Compactness and dichotomy in nonlocal shape optimization
aa r X i v : . [ m a t h . A P ] J un COMPACTNESS AND DICHOTOMY IN NONLOCAL SHAPEOPTIMIZATION
ENEA PARINI AND ARIEL SALORT
Abstract.
We prove a general result about the behaviour of minimizing sequences for nonlocalshape functionals satisfying suitable structural assumptions. Typical examples include functionsof the eigenvalues of the fractional Laplacian under homogeneous Dirichlet boundary conditions.Exploiting a nonlocal version of Lions’ concentration-compactness principle, we prove that eitheran optimal shape exists, or there exists a minimizing sequence consisting of two “pieces” whosemutual distance tends to infinity. Our work is inspired by similar results obtained by Bucur inthe local case. Introduction
A significant task in Shape Optimization consists in proving existence of minimizing sets, ina suitable class, for shape functionals of the kind Ω J (Ω) = F ( λ (Ω) , ..., λ m (Ω)) , where m ∈ N ∗ , Ω ⊂ R N , and λ (Ω) , ..., λ m (Ω) are eigenvalues of some differential operator. Inthe case of the Laplacian under Dirichlet boundary conditions, and J (Ω) = λ k (Ω) , existenceof optimal shapes among all measurable sets with prescribed Lebesgue measure has been achallenging open problem for a long time. Apart from the simpler cases k = 1 and k = 2 ,where the Faber-Krahn inequality implies that the optimal shape is a ball (for k = 1 ) or thedisjoint union of two equal balls (for k = 2 ), for the general case existence in the class of quasi-open sets has been proven only recently by Bucur in [7] and by Mazzoleni and Pratelli in [17]independently. It is still an open problem to identify the optimal shapes for k ≥ , althoughnumerical simulations support some conjectures.When the differential operator under consideration is the fractional Laplacian, defined as ( − ∆) s u ( x ) := C s,N lim ε → Z R N \ B ε ( x ) u ( x ) − u ( y ) | x − y | N +2 s dy, where s ∈ (0 , and C s,N is a normalization constant, the situation is quite different. While theball minimizes again the first eigenvalue under a volume constraint, the problem(1.1) min { λ (Ω) | Ω ⊂ R N , | Ω | = c } , where c > , and | Ω | is the Lebesgue measure of Ω , does not have a solution. Indeed, it wasproven by Brasco and the first author [4] that, for every admissible set Ω , λ (Ω) > λ ( e B ) , where e B is a ball of volume c , and that a minimizing sequence { Ω n } n ∈ N such that λ (Ω n ) → λ ( e B ) is given by the union of two disjoint balls of volume c , such that their mutual distancetends to infinity. This means that, in the nonlocal case, a general existence result as in [7] or [17]can not hold true. On the other hand, if one restricts the minimization to quasi-open sets whichare contained in a fixed open set D ⊂ R N , a generalization of the existence result by Buttazzo and Dal Maso [8] holds true, as shown by Fernández Bonder, Ritorto and the second authorin [11].Inspired by the results obtained in [6] by Bucur, in this paper we prove that, in the case ofthe fractional Laplacian, for a minimizing sequence only two situations can occur: compactness ,which implies, under some assumptions, existence of an optimal shape; or dichotomy , whichmeans that the sequence essentially behaves as the union of two disconnected sets, whose mutualdistance tends to infinity, as in Problem (1.1). To prove the result, we make use of a nonlocalversion of the celebrated concentration-compactness principle of Lions [16]. Although somegeneralizations of Lions’ result to the fractional case are stated in the literature, the proofscontained therein do not seem completely satisfactory, and therefore we prefer to provide ourown proof. Our first main result reads as follows. Theorem 1.1.
Let { u n } n ∈ N be a bounded sequence in H s ( R N ) with R R N | u n | → λ for n → + ∞ .Then there exists a subsequence { n k } k ∈ N such that one of the following three cases occur: (i) Compactness: there exists { y k } k ∈ N ⊂ R N such that ∀ ε > , ∃ R < + ∞ s.t. Z y k + B R | u n k | ≥ λ − ε. (ii) Vanishing: lim k → + ∞ sup y ∈ R N Z y + B R | u n k | = 0 ∀ R > . (iii) Dichotomy: there exists α ∈ (0 , λ ) , such that for all ε > , there exist k ∈ N , { v k } k ∈ N , { w k } k ∈ N ⊂ H s ( R N ) such that, for k ≥ k : k u n k − v k − w k k L ( R N ) ≤ δ ( ε ) → for ε → (cid:12)(cid:12)(cid:12)(cid:12) Z R N | v k | − α (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε, (cid:12)(cid:12)(cid:12)(cid:12) Z R N | w k | − ( λ − α ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε ;dist( supp v k , supp w k ) → + ∞ for k → + ∞ ; (1.2) [ u n k ] H s ( R N ) − [ v k ] H s ( R N ) − [ w k ] H s ( R N ) ≥ − ε. Then, we apply Theorem 1.1 to the sequence of torsion functions w Ω n , where { Ω n } n ∈ N is aminimizing sequence for the shape functional under consideration, which are defined as the weaksolutions of the problems(1.3) (cid:26) ( − ∆) s w Ω n = 1 in Ω n ,w Ω n = 0 in R N \ Ω n . In order to introduce our main result, we recall that a sequence { Ω n } n ∈ N of s -quasi open setsof uniformly bounded Lebesgue measure is said to γ -converge to the s -quasi open set Ω if thesolutions w Ω n of (1.3) strongly converge in L ( R N ) to the solution w Ω ∈ H s (Ω) of the problem (cid:26) ( − ∆) s w Ω = 1 in Ω ,w Ω = 0 in R N \ Ω (see Section 2 for precise definitions of s -quasi open sets). Moreover, we say that a sequence { Ω n } n ∈ N of s -quasi open sets of uniformly bounded Lebesgue measure weakly γ -converges to the s -quasi open set Ω if the solutions w Ω n of (1.3) converge weakly in H s ( R N ) , and strongly in L ( R N ) , to a function w ∈ H s ( R N ) such that Ω = { w > } . Finally, for a given s -quasi open set OMPACTNESS AND DICHOTOMY IN NONLOCAL SHAPE OPTIMIZATION 3 Ω ⊂ R N of finite measure, we denote by R Ω the resolvent operator of ( − ∆) s , which is defined asthe function R Ω : L ( R N ) → L ( R N ) such that R Ω ( f ) = u , where u is the weak solution of (cid:26) ( − ∆) s u = f in Ω ,u = 0 in R N \ Ω . We can now state our second main result, whose proof follows the ideas of [6].
Theorem 1.2.
Let { Ω n } n ∈ N be a sequence of quasi-open sets of uniformly bounded measure.Then there exists a subsequence, still denoted by the same index, such that one of the followingsituations occurs:(i) Compactness: there exists a (possibly empty) quasi-open set Ω , and a sequence { y n } n ∈ N ⊂ R N , such that y n + Ω n weakly γ -converges to Ω as n → + ∞ .(ii) Dichotomy: there exists a sequence of subsets e Ω n ⊂ Ω n such that k R Ω n − R e Ω n k L ( L ( R N )) → , e Ω n = Ω n ∪ Ω n , where dist (Ω n , Ω n ) → + ∞ and lim inf n → + ∞ | Ω in | > for i = 1 , . Theorem 1.2 gives, as a consequence, an existence result for optimal shapes for minimizationproblems, when the shape functional satisfies some structural assumptions.
Theorem 1.3.
Let A ( R N ) := (cid:8) Ω ⊂ R N | Ω s -quasi open (cid:9) and let J : A ( R N ) → ( −∞ , + ∞ ] be a shape functional satisfying the following assumptions:(i) J is lower semicontinuous with respect to γ -convergence;(ii) J is decreasing with respect to set inclusion: if Ω , Ω ∈ A ( R N ) , Ω ⊂ Ω , then J (Ω ) ≤ J (Ω ) ;(iii) J is invariant by translations;(iv) J is bounded from below.Let c > , and define (1.4) m := inf { J (Ω) | Ω ∈ A ( R N ) , | Ω | = c } . Then, one of the following situations occurs:(i) Existence of an optimal shape: there exists a s -quasi open set ˆΩ ∈ A ( R N ) such that | ˆΩ | = c and J ( ˆΩ) = m .(ii) Dichotomy: there exists a minimizing sequence { Ω n } n ∈ N with | Ω n | = c for every n ∈ N ,such that Ω n = Ω n ∪ Ω n , where Ω n , Ω n are such that dist (Ω n , Ω n ) → + ∞ , lim inf n → + ∞ | Ω in | > for i = 1 , , and J (Ω n ) → m as n → + ∞ . Theorem 1.3 applies in particular to spectral functionals of the kind J (Ω) := F ( λ (Ω) , ..., λ k (Ω)) , where k ∈ N , λ j (Ω) is the j − th eigenvalue of the Dirichlet fractional Laplacian, and F : R k → R ∪ { + ∞} is a functional which is lower semicontinuous and nondecreasing in each variable.In the local case, existence of an optimal shape and the dichotomy situation can occur atthe same time. Indeed, as we have pointed out, the classical Hong-Krahn-Szego inequalityasserts that among all domains of fixed volume, the disjoint union of two equal balls has thesmallest second eigenvalue. However, due to the nonlocal effects of the fractional Laplacian, themutual position of two connected component has influence over the second eigenvalue, implyingnonexistence of an optimal shape. Therefore it makes sense to ask whether existence of an E. PARINI AND A. SALORT optimal shape and dichotomy are two mutually exclusive situations in the nonlocal case. Up toour knowledge, this remains an open question.The manuscript is organized as follows. In section 2 we introduce some preliminary definitionsand notation. Section 3 is devoted to prove the concentration-compactness principle in thefractional setting. In section 4 we define the notion of γ - and weak γ -convergence of sets as wellas some related useful result, and finally in sections 5 and 6 we provide a proof of our mainresults. Acknowledgements.
The authors would like to express their gratitude to Lorenzo Brascoand Marco Squassina for useful discussions. This work was started during a visit of A. S.to Aix-Marseille University in October 2017. The visit was supported by CONICET PIP11220150100036CO. A.S. wants to thank the first author for his hospitality which made thevisit very enjoyable. 2.
Definitions and preliminary results
We begin this section with some definitions.2.1.
Fractional Sobolev spaces and s -capacity of sets. For s ∈ (0 , , the fractional Sobolevspace H s ( R N ) is defined as H s ( R N ) := n u ∈ L ( R N ) | [ u ] H s ( R N ) < + ∞ o , endowed with the norm k · k H s ( R N ) defined by k u k H s ( R N ) := (cid:16) k u k L ( R N ) + [ u ] H s ( R N ) (cid:17) , where [ · ] H s ( R N ) is the Gagliardo seminorm defined as [ u ] H s ( R N ) := (cid:18)Z R N Z R N | u ( x ) − u ( y ) | | x − y | N +2 s dx dy (cid:19) . The Gagliardo seminorm of a function u ∈ H s ( R N ) can also be expressed in terms of its Fouriertransform F u as [ u ] H s ( R N ) = 2 C s,N Z R N | ξ | s |F u ( ξ ) | dξ, where C s,N is the normalization constant in the definition of ( − ∆) s , given by C s,N = (cid:18)Z R N − cos ζ | ζ | N +2 s dζ (cid:19) − (see [9, Proposition 3.4]). Given a measurable set Ω ⊂ R N , for any s ∈ (0 , we define the s -capacity of Ω ascap s (Ω) = inf n [ u ] H s ( R N ) : u ∈ H s ( R N ) , u ≥ a.e. on a neighborhood of Ω o . We say that a property holds s -quasi everywhere if it holds up to a set of null s -capacity. Ameasurable subset Ω ⊂ R N is a s -quasi open set if there exists a decreasing sequence { ω n } n ∈ N ofopen subsets of R N such that cap s ( ω n ) → , as n → + ∞ , and Ω ∪ ω n is open.A function u ∈ H s ( R N ) is said to be s -quasi continuous if for every ε > there exists anopen set G ⊂ R N such that cap s ( G ) < ε and u | R N \ G is continuous. It is well-known that cap s is a Choquet capacity on R N [1, Section 2.2] and for every u ∈ H s ( R N ) there exists a unique s -quasi continuous function ˜ u : R N → R such that ˜ u = u s -quasi everywhere on R N . Therefore OMPACTNESS AND DICHOTOMY IN NONLOCAL SHAPE OPTIMIZATION 5 we will always consider, without loss of generality, that a function u ∈ H s ( R N ) coincides withits s -quasi continuous representative. If u : R N → R is s -quasi continuous, then every superlevelset { u > t } is s -quasi open.For a generic measurable set Ω ⊂ R N , we define the fractional Sobolev space H s (Ω) as H s (Ω) = { u ∈ H s ( R N ) : u = 0 s -q.e. on R N \ Ω } . The following Poincaré’s inequality holds for measurable sets of finite measure.
Proposition 2.1.
Let Ω ⊂ R N be a measurable set of finite Lebesgue measure. Then, thereexists a constant C = C ( s, | Ω | ) > such that, for every u ∈ H s (Ω) , k u k L (Ω) ≤ C [ u ] H s ( R N ) . Proof.
Let u be a function in H s (Ω) and consider the ball Ω ∗ such that | Ω ∗ | = | Ω | . Let v := | u | ∗ be the Schwarz symmetrization of | u | , as defined in [14, Definition 1.3.1]. By [2, Theorem 9.2], v ∈ H s (Ω ∗ ) , and [ v ] H s ( R N ) ≤ [ | u | ] H s ( R N ) ≤ [ u ] H s ( R N ) . By [3, Lemma 2.4], there exists C = C ( s, | Ω | ) > such that k v k L (Ω ∗ ) ≤ C [ v ] H s ( R N ) . Since symmetrization preserve the L -norm, k u k L (Ω) = k v k L (Ω ∗ ) ≤ C [ v ] H s ( R N ) ≤ C [ u ] H s ( R N ) , and the claim follows. (cid:3) The previous proposition leads to a useful compactness result.
Proposition 2.2.
Let Ω ⊂ R N be a measurable set of finite Lebesgue measure. Then, forevery bounded sequence { u n } n ∈ N in H s (Ω) , there exists a subsequence { u n k } k ∈ N and a function u ∈ H s (Ω) such that u n k → u in L (Ω) .Proof. The proof can be performed as in [3, Theorem 2.7], using the Poincaré inequality statedin Proposition 2.1. (cid:3)
Given an s -quasi open set Ω of finite Lebesgue measure and f ∈ L ( R N ) we denote by R Ω the resolvent operator of the fractional Laplacian with Dirichlet boundary conditions, that is, R Ω : L ( R N ) → L ( R N ) and R Ω ( f ) = u , where u is the weak solution of(2.1) (cid:26) ( − ∆) s u = f in Ω ,u = 0 in R N \ Ω . In particular, w Ω = R Ω (1) . It is easy to check that R Ω defines a continuous compact, self-adjointlinear operator from L ( R N ) in itself. We denote by k · k L ( L ( R N )) the corresponding operatornorm. Given an s -quasi open set Ω , we say that λ is an eigenvalue of the fractional Laplacian ifthere exists a nontrivial function u ∈ H s (Ω) , called eigenfunction , which is a weak solution of(2.2) (cid:26) ( − ∆) s u = λu in Ω ,u = 0 in R N \ Ω . According to Courant-Fischer’s min-max principle, for every s -quasi-open set Ω ⊂ R N of finiteLebesgue measure there exists a sequence { λ k (Ω) } k ∈ N of eigenvalues of the fractional Laplacian,satisfying < λ (Ω) < λ (Ω) ≤ · · · ≤ λ k (Ω) → + ∞ as k → + ∞ . E. PARINI AND A. SALORT
The first eigenvalue λ (Ω) is characterized as λ (Ω) = inf u ∈ H s (Ω) \{ } [ u ] H s ( R N ) k u k L ( R N ) and the associated first eigenfunction is unique (up to multiplicative constant) and strictly pos-itive (or negative) in Ω .Eigenfunctions satisfy the following regularity property. Proposition 2.3.
Let Ω ⊂ R N be a quasi-open set of finite Lebesgue measure, and let u ∈ H s (Ω) be an eigenfunction of the fractional Laplacian. Then, u ∈ L ∞ (Ω) .Proof. The proof can be performed as in [12, Theorem 3.2] taking into account Theorems 6.5and 6.9 from [9]. (cid:3) The concentration-compactness principle
In this section we prove Theorem 1.1.
Proof of Theorem 1.1.
All the assertions of this theorem, with exception of (1.2), follow fromthe classical concentration-compactness lemma [16, Lemma 1.1]. To prove (1.2), we suitablymodify [16, Lemma III.1]. Let ε > , and let R > be chosen as in [16, Lemma III.1]. Let usdefine two cut-off functions ϕ, ψ ∈ C ∞ c ( R N ) satisfying ≤ ϕ, ψ ≤ , ϕ ≡ on B , ϕ ≡ on R N \ B and ψ ≡ on B , ψ ≡ on R N \ B . Denote by ϕ R , ψ R the functions defined by(3.1) ϕ R ( x ) := ϕ (cid:16) xR (cid:17) , ψ R ( x ) := ψ (cid:16) xR (cid:17) . For any function u ∈ H s ( R N ) with [ u ] H s ( R N ) ≤ M we have Z R N Z R N | ϕ R ( x ) u ( x ) − ϕ R ( y ) u ( y ) | | x − y | N +2 s dx dy = Z R N Z R N | ϕ R ( x ) u ( x ) + ϕ R ( x ) u ( y ) − ϕ R ( x ) u ( y ) − ϕ R ( y ) u ( y ) | | x − y | N +2 s dx dy = Z R N Z R N | ϕ R ( x ) | | u ( x ) − u ( y ) | | x − y | N +2 s dx dy + Z R N Z R N | u ( y ) | | ϕ R ( x ) − ϕ R ( y ) | | x − y | N +2 s dx dy + 2 Z R N Z R N ϕ R ( x ) u ( y )[ ϕ R ( x ) − ϕ R ( y )][ u ( x ) − u ( y )] | x − y | N +2 s dx dy. By the computations in [5, Lemma A.2], it is possible to estimate Z R N Z R N | u ( y ) | | ϕ R ( x ) − ϕ R ( y ) | | x − y | N +2 s dx dy ≤ CR s , where C only depends on k∇ ϕ k ∞ and k u k L ( R N ) . OMPACTNESS AND DICHOTOMY IN NONLOCAL SHAPE OPTIMIZATION 7
Moreover, the Cauchy-Schwarz inequality together with the last inequality gives that Z R N Z R N ϕ R ( x ) u ( y )[ ϕ R ( x ) − ϕ R ( y )][ u ( x ) − u ( y )] | x − y | N +2 s dx dy ≤ (cid:18)Z R N Z R N | u ( y ) | | ϕ R ( x ) − ϕ R ( y ) | | x − y | N +2 s dx dy (cid:19) (cid:18)Z R N Z R N | ϕ R ( x ) | | u ( x ) − u ( y ) | | x − y | N +2 s dx dy (cid:19) ≤ (cid:18)Z R N Z R N | u ( y ) | | ϕ R ( x ) − ϕ R ( y ) | | x − y | N +2 s dx dy (cid:19) (cid:18)Z R N Z R N | u ( x ) − u ( y ) | | x − y | N +2 s dx dy (cid:19) ≤ CR s , where C only depends on k∇ ϕ k ∞ , k u k L ( R N ) , and [ u ] H s ( R N ) .Similar computations hold true for the quantity Z R N Z R N | ψ R ( x ) u ( x ) − ψ R ( y ) u ( y ) | | x − y | N +2 s dx dy. Therefore it is possible to choose R ≥ R such that, for R ≥ R , and for every n ∈ N , (cid:12)(cid:12)(cid:12)(cid:12) Z R N Z R N | ϕ R ( x ) u n ( x ) − ϕ R ( y ) u n ( y ) | | x − y | N +2 s dx dy − Z R N Z R N | ϕ R ( x ) | | u n ( x ) − u n ( y ) | | x − y | N +2 s dx dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε, (cid:12)(cid:12)(cid:12)(cid:12) Z R N Z R N | ψ R ( x ) u n ( x ) − ψ R ( y ) u n ( y ) | | x − y | N +2 s dx dy − Z R N Z R N | ψ R ( x ) | | u n ( x ) − u n ( y ) | | x − y | N +2 s dx dy (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε. The claim follows defining v k ( x ) = ϕ R ( x − y k ) u n k ( x ) , w k ( x ) = ψ R k ( x − y k ) u n k ( x ) , where y k and R k → + ∞ are defined as in [16, pp 136-137] and observing that Z R N Z R N | u n k ( x ) − u n k ( y ) | | x − y | N +2 s dx dy ≥ Z R N Z R N | ϕ R ( x ) | | u n k ( x ) − u n k ( y ) | | x − y | N +2 s dx dy + Z R N Z R N | ψ R k ( x ) | | u n k ( x ) − u n k ( y ) | | x − y | N +2 s dx dy since ϕ R and ψ R k have disjoint support for k big enough, and therefore | ϕ R ( x ) | + | ψ R k ( x ) | ≤ for every x ∈ R N . (cid:3) Corollary 3.1.
In the dichotomy case, it is possible to find sequences { u (1) k } k ∈ N , { u (2) k } k ∈ N ⊂ H s ( R N ) such that k u n k − u (1) k − u (2) k k L ( R N ) → for k → + ∞ ; Z R N | u (1) k | → α, Z R N | u (2) k | → λ − α for k → + ∞ ;dist( supp u (1) k , supp u (2) k ) → + ∞ for k → + ∞ ; (3.2) lim inf k → + ∞ (cid:16) [ u n k ] H s ( R N ) − [ u (1) k ] H s ( R N ) − [ u (2) k ] H s ( R N ) (cid:17) ≥ . E. PARINI AND A. SALORT γ -convergence of sets In this section we introduce the notions of γ -convergence and weak γ -convergence of sets, andwe prove some useful results leading to our main theorem. Proposition 4.1.
Let { u n } n ∈ N be a sequence in H s ( R N ) such that u n ⇀ u weakly in H s ( R N ) as n → + ∞ . Then, for every function ϕ ∈ W , ∞ ( R N ) , it holds that ϕu n ∈ H s ( R N ) for every n ∈ N , and ϕu n ⇀ ϕu weakly in H s ( R N ) as n → + ∞ .Proof. The sequence { u n } n ∈ N is uniformly bounded in H s ( R N ) . Moreover, since the embedding H s ( B r ) ֒ → L ( B r ) is compact for every r > , it follows that u n → u strongly in L ( B r ) for every r > . Arguing as in [9, Lemma 5.3], we have that the sequence { ϕu n } n ∈ N is also bounded in H s ( R N ) . Therefore, every subsequence { ϕu n k } admits a subsequence { ϕu n kj } which convergesweakly in H s ( R N ) , and almost everywhere in R N , to some v ∈ H s ( R N ) . But u n kj must convergeto u almost everywhere in R N . Therefore, ϕu n kj → ϕu a.e. in R N , and thus v = ϕu . Hence allthe sequence ϕu n converges weakly in H s ( R N ) to ϕu . (cid:3) γ -convergence and continuity of the spectrum. We prove that γ -convergence of s -quasi open sets implies the convergence of their resolvent operators in the L ( L ( R N )) norm. Inparticular we obtain continuity of the spectrum with respect to the γ -convergence. Definition 4.2.
Let { Ω n } n ∈ N be a sequence of s -quasi open sets such that | Ω n | ≤ c for every n ∈ N . We say that { Ω n } n ∈ N γ -converges to the s -quasi open set Ω if the solutions w Ω n ∈ H s (Ω n ) of the problems(4.1) (cid:26) ( − ∆) s w Ω n = 1 in Ω n ,w Ω n = 0 in R N \ Ω n , strongly converge in L ( R N ) to the solution w Ω ∈ H s (Ω) of the problem (cid:26) ( − ∆) s w Ω = 1 in Ω ,w Ω = 0 in R N \ Ω . Remark . We observe that, if { Ω n } n ∈ N are s -quasi open sets, with | Ω n | ≤ c , which γ -convergeto Ω , then w Ω n → w Ω strongly in H s ( R N ) . Indeed, by Propositions 4.9 and 4.10, one has | Ω | ≤ c .Therefore Z Ω n w Ω n − Z Ω w Ω ≤ Z Ω n \ Ω w Ω n + Z Ω n ∩ Ω | w Ω n − w Ω | + Z Ω \ Ω n w Ω ≤ Z Ω n ∪ Ω | w Ω n − w Ω | ≤ (2 c ) k w Ω n − w Ω k L ( R N ) and therefore lim n → + ∞ Z Ω n w Ω n = Z Ω w Ω . Passing to the limit in the weak formulation, we obtain [ w Ω n ] H s ( R N ) = Z Ω n w Ω n → Z Ω w Ω = [ w Ω ] H s ( R N ) and therefore, by reflexivity of H s ( R N ) , w Ω n → w Ω strongly in H s ( R N ) . OMPACTNESS AND DICHOTOMY IN NONLOCAL SHAPE OPTIMIZATION 9
Proposition 4.4.
Let { Ω n } n ∈ N be a sequence of s -quasi open sets of uniformly bounded measure,which γ -converges to the s -quasi open set Ω . Let { u n } n ∈ N be a sequence in H s ( R N ) such that u n ∈ H s (Ω n ) for every n ∈ N , and u n ⇀ u weakly in H s ( R N ) . Then, u n → u strongly in L ( R N ) .Proof. The proof goes as in [6, Theorem 2.1]. Denoting by F u n , F u the Fourier transforms of u n and u respectively, for R > we have that k u n − u k L ( R N ) = Z R N |F u n ( ξ ) − F u ( ξ ) | dξ = Z | ξ |≥ R (1 + | ξ | s ) − (1 + | ξ | s ) |F u n ( ξ ) − F u ( ξ ) | dξ + Z | ξ | Let { Ω n } n ∈ N be a sequence of s -quasi open sets such that | Ω n | ≤ c for every n ∈ N . Suppose that { Ω n } n ∈ N γ -converges to the s -quasi-open set Ω . Then, for every sequence f n ∈ L (Ω n ) converging weakly in L ( R N ) to f ∈ L (Ω) , the solutions u n ∈ H s ( R N ) of theproblems (cid:26) ( − ∆) s u n = f n in Ω n ,u n = 0 in R N \ Ω n , strongly converge in L ( R N ) to the solution u ∈ H s ( R N ) of the problem (cid:26) ( − ∆) s u = f in Ω ,u = 0 in R N \ Ω . Proof. Exploiting the weak form of the equations, it is straightforward to see that u n ⇀ u weaklyin H s ( R N ) . By Proposition 4.4, u n → u strongly in L ( R N ) . (cid:3) Proposition 4.6. Let { Ω n } n ∈ N be a sequence of s -quasi open sets such that | Ω n | ≤ c for every n ∈ N . Suppose that { Ω n } n ∈ N γ -converges to the s -quasi open set Ω . Then, the resolvents R Ω n converge to R Ω in L ( L ( R N )) . In particular, for every k ≥ , λ k (Ω n ) → λ k (Ω) as n → + ∞ . Proof. We have to show that lim n → + ∞ sup n k R Ω n ( f ) − R Ω ( f ) k L ( R N ) (cid:12)(cid:12) f ∈ L ( R N ) , k f k L ( R N ) ≤ o = 0 . It is equivalent to prove that, for every sequence { f n } n ∈ N such that k f n k L ( R N ) = 1 , the followinglimit holds lim n → + ∞ k R Ω n ( f n ) − R Ω ( f n ) k L ( R N ) = 0 . Let { f n } n ∈ N be such a sequence. Without loss of generality, we can suppose that there exists f ∈ L ( R N ) such that f n ⇀ f in L ( R N ) . By the triangular inequality we get lim sup n → + ∞ k R Ω n ( f n ) − R Ω ( f n ) k L ( R N ) ≤ lim sup n → + ∞ k R Ω n ( f n ) − R Ω ( f ) k L ( R N ) + lim sup n → + ∞ k R Ω ( f n ) − R Ω ( f ) k L ( R N ) . The first term in the previous inequality is equal to zero by Corollary 4.5, while the second termis also zero since the injection H s (Ω) → L (Ω) is compact due to Proposition 2.1. By [10, LemmaXI.9.5], we have, for every k ≥ ,(4.2) (cid:12)(cid:12)(cid:12)(cid:12) λ k (Ω n ) − λ k (Ω) (cid:12)(cid:12)(cid:12)(cid:12) ≤ k R Ω n − R Ω k L ( L ( R N )) and hence λ k (Ω n ) → λ k (Ω) as n → + ∞ , concluding the proof. (cid:3) Remark . When Ω = ∅ quasi-everywhere, by definition H s (Ω) = { } , R Ω is the null operator,and formally λ k (Ω) = + ∞ for every k ≥ . In this case, (4.2) becomes(4.3) ≤ λ k (Ω n ) ≤ k R Ω n k L ( L ( R N )) . In other words, if Ω n γ -converges to the empty set, then λ k (Ω n ) → + ∞ for every k ≥ .Conversely, if Ω is a s -quasi open set such that w Ω = 0 , then ( − ∆) s w Ω = 0 in Ω , and therefore Ω = ∅ quasi-everywhere. OMPACTNESS AND DICHOTOMY IN NONLOCAL SHAPE OPTIMIZATION 11 Weak γ -convergence. Since A ( R N ) is not compact in the topology of γ -convergence, weintroduce the notion of weak γ -convergence for which A ( R N ) is sequentially compact.In this section we prove that a functional J defined in A ( R N ) which is l.s.c. with respect tothe γ -convergence is also l.s.c. with respect to the weak γ -convergence if it is assumed to bedecreasing with respect to the inclusion of sets. Definition 4.8. Let { Ω n } n ∈ N be a sequence of s -quasi open sets. We say that { Ω n } n ∈ N weakly γ -converges to the s -quasi open set Ω if the solutions w n ∈ H s ( R N ) of the problems(4.4) (cid:26) ( − ∆) s w Ω n = 1 in Ω n ,w Ω n = 0 in R N \ Ω n , converge weakly in H s ( R N ) , and strongly in L ( R N ) , to a function w ∈ H s ( R N ) such that Ω = { w > } . Proposition 4.9. Let { Ω n } n ∈ N be a sequence of s -quasi open sets of uniformly bounded measure,which weakly γ -converges to the s -quasi open set Ω . Then, | Ω | ≤ lim inf n → + ∞ | Ω n | . Proof. Let m := lim inf n → + ∞ | Ω n | . Up to extracting a subsequence, we can suppose that m =lim n → + ∞ | Ω n | . Let w Ω n ∈ H s (Ω n ) be the sequence of torsion functions defined in (4.4). Since w Ω n → w strongly in L ( R N ) , there exists a subsequence w Ω nk such that w Ω nk converges almosteverywhere in R N to w . Since Ω = { w > } , it holds χ Ω ≤ lim inf k → + ∞ χ Ω nk almost everywherein R N . By Fatou’s Lemma, | Ω | = Z R N χ Ω ≤ lim inf k → + ∞ Z R N χ Ω nk = m as required. (cid:3) Proposition 4.10. Let { Ω n } n ∈ N be a sequence of s -quasi open sets of uniformly bounded measurewhich γ -converges to the s -quasi open set Ω . Then { Ω n } n ∈ N weakly γ -converges to Ω .Proof. The proof can be performed as in [13, Remark 4.7.8]. (cid:3) Lemma 4.11. Suppose that { Ω n } n ∈ N is a sequence of s -quasi open sets of uniformly boundedmeasure which weakly γ -converges to the s -quasi open set Ω . Let { u n } n ∈ N be a sequence offunctions in H s ( R N ) such that u n ∈ H s (Ω n ) for every n ∈ N , and u n ⇀ u weakly in H s ( R N ) .Then, u ∈ H s (Ω) .Proof. The proof can be performed as in [13, Lemma 4.7.10] (cid:3) Lemma 4.12. Let { Ω n } n ∈ N be a sequence of s -quasi open sets of uniformly bounded measure,which weakly γ -converges to the s -quasi open set Ω . Then, there exists an increasing sequenceof positive integers { n k } k ∈ N and a sequence of quasi-open sets { C k } k ∈ N such that Ω n k ⊂ C k forevery k ∈ N , and { C k } k ∈ N γ -converges to Ω .Proof. The proof can be performed as in [13, Lemma 4.7.11], where Lemma 4.11 should be usedinstead of [13, Lemma 4.7.10]. (cid:3) Finally, we state the main result of this section. Proposition 4.13. Let J : A ( R N ) → ( −∞ , + ∞ ] be a functional satisfying:(i) J is decreasing with respect to the inclusion of sets; (ii) J is lower semicontinuous with respect to the γ -convergence.Then J is lower semicontinuous with respect to the weak γ -convergence.Proof. Let { Ω n } n ∈ N be a sequence of s -quasi open sets of uniformly bounded measure, whichweakly γ -converges to the s -quasi open set Ω . By Lemma 4.12, there exists an increasing sequenceof positive integers { n k } k ∈ N and a sequence of quasi-open sets { C k } k ∈ N such that lim n → + ∞ J (Ω n k ) = lim inf n → + ∞ J (Ω n ) , Ω n k ⊂ C k for every k ∈ N , and { C k } k ∈ N γ -converges to Ω . Since J is decreasing with respect tothe inclusion of sets, J (Ω) ≤ lim inf k → + ∞ J ( C k ) ≤ lim inf k → + ∞ J (Ω n k ) = lim inf n → + ∞ J (Ω n ) . The proof is concluded. (cid:3) Proof of Theorem 1.2 In the following, { Ω n } n ∈ N will be a sequence of s -quasi open sets of uniformly bounded mea-sure. The proof of Theorem 1.2, which will be performed in several steps, is based on thebehavior of the sequence { w Ω n } n ∈ N according to the concentration-compactness principle statedin Proposition 1.1. Without loss of generality, we can suppose that R R N | w Ω n | → λ as n → + ∞ for some λ > .5.1. Compactness for w Ω n . Assume that { w Ω n } n ∈ N is in the compactness case, that is, upto some subsequence still denoted with the same index, and some translations, the sequence { w Ω n } n ∈ N converges strongly in L ( R n ) to some w ∈ H s ( R N ) . Then, by definition, Ω n weakly γ -converges to the set Ω := { w > } .5.2. Vanishing for w Ω n . In the spirit of [15] we prove the following lemma. Lemma 5.1. Let A and B be two measurable sets. Then there exists z ∈ R N such that, if A z = z + A , λ ( A z ∩ B ) ≤ λ ( A ) + λ ( B )) . Proof. The roles of u and v were reversed, and also x and z . Let z ∈ R N be arbitrary and let u and v be positive first eigenfunctions on A and B respectively, normalized such that k u k L ( A ) = k v k L ( B ) = 1 . By regularity, the function u z defined by u z ( x ) = u ( z − x ) satisfies u z ∈ H s ( A z ) ∩ L ∞ ( A z ) , and v ∈ H s ( B ) ∩ L ∞ ( B ) . The function w z defined as w z ( x ) = u ( x − z ) v ( x ) belongs to H s ( A z ∩ B ) ∩ L ∞ ( A z ∩ B ) . Define T ( z ) := Z R N Z R N | w z ( x ) − w z ( y ) | | x − y | N +2 s dx dy, D ( z ) := Z R N | w z ( x ) | dx. It holds that Z R N D ( z ) dz = Z R N Z R N | w z ( x ) | dx dz = Z R N Z R N | u ( x − z ) v ( x ) | dx dz = 1 . OMPACTNESS AND DICHOTOMY IN NONLOCAL SHAPE OPTIMIZATION 13 Moreover, | w z ( x ) − w z ( y ) | = | u ( x − z ) v ( x ) − u ( y − z ) v ( y ) | = | u ( x − z ) v ( x ) − u ( x − z ) v ( y ) + u ( x − z ) v ( y ) − u ( y − z ) v ( y ) | = | u ( x − z ) | | v ( x ) − v ( y ) | + | v ( y ) | | u ( x − z ) − u ( y − z ) | + 2 u ( x − z ) v ( y )[ v ( x ) − v ( y )][ u ( x − z ) − u ( y − z )] . Using the elementary inequality ab ≤ a + b , the last term in the inequality above can bebounded as | u ( x − z ) | | v ( x ) − v ( y ) | + | v ( y ) | | u ( x − z ) − u ( y − z ) | , and from the last two expressions we get | w z ( x ) − w z ( y ) | ≤ (cid:0) | u ( x − z ) | | v ( x ) − v ( y ) | + | v ( y ) | | u ( x − z ) − u ( y − z ) | (cid:1) . Thus T ( z ) ≤ Z R N Z R N | u ( x − z ) | | v ( x ) − v ( y ) | | x − y | N +2 s dx dy + 2 Z R N Z R N | v ( y ) | | u ( x − z ) − u ( y − z ) | | x − y | N +2 s dx dy. Then, integrating over z and performing a change of variables, since u and v are normalized in L norm, we get Z R N T ( z ) dz ≤ λ ( A ) + λ ( B )) := Λ . Therefore, R R N [ T ( z ) − Λ D ( z )] dz ≤ , hence ≤ T ( z ) ≤ Λ D ( z ) on a set of positive measure.From the definitions of T , D and Λ the lemma follows. (cid:3) Assume that { w Ω n } n ∈ N is in the vanishing case, that is, for all R > it holds that lim n → + ∞ sup y ∈ R N Z y + B R | w Ω n | = 0 . Since the sequence { w Ω n } n ∈ N ⊂ H s ( R N ) , we can assume that w Ω n ⇀ w weakly in H s ( R N ) . Fix ε > . By Lemma 5.1, there exists R > and a sequence { y n } n ∈ N in R N such that(5.1) λ (( y n + Ω n ) ∩ B R ) ≤ λ (Ω n ) + ε. From the weak maximum principle it follows that w y n +Ω n ≥ w ( y n +Ω n ) ∩ B R ≥ , and then, thevanishing assumption on w Ω n gives that lim n → + ∞ Z B R | w ( y n +Ω n ) ∩ B R | = 0 . This means that w ( y n +Ω n ) ∩ B R → strongly in L ( R N ) , and therefore ( y n +Ω n ) ∩ B R γ − convergesto the empty set. By Remark 4.7, λ (( y n + Ω n ) ∩ B R ) → + ∞ as n → + ∞ . By (5.1) we obtain that λ (Ω n ) → + ∞ as n → + ∞ . From the Poincaré inequality given in Proposition 2.1 we find that k w Ω n k L (Ω n ) ≤ λ (Ω n ) [ w Ω n ] H s ( R N ) → as n → + ∞ since w Ω n ∈ H s ( R N ) is bounded in H s ( R N ) . Finally, by Proposition 4.6 and Remark 4.7 weobtain that k R Ω n k L ( L ( R N )) → . By definition, the sequence { Ω n } n ∈ N γ -converges, and henceweakly γ -converges, to the empty set.5.3. Dichotomy for w Ω n . Finally, suppose that w Ω n is in the dichotomy case. That means thatit is possible to find two sequences { u n } n ∈ N and { v n } n ∈ N of nonnegative functions in H s (Ω n ) and a number α ∈ (0 , λ ) such that, up to a subsequence, k w Ω n − u n − v n k L ( R N ) → as n → + ∞ ; Z R N u n → α, Z R N v n → λ − α for n → + ∞ ;dist( supp u n , supp v n ) → + ∞ for n → + ∞ ; (5.2) lim inf n → + ∞ (cid:16) [ w Ω n ] H s ( R N ) − [ u n ] H s ( R N ) − [ v n ] H s ( R N ) (cid:17) ≥ . We define the following sets(5.3) Ω n := { u n > } , Ω n := { v n > } , e Ω n := Ω n ∪ Ω n , and then e Ω n is a quasi-open set contained in Ω n .The proof of the claims in the dichotomy case will be a consequence of the following threelemmas. Lemma 5.2. The sequence of sets (5.3) satisfies lim inf n → + ∞ | Ω in | > for i = 1 , . Proof. Suppose by contradiction that, for instance, lim inf n → + ∞ | Ω n | = 0 . The functions w Ω n are uniformly bounded in L ∞ by [4, Theorem 3.1], and therefore, by construction, also thefunctions u n are uniformly bounded in L ∞ . But then, R R N u n → , which contradicts the factthat R R N u n → α > . (cid:3) Lemma 5.3. With the previous notation, we have that k w Ω n − w e Ω n k H s ( R N ) → as n → + ∞ . Proof. We observe that w e Ω n is the orthogonal projection of w Ω n on the space H s ( e Ω n ) . Indeed,let us consider the functional F : H s ( ˜Ω n ) → R defined by F ( v ) = 12 [ w Ω n − v ] H s ( R N ) . Observe that F ( v ) = 12 [ w Ω n ] H s ( R N ) + 12 [ v ] H s ( R N ) − Z R N Z R N ( w Ω n ( x ) − w Ω n ( y ))( v ( x ) − v ( y )) | x − y | N +2 s dxdy. Using the weak formulation of w Ω n we have that F ( v ) = 12 [ w Ω n ] H s ( R N ) + 12 [ v ] H s ( R N ) − Z e Ω n v. Then, the functional F will be minimized for v = w ˜Ω n , since w ˜Ω n minimizes the functional v 12 [ v ] H s ( R N ) − Z e Ω n v. OMPACTNESS AND DICHOTOMY IN NONLOCAL SHAPE OPTIMIZATION 15 Hence, Z R N Z R N | w Ω n ( x ) − w Ω n ( y ) − w e Ω n ( x ) + w e Ω n ( y ) | | x − y | N +2 s dx dy ≤ Z R N Z R N | w Ω n ( x ) − w Ω n ( y ) − ( u n + v n )( x ) + ( u n + v n )( y )) | | x − y | N +2 s dx dy = [ w Ω n ] H s ( R N ) + [ u n + v n ] H s ( R N ) − Z R N Z R N [ w Ω n ( x ) − w Ω n ( y )][( u n + v n )( x ) − ( u n + v n )( y )] | x − y | N +2 s dx dy = Z R N w Ω n + [ u n + v n ] H s ( R N ) − Z R N ( u n + v n )= 2 (cid:18)Z R N w Ω n − Z R N ( u n + v n ) (cid:19) + [ u n + v n ] H s ( R N ) − [ w Ω n ] H s ( R N ) . Observe that (cid:12)(cid:12)(cid:12)(cid:12) Z R N w Ω n − Z R N ( u n + v n ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ | Ω n | k w Ω n − ( u n + v n ) k L ( R N ) → as n → + ∞ . Moreover, using the fact that [ u n + v n ] H s ( R N ) ≤ [ u n ] H s ( R N ) + [ v n ] H s ( R N ) since theyare nonnegative functions, we obtain from (5.2) that lim sup n → + ∞ (cid:16) [ u n + v n ] H s ( R N ) − [ w Ω n ] H s ( R N ) (cid:17) ≤ and therefore [ w Ω n − w e Ω n ] H s ( R N ) → as n → + ∞ . By Proposition 2.1, there exists C > such that, for every n ∈ N , k w Ω n − w e Ω n k L ( R N ) = k w Ω n − w e Ω n k L (Ω n ) ≤ C [ w Ω n − w e Ω n ] H s ( R N ) and hence k w Ω n − w e Ω n k H s ( R N ) → as n → + ∞ . (cid:3) Lemma 5.4. Let ˜Ω ⊂ Ω ⊂ R N two sets of finite measure. There exists a constant C = C ( | Ω | , N ) > and α > such that k R Ω − R ˜Ω k L ( L ( R N )) ≤ C k w Ω − w ˜Ω k αL ( R N ) . Proof. Let < s < be fixed. Observe that if u, v ∈ H (Ω) are the unique solutions of ( − ∆) s u = f in Ω , ( − ∆) s v = 1 in Ω , respectively, using v and u as test functions in the weakformulation of the two previous equations, respectively, we get Z R N Z R N ( u ( x ) − u ( y ))( v ( x ) − v ( y )) | x − y | N +2 s dx dy = Z Ω f w = Z Ω u, that is, R Ω f w Ω = R Ω R ( f ) . The previous computation gives that Z Ω R Ω ( f ) − R ˜Ω ( f ) = Z Ω f ( w Ω − w ˜Ω ) . By [4, Theorem 3.1], for N < s we have(5.4) k R Ω ( f ) k L ∞ (Ω) ≤ C ( N, | Ω | ) k f k L (Ω) , and then, by using (5.4) and Hölder’s inequality we get k R Ω ( f ) − R ˜Ω ( f ) k L (Ω) ≤ k R Ω ( f ) − R ˜Ω ( f ) k L ∞ (Ω) k R Ω ( f ) − R ˜Ω ( f ) k L (Ω) ≤ C k f k L (Ω) k f ( w Ω − w ˜Ω ) k L (Ω) ≤ C k f k L (Ω) k w Ω − w ˜Ω k L (Ω) . The case N ≥ s will follow by an interpolation argument. For that end, consider p > , N ≥ s and f ∈ L p (Ω) , f ≥ . By using again [4, Theorem 3.1] and Hölder’s inequality we get k R Ω ( f ) − R ˜Ω ( f ) k L p (Ω) ≤ C k f k L p (Ω) k w Ω − w ˜Ω k p L p ′ (Ω) for a suitable constant C depending only on p , N and | Ω | , that is, k R Ω − R ˜Ω k L ( L p ( R N )) ≤ C k w Ω − w ˜Ω k p L p ′ (Ω) . Now, let R ∗ Ω and R ∗ ˜Ω be the adjoint operators of R Ω and Ω ˜Ω , respectively, which are defined from L p ′ (Ω) in itself. Since the L p ′ norm of R ∗ Ω − R ∗ ˜Ω coincides with the L p norm of R Ω − R ˜Ω , we get k R ∗ Ω − R ∗ ˜Ω k L ( L p ′ ( R N )) ≤ C k w Ω − w ˜Ω k p L p ′ (Ω) . Since R Ω and R ˜Ω are self-adjoint on L (Ω) , keeping the same notation for R A , R ˜Ω and theirextension on L p ′ (Ω) , we obtain that R Ω − R ˜Ω : L p ′ (Ω) → L p ′ (Ω) and k R Ω − R ˜Ω k L ( L p ′ ( R N )) ≤ C k w Ω − w ˜Ω k p L p ′ (Ω) . Finally, from the Riesz-Thorin interpolation theorem and since < p ′ < , we obtain that k R Ω − R ˜Ω k L ( L ( R N )) ≤ k R Ω − R ˜Ω k L ( L p ( R N )) k R Ω − R ˜Ω k L ( L p ′ ( R N )) ≤ C k w Ω − w ˜Ω k p L p ′ (Ω) ≤ C | Ω | − p ′ p ′ p k w Ω − w ˜Ω k p L (Ω) which ends the proof. (cid:3) Proof of Theorem 1.3 Let { Ω n } n ∈ N ⊂ A ( R N ) be a minimizing sequence for Problem (1.4), satisfying | Ω n | = c forevery n ∈ N , and J (Ω n ) → m as n → + ∞ . By Theorem 1.2, we have two possible cases:(i) there exists a subsequence, still denoted by { Ω n } n ∈ N , and a set Ω ∈ A ( R N ) , such that,up to some translations, { Ω n } n ∈ N weakly γ -converges to Ω . Since J is invariant bytranslations, the sequence will be again a minimizing sequence for J . By Proposition 4.9, | Ω | ≤ c . Let ˆΩ ∈ A ( R N ) be such that Ω ⊂ ˆΩ and | ˆΩ | = c . Since J is decreasing withrespect to set inclusion, and by Propositions 4.6 and 4.13, m ≤ J ( ˆΩ) ≤ J (Ω) ≤ lim inf n → + ∞ J (Ω n ) = m. 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Parini) Aix Marseille Univ, CNRS, Centrale Marseille, I2M, 39 Rue Frédéric Joliot Curie,13453 Marseille CEDEX 13, France E-mail address : [email protected] URL : (A. Salort) Departamento de Matemática, FCEN – Universidad de Buenos Aires and IMAS –CONICET, Buenos Aires, Argentina E-mail address : [email protected] URL ::