

_loc¹

( R

ⁿ

) such that ∇ u

n

= 0 on ( R

ⁿ

\ Ω)

^ǫn

and u

n

→ u in H

¹

(Ω). Consequently, there exists c

n

such that u

n

− c

n

∈ H

₀¹

(D), and we may see the limit u

n

→ u in H

¹

(Ω) for the extensions of u

n

in H

_loc¹

( R

ⁿ

). Since ∇ u

n

= 0 a.e.

on ( R

ⁿ

\ Ω)

^ǫn

we have that ∇ u

n L²(Rⁿ)

⇀ ∇e u, u e ∈ H

_loc¹

( R

ⁿ

). Since R

Rⁿ

|∇ u

n

|

²

dx = R

Ω

|∇ u

n

|

²

dx we get that R

Rⁿ

|∇ u

n

− ∇e u |

²

dx −→ 0, where

∇e u =

∇ u on Ω

0 on Ω

^c

. (3.1)

Up to a sequence (still denoted using the same index) and since D is bounded and connected, we have u

n

L²(D)

−→ e u and e u = u on Ω. Let K be a connected component of R

ⁿ

\ Ω.

We get that u e is quasi everywhere constant on K . In particular, u e equals the constant c on the unbounded component, so that e u − c ∈ H

₀¹

(D).

We notice that if Ω

₁

and Ω

₂

differ on a set of zero capacity, it is possible that H

_cond¹

(Ω

₁

) and H

_cond¹

(Ω

2

) are not the same! This is not a real problem for the the study of isoperimetric inequalities, as will be observed in the sequel, but is a hard difficulty for the study of the shape stability question (which remains open).

We introduce the following closed subspace in H

_cond¹

(Ω) U

^cond

(Ω) = { u ∈ H

_cond¹

(Ω) |

Z

Ω

udx = 0 } .

The mapping u −→ k∇ u k

^L2(Ω)

is a norm on U

^cond

(Ω). Moreover, the Poincar´e inequality Z

Ω

| u |

²

dx ≤ C(D) Z

Ω

|∇ u |

²

dx

10

holds with a constant C(D) depending only on D. Indeed, we can write for a suitable value

c, Z

D

(u − c)

²

dx ≤ C(D) Z

D

|∇ (u − c) |

²

dx, since u − c ∈ H

₀¹

(D). Consequently

Z

Ω

(u − c)

²

dx ≤ C(D) Z

Ω

|∇ u |

²

dx.

Since R

Ω

udx = 0 we get R

Ω

u

²

dx ≤ R

Ω

(u − c)

²

dx ≤ C(D) R

D

|∇ u |

²

dx. The eigenvalues of the Laplace operator with conductivity boundary condition are defined as the eigenvalues of the operator

A

Ω

: L

²₀

(Ω) −→ L

²₀

(Ω) defined by A

Ω

f = u where u solves the problem

u ∈ U

^cond

(Ω) Z

Ω

∇ u · ∇ φdx = Z

Ω

f φdx ∀ φ ∈ U

^cond

(Ω).

Alternatively, u minimizes

u∈U

min

cond(Ω)

1 2

Z

Ω

|∇ u |

²

dx − Z

Ω

f udx.

Proposition 3.1 The operator A

Ω

is self adjoint, positive and compact.

Proof The continuity of A

Ω

is a consequence of the Poincar´e inequality in U

^cond

(Ω). Its compactness comes from the compact embedding U

^cond

(Ω) ֒ → L

²₀

(Ω), while selfadjointess is

inherited from the Laplacian. 2

A direct consequence of Proposition 3.1 is that the spectrum of conductivity b.c-Laplacian consists on a sequence of eigenvalues

0 < λ

^cd₁

(Ω) ≤ λ

^cd₂

(Ω) ≤ ....

It is straightforward to observe that

λ

^cd_k

(tΩ) = t

⁻²

λ

^cd_k

(Ω).

The monotonicity property Ω

1

⊆ Ω

2

= ⇒ λ

^cd_k

(Ω

2

) ≤ λ

^cd_k

(Ω

1

) is a consequence of the min max principle:

λ

^cd_k

(Ω) = min

V^k∈Ucond(Ω)\{0}

max

u∈V^k\{0}

R

Ω

|∇ u |

²

dx R

Ω

u

²

dx .

For every V

^k

⊆ U

^cond

(Ω

1

), V

^k

= span { v

1

, ...v

n

} we consider v

k

extended to elements v

_k^∗

of H

_cond¹

(Ω

2

) and v

_k^∗

− c

k

∈ U

^cond

(Ω

2

). The extension is done as follows: consider a sequence v

_k^ε

converging in H

_cond¹

(Ω

1

) to v

k

, such that ∇ v

_k^ε

= 0 on ( R

^N

\ Ω

1

)

^ε

. We extend v

_k^ε

by suitable constants to elements (v

^ε_k

)

^∗

H

_cond¹

(Ω

2

) and define v

_k^∗

as the limit of (v

_k^ε

)

^∗

.

11

It is straightforward to observe that R

Ω1

|∇ v

k

|

²

dx R

Ω1

v

_k²

dx ≥ R

Ω2

|∇ (v

_k^∗

− c

k

) |

²

dx R

Ω2

| v

^∗_k

− c

k

|

²

dx . Indeed, R

Ω2

| v

^∗_k

− c

k

|

²

dx ≥ R

Ω1

| v

¹_k

− c

k

|

²

dx ≥ R

Ω1

| v

k

|

²

dx.

Proposition 3.2 For every k ∈ N the following inequality holds true λ

k−p+1

(Ω) ≤ λ

^ct_k−p+1

(Ω) ≤ λ

^cd_k

(Ω) ≤ λ

^ct_k

(Ω) ≤ λ

k+1

(Ω) where p ≥ 1 is the number of the connected components of Ω

^c

Proof The first and the last inequalities are proved in [10]. The third inequality is a consequence of the min-max formula and the inclusion U

^const

(Ω) ⊆ U

^cond

(Ω).

For the second inequality, let us consider a k-dimensional subspace S

k

⊂ U

^cond

(Ω). It is easy to notice that S

k

has a k − p + 1 dimensional subspace S

k−p+1

⊂ U

^const

(Ω). Conse- quently, the min-max formula implies the second inequality. 2 The shape stability question for λ

^cd_k

is much more difficult than for λ

^ct_k

. No general continuity result could be obtained, since many small pieces carrying different constants may produce a relaxation process into the limit process, which is not clear how to handle. It turns out that this relaxation process is close to the relaxation of Neumann problems, which is an open question.

Proposition 3.3 In two dimensions of the space, assume that Ω

n

⊆ D, ♯Ω

^c_n

≤ M. If Ω

n

H^c

−→ Ω and | Ω

n

| −→ | Ω | then λ

^cd_k

(Ω

n

) −→ λ

^cd_k

(Ω).

The proof is a consequence of the Mosco convergence U

^cond

(Ω

n

) to U

^cond

(Ω). The difficulty is to handle the constant values of varying functions on merging connected components. We refer the reader to [4] for a discussion of this topic.

4 The isoperimetric inequality for the first conductiv- ity eigenvalue

In [10], the authors proved the following isoperimetric eigenvalue for λ

^ct₁

: λ

^ct₁

(B

₁

∪ B

₂

) ≤ λ

^ct₁

(Ω)

where Ω is a bounded open set of measure m and B

1

, B

2

are two disjoint balls of measure m/2. The main idea consists in rearranging in the sense of Schwarz (u − c)

⁺

and (u − c)

⁻

, for some test function u ∈ U

^const

(Ω), such that u − c ∈ H

₀¹

(Ω). This rearrangement leads to searching the minimizer only among the union of two disjoint ball of total measure equal to m, and an analysis depending on the radius drives to the conclusion that the balls have to be equal.

12

In the sequel we shall prove a similar isoperimetric inequality for the conductivity eigen- value. The new difficulty is that on the boundary of Ω, the test function has different constant values, so that several regions with flat parts may appear in the rearrangement procedure. Consequently, a second rearrangement process has to be introduced in order to eliminate the flat parts.

Theorem 4.1 Let Ω be a bounded open set of R

^N

. Then λ

^cd₁

(B

₁

∪ B

₂

) ≤ λ

^cd₁

(Ω) where B

1

and B

2

are two disjoint balls of measure

^|Ω|₂

.

Proof Let u ∈ U

^cond

(Ω) be a test function such that ∇ u = 0 on ( R

ⁿ

\ Ω)

^ǫ

. In particular, this implies the existence of a finite number of flat parts of u, corresponding to the (finite number of) connected components of ( R

ⁿ

\ Ω)

^ǫ

.

Let c ∈ R such that u − c ∈ H

¹

( R

ⁿ

) (i.e. u = c on the unbounded component of ( R

ⁿ

\ Ω)

^ǫ

).

We make a Schwarz rearrangement of (u − c)

⁺

and (u − c)

⁻

(see for instance [11, 12]). Let us denote u

⁺

= [(u − c)

⁺

]

^∗

and u

⁻

= [(u − c)

⁻

]

^∗

. If (u − c)

⁺

= c

1

on a connected component K of ( R

ⁿ

\ Ω)

^ǫ

, after the rearrangement procedure there is an annulus K

0,r1,r2

with measure at least the measure of | K | , where u

⁺

equals c

1

.

The set Ω with the test function u is replaced by the set Ω

1

∪ Ω

2

with the test function c − u

⁻

, c +u

⁺

obtained in the following way. We remove out from the flat part of value c

1

the annulus K

_0,r1,r^′

1

such that r

^′₁

≤ r

2

, and | K

_0,r1,r^′

1

| = | K ∩ ( R

ⁿ

\ Ω) | . Making this procedure for every flat part of u, (without c), we construct the new test function u e consisting on c − u

⁻

on Ω

1

and c + u

⁺

on Ω

2

, where Ω

1

and Ω

2

are respectively unions of concentric annuli. In this way, | Ω

₁

∪ Ω

₂

| = | Ω | and R

Ω1∪Ω2

udx e = 0, R

Ω1∪Ω2

e u

²

dx = R

Ω

u

²

dx. Moreover e u ∈ U

^cond

(Ω

₁

∪ Ω

₂

) and R

Ω1∪Ω2

|∇e u |

²

dx ≤ R

Ω

|∇ u |

²

dx.

At this point, we have to search the minimizer only among configurations of the form ( u, e Ω

1

, Ω

2

). We make a new rearrangement procedure which will keep constant the measures of the level sets of e u and diminish the Dirichlet energy, in order to create a test function in U

^const

. Consequently, we will be able to use the isoperimetric inequality of Greco and Lucia and conclude the proof. In fact, we will stretch the annuli in order to eliminate the flat parts of u e which are not counted in the measure | Ω

₁

∪ Ω

₂

| .

It is enough to prove the following result corresponding to one annulus. Let u ∈ H

₀¹

(B (0, r

2

)), u = 1 on B(0, r

1

) be a positive radial function symmetric in the sense of Schwarz. For r

₂^′

< r

2

such that | B (0, r

₂^′

) | ≥ | K

0,r1,r2

| , we introduce the function c : [r

1

, r

2

] −→

[r

₁^′

, r

₂^′

], c(r) = p

ⁿ

(r

₂^′

)

ⁿ

− r

₂ⁿ

+ r

ⁿ

, such that | K

_0,c(r),r^′

2

| = | K

0,r,r2

| . We define the rearrange- ment of u by { u

^∗

> c } = B(0, c(r)) if c is such that { u > c } = B (0, r). Clearly, since the measures of the level sets are constant, we have that

∀ 1 ≤ p < + ∞ Z

K_0,r1,r2

u

^p

dx = Z

K0,r′ 1,r′

2

(u

^∗

)

^p

dx.

We evaluate the behavior of the gradient. Since u is radial, we can use spherical coordinates to express R

K0,r1,r2

|∇ u |

²

dx. We notice that u

^∗

∈ H

₀¹

(B(0, r

₂^′

)), since u

^∗

(c(r), θ

1

, ..., θ

n−1

) =

13