Spectral optimization problems with internal constraint

(1)

Ann. I. H. Poincaré – AN 30 (2013) 477–495

www.elsevier.com/locate/anihpc

Spectral optimization problems with internal constraint

Dorin Bucur

^a

, Giuseppe Buttazzo

^b,^∗

, Bozhidar Velichkov

^c

aLaboratoire de Mathématiques (LAMA), Université de Savoie, Campus Scientifique, 73376 Le-Bourget-Du-Lac, France bDipartimento di Matematica, Università di Pisa, Largo B. Pontecorvo 5, 56127 Pisa, Italy

cScuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, Italy Received 12 September 2011; received in revised form 8 October 2012; accepted 19 October 2012

Available online 26 October 2012

Abstract

We consider spectral optimization problems with internal inclusion constraints, of the form min

λ_k(Ω):D⊂Ω⊂R^d, |Ω| =m ,

where the setDis fixed, possibly unbounded, andλ_k is thek-th eigenvalue of the Dirichlet Laplacian onΩ. We analyze the existence of a solution and its qualitative properties, and rise some open questions.

MSC:49J45; 49R05; 35P15; 47A75; 35J25

Keywords:Shape optimization; Capacity; Eigenvalues; Sobolev spaces; Concentration-compactness

1. Introduction

A spectral optimization problem is a minimization problem of the form min

J (Ω): Ω∈A

(1.1) whereJ is a cost functional depending on the spectrum of an elliptic operator defined on the (quasi) open setΩ and Ais a class of admissible domains. A wide literature on the subject is available, dealing with existence, regularity, necessary conditions of optimality, relaxation, explicit solutions and numerical computations of the optimal shapes.

We quote for instance the books[7,18,19]and the articles [2,9,17], where the reader may find a complete list of references on the field.

The simplest situation for the existence of a solution of problem (1.1) occurs when the class of admissible domains Asatisfies anexternalinclusion constraint, i.e. consists on quasi-open sets which are supposeda prioricontained in a givenboundedopen setDof the Euclidean spaceR^d,

A= {Ω: Ω⊂D, Ωquasi-open}.

* Corresponding author.

E-mail addresses:[email protected](D. Bucur),[email protected](G. Buttazzo),[email protected](B. Velichkov).

URLs:http://www.lama.univ-savoie.fr/~bucur/(D. Bucur),http://www.dm.unipi.it/pages/buttazzo/(G. Buttazzo).

http://dx.doi.org/10.1016/j.anihpc.2012.10.002

(2)

In this case a general existence result, due to Buttazzo and Dal Maso (see [11]), states that problem (1.1), with the additional constraint|Ω|mon the Lebesgue measure of the competing domains, admits a solution provided the cost functionalJsatisfies the following conditions:

(i) J is lower semicontinuous for theγ-convergence, suitably defined;

(ii) J is monotone decreasing for the set inclusion.

When the surrounding boxDis unbounded the existence result above is no longer true, as some simple examples show. In the case D=R^d a quite different approach to the proof of the existence of optimal domains has been considered by Bucur in [5,6], using a refined argument related to the Lions concentration-compactness principle (see [22]), and by Mazzoleni and Pratelli in[21]using a more direct approach. However, the latter approach only works in the caseD=R^d, while the concentration-compactness approach seems more flexible for our purposes.

In this paper we consider problem (1.1) where the admissible classAis defined through aninternalconstraint:

A=

Ω: D⊂Ω⊂R^d, Ωquasi-open, |Ω|m

, (1.2)

where D is a fixed quasi-open set of finite measure, possibly unbounded. We consider mainly the cases J (Ω)= λk(Ω); the case of general monotone decreasing functionals is at present still open (see Section6).

In spite of its simplicity, even for cost functionals likeJ (Ω)=λ₁(Ω), the existence proof is rather involved, and several interesting questions arise. For this functional, together with the existence of a solution, we prove some global properties for the optimal set: it has to lie at a finite distance fromD(in particular the optimal set is bounded, provided D is bounded), it has finite perimeter outsideD, it is an open set as soon as its measure is strictly greater than the measure of (the quasi-connected) D. Local regularity properties, outsideDare not discussed here, being similar to the bounding box situation, and we refer the reader for instance to [4]. We discuss as well the existence question forJ (Ω)=λ_k(Ω). We refer the reader to[6]and to[21]for the analysis of these functionals in the absence of any inclusion constraint inR^d.

It is convenient for our purposes to consider also the problem min

λk(Ω)+Λ|Ω|: D⊂Ω⊂R^d, Ωquasi-open

, (1.3)

where the measure constraint |Ω|mis replaced by the Lagrange multiplier penalizationΛ|Ω|. The relations between the constrained problem

min

λk(Ω): D⊂Ω⊂R^d, Ωquasi-open,|Ω|m

, (1.4)

and the penalized version (1.3) have been analyzed fork=1 in[4], while for generalkonly a partial result is available (see Lemma5.10), which is enough for our purposes.

The existence of an optimal domain for problem (1.4), as well as for its penalized version (1.3), is proved in Theorem4.7.

2. Notations and preliminaries

We introduce here the main tools we use; further details can be found for instance in[7,9].

In the sequel, we will work in the Euclidean spaceR^dwithd2. Given a subsetE⊂R^d we define the capacity ofEby

cap(E)=inf

|∇u|²dx+

u²dx: u∈UE

,

whereUEis the set of all functionsuof the Sobolev spaceH¹(R^d)such thatu1 almost everywhere in a neighbor- hood ofE. If a propertyP (x)holds for allx∈Eexcept for the elements of a setZ⊂Ewith cap(Z)=0, we say thatP (x)holdsquasi-everywhere(shortlyq.e.) onE, whereas the expressionalmost everywhere(shortlya.e.) refers, as usual, to the Lebesgue measure, that we often denote by| · |.

A subsetΩofR^dis said to bequasi-openif for everyε >0 there exists an open subsetΩ_ε ofR^d, withΩ⊂Ω_ε, such that cap(Ωε\Ω) < ε. Similarly, a function f :R^d→R is said to be quasi-continuous (resp.quasi-lower

(3)

semicontinuous) if there exists a decreasing sequence of open sets(ω_n)_n>0such that limn→∞cap(ωn)=0 and the restrictionf_noff to the setω^c_nis continuous (resp. lower semicontinuous). It is well known (see for instance[23]) that every function u∈H¹(R^d)has a quasi-continuous representativeu, which is uniquely defined up to a set of˜ capacity zero, and given by

˜

u(x)=lim

ε→0

1

|Bε(x)|

Bε(x)

u(y) dy,

whereB_ε(x)denotes the ball of radiusεcentered at x. We often identify the functionu with its quasi-continuous representative u; in this way, we have that quasi-open sets can be characterized as the sets of strict positivity of˜ functions inH¹(R^d)and that the capacity can be equivalently defined by

cap(E)=min

|∇u|²dx+

u²dx: u∈H¹ R^d

, u1 q.e. onE

.

The closureDof a quasi-open setDdepends on the representative set ofDwhich is only defined up to a set of capacity zero. A canonical minimal representative ofDcan be defined as

D=

cap(N )=0

{Cclosed: C⊇D\N},

which containsDq.e. since it can be reduced to a countable intersection.

For every quasi-open setΩ⊂R^dwe denote byH₀¹(Ω)the space of all functionsu∈H¹(R^d)such thatu=0 q.e.

onR^d\Ω, with the Hilbert space structure inherited fromH¹(R^d), u, v_H¹

0(Ω)= u, vH¹(R^d)=

∇u∇v dx+

uv dx.

The usual properties of Sobolev functions on open sets extend to quasi-open sets.

LetΩbe a quasi-open set of finite measure. ByR_Ω we denote the resolvent operator of the Laplace equation with Dirichlet boundary condition,

R_Ω :L² R^d

→L² R^d

,

whereRΩ(f )is the weak solution of the equation −u=f ∈L²(R^d),

u∈H₀¹(Ω).

It is well known that R_Ω is a compact, self adjoint and positive operator, so its spectrum consists on a discrete decreasing sequence of eigenvalues, which we denote (counting the multiplicities) by 1/λk(Ω). From now on, we call λ_k(Ω)the eigenvalues of the Dirichlet Laplacian.

3. Theγ-convergence

We endow the admissible class of domains with the following notion of convergence.

Definition 3.1. Let (Ωn)n be a sequence of quasi-open sets of uniformly bounded measure. We say that Ωn

γ-converges to Ω if the resolvent operatorsRΩ_n converge to the resolvent operator RΩ in the operator norm of L(L²(R^d)).

Theγ-convergence is metrizable but not compact (see for instance[7, Chapter 4]and[12]). This convergence is very strong and the eigenvaluesλ_k(Ω)turn out to beγ-continuous. Its (local) compactification has been characterized in[13]as a space of measures.

We denote byM0the set of capacitary measures onR^d, that is the set of all Borel measures, possibly taking the value+∞, vanishing on all sets of zero capacity. Observe that for each Borel setSthe measure

(4)

∞S(B)=

0 if cap(B∩S)=0, +∞ otherwise

is a capacitary measure.

For each capacitary measureμ, we define the linear vector space H_μ¹=H¹

R^d

∩L² R^d, μ

=

u∈H¹ R^d

:

R^d

|u|²dμ <∞

.

TakingΩa quasi-open set,S=Ω^candμ= ∞SgiveH_μ¹=H₀¹(Ω). In[10]it was shown that the spaceH_μ¹, endowed with the scalar product

u, v =

R^d

∇u∇v dx+

R^d

uv dx+

R^d

uv dμ,

is a Hilbert space. Moreover, the spaceH_μ¹is separable when seen as a subset of the separable metric spaceH¹(R^d).

If{u_n}n0⊂H_μ¹is a dense countable subset, then we define the regular set of the capacitary measureμ∈M0as Ω_μ=

n0

{u_n=0}.

Notice that ifμ= ∞S, we haveΩ_μ=S^c. If the setΩ_μhas finite Lebesgue measure, then u²=

R^d

|∇u|²dx+

R^d

|u|²dμ,

is an equivalent norm onH_μ¹. We define the resolventRμas the map Rμ:L²

R^d

→L² R^d

,

which associates to each functionf ∈L²(R^d)the solutionuof the relaxed problem formally written as

−u+μu=f, u∈H_μ¹,

which has to be rigorously defined in the weak form

⎧⎪

⎪⎨

⎪⎪

⎩

R^d

∇u∇ϕ dx+

R^d

uϕ dμ=

R^d

f ϕ dx ∀ϕ∈H_μ¹,

u∈H_μ¹.

Ifμis a capacitary measure with regular set of finite Lebesgue measure thenR_μis a compact, self adjoint, positive operator and we denote byλ_k(μ)the eigenvalues ofR⁻_μ¹. In this case the constant function 1 is in the dual space(H_μ¹) andRμcan be extended to an operator from(H_μ¹)toH_μ¹, so we can definewμ:=Rμ(1)and we haveΩμ= {wμ>0}

up to zero capacity sets.

We consider the following relation of equivalence onM0: μ1∼μ2 ⇐⇒ μ1(Ω)=μ2(Ω), ∀Ωquasi-open.

From now on, we work with the quotient setM0/∼which we still denote byM0and we call its elements capacitary measures. We introduce the following convergence onM0:

Definition 3.2.We say thatμnγ-converges toμ, if the sequence of regular setsΩμ_nis of uniformly bounded Lebesgue measure and

R_μ_n^L^(L

2(R^d))

−→ R_μ.

(5)

Remark 3.3.With the definition above, we have the equivalence μ_n−→^γ μ ⇐⇒ (w_μ_n)_n₀converges inL²

R^d tow_μ.

Indeed, for the “⇐” implication, we refer to[5, Proposition 3.3]. For the direct implication, the proof is immediate.

On the one hand, we have

R_μ_n(1Ω_μn)−R_μ(1Ω_μn)→0 inL² R^d

, and on the other hand

R_μ_n(1Ωμ)−R_μ(1Ωμ)→0 inL² R^d

. Making the difference we get that

R_μ_n(1Ω_μn)−R_μ_n(1Ωμ)+R_μ(1Ωμ)−R_μ(1Ω_μn)

L²(R^d)→0 and using the maximum principle we conclude with

R_μ_n(1)−R_μ(1)

L²(R^d)→0.

Definition 3.4.We say that the sequence of capacitary measuresμn γloc-converges to the capacitary measureμ, if for each bounded open setω⊂R^d, we have that the sequence of capacitary measuresμn∨ ∞ω^c γ-converges to the capacitary measureμ∨ ∞ω_c.

Remark 3.5.In[3, Definition 2.7]theγloc-convergence introduced above was calledγ-convergence (see also[13]) and was related to theΓ-convergence inL²(R^d)of the integral functionals

F_μ(u, ω)=

R^d

|∇u|²dx+

R^d

u²dμ+χ_H1 0(ω)(u),

for each bounded open setω⊂R^d, where χ_H1

0(ω)(u)=

0 ifu∈H₀¹(ω), +∞ otherwise.

In[13], it was also proven that with this convergence the spaceM0is metrizable[13, Theorem 4.9]and compact[13, Theorem 4.14].

For eacht >0, we will denote withM^t₀the following set of capacitary measures M^t₀=

μ∈M0: |Ω_μ|t .

Proposition 3.6.The setM^t₀endowed with the metric dγ(μ1, μ2)= wμ₁−wμ₂L²(R^d),

is a complete metric space.

Proof. Let(μ_n)_n₀be a sequence such that|Ω_μ_n|tand(w_μ_n)_n₀converges strongly inL²(R^d)to some function w∈H¹(R^d). By the compactness of theγloc-convergence, each subsequence of(μ_n)_n₀has aγloc-convergent subsequence, which we still denote byμ_nand whose limit isμ. By Remark 5.6 in[5], we have thatw=w_μ,|Ω_μ|t andμ_n γ-converges toμ. Sinceμis uniquely determined by the relationw=w_μ(see[14, Theorem 5.1]), we have the thesis. 2

The proposition below deals with the continuity ofλ_k with respect to theγ-convergence.

(6)

Proposition 3.7. Consider a sequence (Ω_n)_n₀ of quasi-open sets of uniformly bounded measure such that Ω_n γ-converges to the capacitary measureμwith regular setΩ_μ. Then, for everyk1

λ_k(Ω_μ)λ_k(μ)= lim

n→∞λ_k(Ω_n).

Proof. By Remark3.3,RΩ_n→Rμin the operator norm ofL(L²(R^d)), and so we have λ_k(Ω_n)→λ_k(μ).

The inequality

λ_k(Ω_μ)λ_k(μ),

now follows as a consequence of the inequality of the measures∞Ω_μ^c(B)μ(B), for each quasi-open setB, in the min – max definition of the eigenvalues. 2

4. Existence of an optimal set

A fundamental tool allowing to understand the behavior of a minimizing sequence in R^d is the concentration- compactness result (see [5, Theorem 2.2]) for the resolvent operators. We adapt it below in order to manage the internal constraint. The main changes deal with the compactness situation, where translations disappear.

Theorem 4.1.Let(Ω_n)_n₀be a sequence of quasi-open sets of uniformly bounded measure, all containing a given quasi-open setD. Then, there exists a subsequence, still denoted by(Ω_n)_n₀, such that one of the following situations occurs.

(i) Compactness.The sequence(Ω_n)_n₀γ-converges to a capacitary measureμandR_Ω_n converges in the uniform operator topology ofL²(R^d)toRμ. Moreover, we have thatD⊂Ωμ.

(ii) Dichotomy.There exists a sequence of subsetsΩ˜_n⊆Ω_n, such that:

• R_Ω_n−R_Ω_˜

n_L_(L²₍_R^d_),L²₍_R^d₎₎→0;

• ˜Ω_nis a union of two disjoint quasi-open setsΩ˜_n=Ω_n⁺∪Ω_n⁻;

• d(Ω_n⁺, Ω_n⁻)→ ∞;

• lim infn→∞|Ω_n^±|>0;

• lim sup_n_→∞|Ω_n⁺∩D| =0orlim sup_n_→∞|Ω_n⁻∩D| =0.

Proof. Since(Ωn)n1is a sequence of quasi-open sets of uniformly bounded measure we can apply[5, Theorem 2.2].

If the compactness situation holds in [5, Theorem 2.2], then one can take the sequencey_n introduced there to be convergent. In fact, suppose that y_n is divergent and notice that the solutionw_D₊_y_n is justw_D translated to the left byy_n. By the maximum principle, we have thatw_Ω_n₊_y_nw_D₊_y_nand so

w_D₊_y_nw_Ω_n₊_y_ndx

w_D² dx >0.

Sincey_n→ ∞, we have thatw_D₊_y_n0 weakly inL². By the strong convergence ofw_Ω_n₊_y_nwe have

w_D₊_y_nw_Ω_n₊_y_ndx→0,

which is a contradiction and so we have that yn is bounded and thus, we can extract a convergent subsequence. It remains to prove that we can takey_n=0, for everyn∈N. Lety_n→yand setw=L²-limn→∞w_Ω_n₊_y_n. We have

wΩ_n−w(· −y)

L²(R^d)wΩ_n−w(· −yn)

L²(R^d)+w(· −yn)−w(· −y)

L²(R^d)

w_Ω_n₊_y_n−w

L²(R^d)+w(· −y_n)−w(· −y)

L²(R^d),

and both last terms converge to zero asn→ ∞. Thus, by Proposition3.6, we have thatw_Ω_nγ-converges to a capacitary measureμandw(· −y)=w_μ. Moreover, sincew_Ω_nw_Dfor everyn∈N, we have that alsow_μw_Dand so D⊂Ω_μa.e.

(7)

If the dichotomy occurs in[5, Theorem 2.2], we have only to show that we can choose a subsequence such that lim sup_n_→∞|Ω_n⁺∩D| =0 or lim sup_n_→∞|Ω_n⁻∩D| =0. In fact, sinced(Ω_n⁺, Ω_n⁻)→ ∞, we have that one of the sequences of characteristic functions 1_Ω+

n or 1_Ω−

n has a subsequence, which converges weakly inL²(R^d)to zero.

Taking into account that 1D∈L²(R^d), we have the thesis. 2

We study the existence of a solution for problem (1.4). We notice that if a solutionΩof problem (1.4) exists, then necessarily the measure ofΩ is precisely equal tom. Indeed, assume by contradiction thatΩ is an optimal set with measure strictly less thanm. Since the mapt→ |t Ω∪D|is continuous, we can choose somet >1 such that the set Ω_t =t Ω∪Dis still of measure less thanm. Butλ_k(Ω_t)λ_k(tΩ)=_t¹2λ_k(Ω)and so, thek-th eigenvalue strictly diminishes, which contradicts the optimality ofΩ.

In the sequel we study problem (1.4) for anyk; we will see that the most complete result is fork=1, while the casek2 requires some additional assumptions on the internal constraintD(see Theorem4.7).

Theorem 4.2.LetDbe a quasi-open set and letm|D|. Then, the problem min

λ₁(Ω):Ω quasi-open, D⊂Ω, |Ω|m

, (4.1)

has at least one solution.

Proof. We consider a minimizing sequence(Ω_n)_n₁with the property that lim infn→∞|Ω_n|is minimal. Clearly, this value cannot be equal to zero. According to Theorem4.1, if we are in the compactness situation, for a subsequence (still denoted with the same indices) there exists a measureμsuch thatΩ_nγ-converges toμ. Moreover, the regular setΩ_μis admissible since|Ω_μ|mandD⊂Ω_μ. By Proposition3.7we obtain thatΩ_μis a solution of (4.1).

If we are in the dichotomy situation, we get a contradiction. SinceΩ_n⁺andΩ_n⁻ are at positive distance, we may assume thatλ1(Ω_n⁺∪Ω_n⁻)=λ1(Ω_n⁺). Then, the sequenceΩ_n⁺∪Dis also minimizing since|λ1(Ω_n⁺)−λ1(Ωn)| →0 (see[5, Proposition 3.7]), but either

lim inf

n→∞ Ω_n⁺∪D<lim inf

n→∞ |Ω_n|,

or|Ω_n⁻\D| →0. The first assertion is in contradiction with our assumption on the choice of a least measure minimizing sequence. The second assertion is also impossible, since it implies thatd(Ω_n⁺,{0})→ +∞, otherwise the measure ofDwould be infinite. Consequently, since the measure ofDis finite, we get that|Ω_n⁺∩D| →0 and consider the ballBof measure equal to lim sup|Ω_n⁺|. Therefore,B∪Dis a solution for every position of the ballB. In particular, this leads to a contradiction if the ball intersects, but not cover, a quasi-connected component ofD. 2

Remark 4.3.Let us notice that from every minimizing sequence we can extract aγ-convergent subsequence. The basic observation is that any minimizing sequence for which lim infn→∞|Ω_n|is minimal leads to an optimal set, which necessarily has the measure equal tom. Since the Lebesgue measure is lower semicontinuous for theγ-convergence, this means that any minimizing sequence should satisfy limn→∞|Ω_n| =m excluding the dichotomy in the proof above.

In the sequel we show a result which gives a rather explicit behavior of a minimizing sequence for problem (1.4).

For everym >0 we introduce the value λ^∗_k(m)=inf

λ_k(Ω): Ω quasi-open, |Ω|m .

Following[6], there exists a bounded quasi-open setΩ^m, with measure equal tomsuch thatλk(Ω^m)=λ^∗_k(m).

Theorem 4.4.LetDbe a quasi-open set and letm|D|. Fork∈N,k2, we define α_k=inf

λ_k(Ω): Ωquasi-open, D⊂Ω,|Ω|m . One of the following assertions holds:

(8)

(i) problem(1.4)has a solution;

(ii) there existl∈ {1, . . . , k−1}and an admissible quasi-open setΩ such thatα_k=λ_k₋_l(Ω)=λ^∗_l(m− |Ω|);

(iii) there existsl∈ {1, . . . , k−1}such thatα_k=λ^∗_l(m− |D|) > λ_k₋_l(D).

Proof. Let us consider a minimizing sequence(Ω_n)_n₁with the property that lim infn→∞|Ω_n|is minimal. If compactness occurs in Theorem4.1, then the existence of a solution follows as in Theorem4.2.

If dichotomy occurs, as in Theorem4.2we may assume that Ω_n⁺→α⁺, Ω_n⁻→α⁻, Ω_n⁺∩D→0.

Then, up to a subsequence there existsl∈ {1, . . . , k−1}such that one of the two possibilities below holds:

(A) |λk(Ωn)−λk−l(Ω_n⁻)| →0 andλl(Ω_n⁺)λk−l(Ω_n⁻)λl+1(Ω_n⁺);

(B) |λ_k(Ω_n)−λ_l(Ω_n⁺)| →0 andλ_k₋_l(Ω_n⁻)λ_l(Ω_n⁺)λ_k₋_l₊₁(Ω_n⁻).

We may take the maximallwith such a property. We use now an induction argument as follows. Fork=1 as proved in Theorem4.2, dichotomy does not occur, so the compactness gives (i). Assume that for 1, . . . , k−1 Theorem4.4is true. We prove it fork. If compactness occurs, then (i) holds. If dichotomy occurs and we are in situation (A) we get that(Ω_n⁻∪D)_nis minimizing for thek−leigenvalue with the inclusion constraint and the corresponding measure m−α⁺lim infn→∞|Ω_n⁻∪D|. Sincel is maximal with this property, for the sequence(Ω_n⁻∪D)_n dichotomy cannot occur again, so finally (ii) holds.

If (B) occurs, then|Ω_n⁻\D| →0 and we are in situation (iii). 2

Remark 4.5. Theorem 4.4gives a complete description of the behavior of a minimizing sequence forλ_k,k2.

Assertion (i) implies the existence of a solution. As well ifDhas some suitable geometric properties (for instance if Dis bounded), both alternatives (ii) and (iii) lead to the existence of a solution. In fact, if for instance (ii) occurs, the setΩ∪Ω^m^−|^Ω^|is a minimizer provided thatΩ∩Ω^m^−|^Ω^|= ∅. Similarly, for (iii) the setD∪Ω^m^−|^D^|is a minimizer, provided thatD∩Ω^m^−|^D^|= ∅.

The result below is analogous to Theorem4.2and Theorem4.4, corresponding to the problem (1.3). We omit the proof, since it is the same as that of Theorem4.2and Theorem4.4.

Theorem 4.6.LetDbe a quasi-open set of finite measure and letΛ >0. Fork∈N, we define β_k=inf

λ_k(Ω)+Λ|Ω|: Ωquasi-open, D⊂Ω .

Suppose thatΩ_nis a minimizing sequence forβ_ksuch that there exists the limitm:=limn→∞|Ω_n|. Ifk=1, then the problem(1.3)has solution. Ifk2, then one of the following assertions holds:

(i) problem(1.3)has a solution;

(ii) there existl∈ {1, . . . , k−1}and an admissible quasi-open setΩ such thatβ_k=λ_k₋_l(Ω)=λ^∗_l(m− |Ω|);

(iii) there existsl∈ {1, . . . , k−1}such thatβ_k=λ^∗_l(m− |D|) > λ_k₋_l(D).

We conclude summarizing the assertions above into the following existence result, whose proof will be given in Section5.

Theorem 4.7.LetD⊂R^d be a quasi-open set of finite measure such that

for anyR >0, there existsx∈R^dsuch thatB_R(x)∩D= ∅. (4.2) Then, for anyk >0andΛ >0, there exists a solution of(1.3). Moreover, ifDsatisfies in addition also the assumption

lim sup

t→1⁺

|D\t D| t−1 <∞,

then problem(1.4)admits a solution, for anyk >0andm|D|.

(9)

Remark 4.8.The assumption (4.2) is crucial for the proof of Theorem4.7. We do not know whether the existence of an optimal domain occurs without it.

5. Qualitative properties of the optimal sets

A natural question that arises in the shape optimization problems with constraints like (4.1) is to understand the influence of the inclusion domainD on the optimal sets: does boundedness and/or convexity ofDimply the same properties on the optimal set? As we shall see, the answer is positive for the boundedness constraint, but negative for the convexity constraint.

5.1. Regularity of the optimal set fork=1

In this section we deal with the penalized version of problem (4.1) min

λ₁(Ω)+Λ|Ω|: Ω⊂R^d, Ω quasi-open,D⊂Ω

, (5.1)

for someΛ >0. For the local equivalence of the two problems we refer the reader to[4]. As well, we refer the reader to[4]for a complete analysis of a similar problem, in which the internal constraintD⊂Ω is replaced by an external constraintΩ⊂D, with a bounded open setD.

In the case of internal constraint, new behaviors can be noticed with respect to[4]; we prove that the optimal set of (5.1) is open even ifDis only quasi-open, provided thatDis quasi-connected and the optimal set has a measure strictly greater than|D|.

Definition 5.1.We say that the quasi-open setDis quasi-connected if for every pair of non-empty quasi-open setsA₁ andA₂having intersection of positive capacity withDand such thatD⊂A₁∪A₂, we get cap(A1∩A₂) >0.

The quasi-connectedness has a topological counterpart. Indeed, a quasi-open, quasi-connected setA has a fine interior (which differs fromAby a set of zero capacity) which is finely connected (the fine topology being the coarsest topology making all superharmonic functions continuous). A non-negative superharmonic function inH₀¹(A)withA finely connected, is either equal to 0 or is strictly positive (see[8,16,20]).

Example 5.2.The assumptionDquasi-connected is essential to obtain that the optimal setΩ is open even whenD is only quasi-open. Indeed, considerD=B₀∪D₀, whereB₀is a large ball andD₀is a quasi-open and not open set, whose distance fromB₀is large enough and whose measure is less than |B₀|. It is easy to see that in this case the optimal set is of the formB∪D₀, whereBis a ball containingB₀.

Remark 5.3.In spite of the example above, if D is an arbitraryopen set, then every optimal setΩ for problem (5.1) is open. Indeed, a careful inspection of the proof of Proposition5.6below gives that {u >0}is open; since Ω= {u >0} ∪Dwe have thatΩ is open.

In the following, without loss of generality we assume thatΛ=1.

Remark 5.4.The existence of a solution to (5.1) follows by the same argument we used in the proof of Theorem4.2 and so we omit the proof.

LetDbe a quasi-open, quasi-connected set of finite measure. LetΩbe a solution of problem (5.1), letλ:=λ₁(Ω), and letu:=u_Ω be the first normalized eigenfunction:

−u=λu,

u∈H₀¹(Ω), uL²=1.

AsDis quasi-connected, ifΩ is optimal, thenuis a solution of the minimization problem min

|∇v|²dx

v²dx +{v >0}: v∈H¹ R^d

, v0, D⊂ {v >0}

. (5.2)

(10)

The following lemma has a proof similar to[1, Lemma 3.2]and[4, Lemma 3.1], so we omit it.

Lemma 5.5.Letube a supersolution of problem(5.2), in the sense that for anyv∈H¹(R^d)such thatvu, we have |∇v|²dx

v²dx +{v >0}

|∇u|²dx

u²dx +{u >0}.

Then there is a constantC depending only on the dimensiond such that for eachr >0, the following implication holds:

Cr 1

|∂Br|

∂Br

u dH^d⁻¹ ⇒ B_r⊂ {u >0}. (5.3)

The next proposition follows the approach first introduced in[1]; nevertheless, we give the proof below to stress the fact that the quasi-open internal constraint does not change the argument too much.

Proposition 5.6.LetDbe a quasi-open, quasi-connected set of finite measure. Every solutionΩ of problem(5.1)is an open set up to a set of capacity0.

Proof. SinceDis quasi-connected, the optimal domainΩ is quasi-connected too. Indeed, otherwiseΩ would have at least two quasi-connected components, one of which containsD. Then:

(i) eitherλ₁(Ω)is realized on a component not containingD, in which caseD∪B, for a suitable ballB, is a better competitor;

(ii) orλ₁(Ω)is realized on the componentΩ₁containingD, in which case replacingΩby a suitable enlargement of Ω1gives again a better competitor.

Letube a solution of (5.2). We prove that ifu(x) >0, thenuis positive in a small ball centered atx. Without loss of generality, we can suppose thatx=0 and that 0 is a regular point in the sense that

u(0)=lim

r→0

1

|B_r(0)|

B_r(0)

u(y) dy.

Denote byϕ_r the solution of −ϕr=1,

ϕ_r∈H₀¹(B_r), (5.4)

whereB_r denotes the ball centered in 0 of radiusr. An explicit computation gives ϕ_r(y)=r²− |y|²

2d .

Sinceu0 andu+λu=0 on{u >0}, we have thatu+λu0 in distributional sense onR^d. Indeed, arguing as in[19, Lemma 7.2.5], we consider, for any non-negativeϕ∈H¹(R^d), the test functionp_n(u)ϕ∈H₀¹({u >0}), where p_n(t)is 0, fort0, 1, fort1/nand equalsnt, fort∈ [0,1/n]. An explicit calculation gives

0=lim sup

n→∞

u+λu, p_n(u)ϕ

u+λu, ϕ.

As a consequence, using the boundedness ofu(see[15]), we obtain

u− u_∞λϕ_r

−λu+λu_∞0

on each ballB_r, so the functionu− u_∞λϕ_r is subharmonic onB_r. By Poisson’s formula, we have

(11)

u(0)− u_∞λϕ_r(0)C(d) 1

|∂Br|

∂Br

u(y) dH^d⁻¹(y), u(0)− u_∞λC₁r²C(d) 1

|∂B_r|

∂Br

u(y) dH^d⁻¹(y).

Suppose thatu(0) >0. Then, choosingrsmall enough, we have u(0)2C(d)

|∂B_r|

∂Br

u(y) dH^d⁻¹(y).

Now chooseCas in Lemma5.5andrsuch that 2rCC(d)u(0). Then Cr 1

|∂B_r|

∂B_r

u(y) dH^d⁻¹(y),

and sou >0 onB_r. 2

Remark 5.7.Alternatively, one can formulate the proposition above, requiring that the inclusionD⊂Ωholds quasi- everywhere, and in this case the optimal sets{u >0}in (5.2) are open anduis continuous. In fact, in[1]it is proven a stronger result on the Lipschitz continuity ofu, even if this does not provide a higher regularity of the optimal setΩ. Remark 5.8.The regularity of the free parts of the boundary is the same as in[4, Theorem 1.2], being independent of the fact that the inclusion constraint is internal or external.

Remark 5.9.IfDis a quasi-open set such that there does not exist an open set containingD and having the same Lebesgue measure, then Proposition5.6asserts that the measure of any optimal set is strictly greater than the measure ofD.

In general, this is not the case ifDis an open set. Indeed, following a simple computation one can considerDto be a ballBand take a constantΛlarge enough, so that the optimal set isBitself. More generally, if the partial metric derivative of the first eigenvalue onDis finite, i.e.

λ₁(D):= lim sup

|E\D|→0, E⊃D

λ₁(D)−λ₁(E)

|E\D| <+∞,

then for every Λ > λ₁(D) there exists Λ> Λ such that the optimal solution of (5.1) with Λ is D. Indeed, by contradiction for everyΛ > λ₁(D)there existsε >0 such that for everyΩ⊃Dsuch that|Ω||D| +εwe have

λ₁(D)+Λ|D|λ₁(Ω)+Λ|Ω|.

Then, replacingΛwithΛ> Λsuch thatΛ^λ¹^(D)_ε , we get thatDis a global minimizer.

5.2. Bounded constraint implies bounded minimizers

In this paragraph we consider the penalized version (1.3). In fact the following lemma gives a relation between the solutions of the constrained problem (1.4) and the subsolutions of the Lagrange multiplier penalized version (1.3).

Adapting a notion introduced in[6], we say thatΩ^∗is a shape subsolution forλkif there existsc >0 such that λ_k

Ω^∗

+cΩ^∗λ_k(Ω)+c|Ω| ∀D⊂Ω⊂Ω^∗. (5.5)

For simplicity, we consider the internal constraintDregular enough such that lim sup

t→1⁺

|D\t D|

t−1 <∞. (5.6)

This condition is for instance satisfied ifDis bounded and Lipschitz, or ifDis star shaped.

(12)

Lemma 5.10.Suppose that the internal constraintDsatisfies(5.6)and assume thatΩ_mis a solution of min

λ_k(Ω): D⊂Ω⊂R^d, Ω-quasi-open,|Ω|m

. (5.7)

ThenΩ_mis a shape subsolution forλ_k.

Proof. We first notice that|Ω_m| =m. Suppose by contradiction, that for eachε >0, there is some quasi-open setΩ_ε such thatD⊂Ω_ε⊂Ω_mand

λ_k(Ω_ε)+ε|Ω_ε|< λ_k(Ω_m)+ε|Ω_m|. (5.8) By the compactness of the inclusionH₀¹(Ω)⊂L²(Ω), we can suppose, up to a subsequence thatΩε γ-converges to some capacitary measureμ, whose regular setΩμis such that

|Ω_μ|lim inf

ε→0 |Ω_ε|, λ_k(Ω_μ)λ_k(μ)=lim

ε→0λ_k(Ω_ε)λ_k(Ω_m),

where the last inequality is due to (5.8) and Lemma3.7. Thus, we obtain thatΩ_μis a solution of (5.7) and so|Ω_μ| =m andλ_k(Ω_μ)=λ_k(Ω_m).

LetΩ_ε=t_εΩ_ε∪D, wheret_εis such that|Ω_ε| =m. Then, we have that λk(Ωε)+ε|Ωε|< λk(Ωm)+ε|Ωm|

λ_k Ω_ε

+εΩ_ε λk(tεΩε)+ε|tεΩε∪D|

1

t_ε²λk(Ωε)+ε

|tεΩε| + |D\tεΩε|

1

t_ε²λk(Ωε)+ε

|tεΩε| + |D\tεD|

, (5.9)

and so t_ε²−1

t_ε² λ_k(Ω_ε)ε t_ε^d−1

|Ω_ε| + |D\t_εD|

. (5.10)

By hypothesis (5.6) passing to the limit ast_ε→1⁺, there is some constantCsuch that

λ_k(Ω_ε)εC, (5.11)

forεsmall enough. But, by (5.9),λ_k(Ω_ε)→λ_k(Ω_m)and so, we have a contradiction. 2

We give the following technical result for which we refer to[1, Lemma 3.4]and to[4, Lemma 3.1]in the case k=1 and to[6, Lemma 2.3]for the general case.

Lemma 5.11.LetΩbe a shape subsolution forλ_kand letwbe the solution of the equation −w=1∈Ω,

w∈H₀¹(Ω).

Then there exist two constantsC₀andr₀such that for eachx∈R^d such thatd(x, D) > r₀and for eachr < r₀the following implication holds:

wL^∞(Br)C₀r

⇒ (w=0on B^r

2). (5.12)

The proof of this lemma relies on the fact that shape subsolutions forλ_kare localshape subsolutionsfor the energy problem (see[6, Definition 2.1]for more details).