Spectral optimization problems

Spectral Optimization Problems Buttazzo, Giuseppe ∗ December 16, 2010 arXiv:1012.3299v1 [math.OC] 15 Dec 2010 Abstract In this survey paper we present a class of shape optimization problems where the cost function involves the solution of a PDE of elliptic type in the unknown domain. In particular, we consider cost functions which depend on the spectrum of an elliptic operator and we focus on the existence of an optimal domain. The known results are presented as well as a list of still open problems. Related fields as optimal partition problems, evolution flows, Cheeger-type problems, are also considered. Keywords: optimization problems for eigenvalues, shape optimization, capacity, integral functionals, Sobolev spaces, optimality conditions, calculus of variations, relaxation. 2000 Mathematics Subject Classification: 49J45, 49R05, 35P15, 47A75, 35J25 1 Introduction Looking for optimal shapes is a very fascinating field, perhaps due to the fact that a shape is something closer to the human spirit than a function, in which a parametrization procedure is present, making it less direct and intuitive. However, from a mathematical point of view, handling shapes is much more difficult than handling functions, for instance because no vector space structure can be defined on a family of domains, which makes useless most of the functional analysis tools in Banach spaces. A general shape optimization problem reads as  min F (Ω) : Ω ∈ A (1.1) where A is a suitable family of admissible domains and F is a suitable cost function defined on A. Problems of this kind arise in many fields and in many applications, and the literature is very wide, from the classical cases of isoperimetric problems and the Newton problem of the best aerodynamical shape to the most recent applications to elasticity and to spectral optimization. Without any claim to be complete and exhaustive we quote the recent books [3, 16, 26, 67, 70, 80, 82], where the reader can find a lot of shape optimization problems together with all the necessary details and references. A family of shapes is in general more rigid than a family of functions, in the sense that a sequence of shapes may have a limit which is no more a shape; then quite often problems like (1.1) do not have optimal solutions and optimal shapes do not exist, even for very natural cost functions F and admissible domains A. In these situations, in order to perform the analysis of minimizing sequences, a relaxation procedure is needed, enlarging the class A to objects that are only limits of shapes and defining the cost F on these new objects. ∗ Dipartimento di Matematica, [email protected] Università di Pisa, 1 Largo B. Pontecorvo 5, 56127 Pisa, ITALY It is not our goal to describe shape optimization problems and their relaxed formulations in full generality; we will limit ourselves to the cases where the cost F is of the form F (Ω) = J(uΩ ) being uΩ the solution of some elliptic PDE in Ω. Two main cases fall for instance in this scheme: the case of integral functionals and the case of functions of the spectrum of an elliptic operator. Integral functionals. Given a right-hand side f we consider the PDE −∆u = f in Ω, u ∈ H01 (Ω) which provides, for every admissible domain Ω ⊂ Rd , a unique solution uΩ that we assume extended by zero outside of Ω. The cost F (Ω) = J(uΩ ) is obtained by taking Z
j x, u(x) dx J(u) = Rd for a suitable integrand j. Spectral optimization. For every admissible domain Ω consider the Dirichlet Laplacian −∆ which, under mild conditions on Ω, admits a compact resolvent and so a discrete spectrum λ(Ω). The cost is in this case of the form
F (Ω) = Φ λ(Ω) for a suitable function Φ. For instance, taking Φ(λ) = λk we may consider the optimization problem for the k-th eigenvalue of −∆:  min λk (Ω) : Ω ∈ A . We will see that, adding suitable geometrical restrictions to the class A of admissible domains, gives some extra compactness properties and prevents the relaxation of minimizing sequences, thus providing in many cases the existence of optimal domains. In some other cases, the existence of optimal domains will be obtained without any geometrical restriction, as a consequence of some qualitative properties of the cost function. The plan of the paper is as follows. ◦ In Section 2 we introduce all mathematical tools that are necessary to treat the optimization problems in the rest of the paper. In particular capacity, capacitary measures, γ-convergence are recalled, together with their main properties. ◦ Section 3 deals with optimization problems for integral functionals of the form above. We show that in general one cannot expect that an optimal domain exists and the optimum has to be searched in the class of capacitary measures, that is the relaxation of the class of domains. Some necessary conditions of optimality are also provided.
◦ In Section 4 we consider the case of optimization problems for a function Φ λ(Ω) where λ(Ω) is the spectrum of an elliptic operator, that we always take −∆ with Dirichlet conditions at the boundary. ◦ Section 5 deals with the particular case of cost functions which only depends on the
first two eigenvalues, of the form Φ λ1 (Ω), λ2 (Ω) . The particular form of the cost allows to obtain the existence of optimal domains under very mild assumptions on the function Φ. 2 ◦ Section 6 is devoted to optimal partition problems, where the unknown is a partition SN Ω of a given set D in a given number N of domains, and the cost function i i=1 F (Ω1 , . . . , ΩN ) is of a fairly general type. Some interesting cases of spectral costs are presented. ◦ The last Section 7 collects some interesting directions of investigation that merit to be developed. For instance the case of Cheeger-type costs in which the product of two quantities, scaling in a different way, has to be minimized, as well as the study of spectral flows, evolution patterns that follow the gradient flow of a spectral cost functional. The case of boundary conditions different from Dirichlet type is briefly addressed and referred to the existing literature. 2 2.1 Preliminary tools Capacity One of the key tools for treating optimization problems governed by elliptic equations is the notion of capacity. Definition 2.1. Given a subset E of Rd , we define the capacity of E as nZ o Cap(E) = inf |∇u|2 + u2 dx : u ∈ UE , Rd where UE is the set of all functions u of the Sobolev space H 1 (Rd ) such that u ≥ 1 almost everywhere in a neighborhood of E. For every bounded open set D of Rd we also define a local capacity as nZ o cap(E, D) = inf |∇u|2 dx : u ∈ UE , D where now UE is the set of all functions u of the Sobolev space H01 (D) such that u ≥ 1 almost everywhere in a neighborhood of E. Since we are mainly interested in sets with capacity zero the two notions above play the same role and in most of the situations they will be equivalent. If a property P (x) holds for all x ∈ E except for the elements of a set Z ⊂ E with Cap(Z) = 0, we say that P (x) holds quasi-everywhere (shortly q.e.) on E, whereas the expression almost everywhere (shortly a.e.) refers, as usual, to the Lebesgue measure. A subset A of Rd is said to be quasi-open if for every ε > 0 there exists an open subset Aε of Rd , such that Cap(Aε ∆A) < ε, where ∆ denotes the symmetric difference of sets. Actually, in the definition above we can additionally require that A ⊂ Aε . Similarly, we define quasi-closed sets. The class of all quasi-open subsets of a given set D will be denoted by A(D). A function f : D → R is said to be quasi-continuous (resp. quasi-lower semicontinuous) if for every ε > 0 there exists a continuous (resp. lower semicontinuous) function fε : D → R such that Cap({f 6= fε }) < ε. It is well known (see for instance [88]) that every function u ∈ H 1 (D) has a quasi-continuous representative ũ, which is uniquely defined up to a set of capacity zero, and given by Z 1 ũ(x) = lim u(y) dy , ε→0 |B(x, ε)| B(x,ε) where B(x, ε) denotes the ball of radius ε centered at x. By an abuse of notation we always identify the function u with its quasi-continuous representative ũ, so that a pointwise 3 condition can be imposed on u(x) for quasi-every x ∈ D. In this way, we have for every E⊂D o nZ |∇u|2 dx : u ∈ H01 (D), u ≥ 1 q.e. on E . cap(E, D) = min D By the identification above, the quasi-open sets can be characterized as the sets of positivity of functions in H 1 (Rd ); more precisely, this means that A ⊂ Rd is quasi-open if and only if there exists a function u ∈ H 1 (Rd ) such that A = {ũ > 0}. For every quasi-open set Ω ⊂ D we denote by H01 (Ω) the space of all functions u ∈ H01 (D) such that u = 0 q.e. on D \ Ω, with the Hilbert space structure inherited from H01 (D), i.e. hu, viH01 (Ω) = hu, viH01 (D) . Note that H01 (Ω) is a closed subspace of H01 (D) (see for instance [76, 88]). Moreover, if Ω is an open set, the previous definition of H01 (Ω) is equivalent to the usual one (see [2]). Most of the well-known properties of Sobolev functions on open sets extend to quasiopen sets; for instance, if Ω1 , Ω2 are two quasi-open sets disjoint in capacity, i.e. with Cap(Ω1 ∩ Ω2 ) = 0, then H01 (Ω1 ∪ Ω2 ) = H01 (Ω1 ) ∩ H01 (Ω2 ) in the sense that u ∈ H01 (Ω1 ∪ Ω2 ) if and only if u|Ω1 ∈ H01 (Ω1 ) and u|Ω2 ∈ H01 (Ω2 ). Remark 2.2. All the definitions above have a natural extension to the Sobolev spaces W01,p (Ω) with 1 < p < +∞. We refer to [65] for a review of the main definitions and properties of the p-capacity; in particular, when p > d, the p-capacity of a point is strictly positive and every W 1,p -function has a continuous representative and therefore, a property which holds p-quasi-everywhere, with p > d, holds in fact everywhere, which makes trivial several shape optimization problems. This is why, from the shape optimization point of view, the most interesting case is when p ∈ (1, d]. 2.2 Γ-convergence for sequences of functionals In this section we recall briefly the definition and the main properties of the Γ-convergence, which revealed to be one of the key tools for handling variational problems. We do not want here to enter into the details of that theory, but only to use it in order to characterize the relaxed form of Dirichlet problems; we refer for all details to the books [20, 52]. Definition 2.3. Given a sequence (Fn ) of functionals from a separable metric space X into R we say that (Fn ) Γ-converges to a functional F if for every x ∈ X 1. ∀xn → x F (x) ≤ lim inf Fn (xn ); 2. ∃xn → x F (x) ≥ lim sup Fn (xn ). n→∞ n→∞ Theorem 2.4. Properties of Γ-convergence. 1. Every Γ-limit is lower semicontinuous on X; S 2. if (Fn ) is equi-coercive on X, that is for every t ∈ R the set n {Fn ≤ t} is relatively compact in X, and (Fn ) Γ-converges to F , then F is coercive too, and so it admits a minimum on X; 3. if xn ∈ argmin Fn and xn → x in X, then x ∈ argmin F ; 4. from every sequence (Fn ) of functionals on X it is possible to extract a subsequence Γ-converging to a functional F on X; 4 5. we have Γ limn (G + Fn ) = G + Γ limn Fn for every functional G which is continuous on X. Here we used the notation argmin F to indicate the set of all minimum points of F on X. The Γ-convergence provides a convergence structure on the class F of all lower semicontinuous functionals on X; however, in several situations the functionals under considerations fulfill an equicoercivity condition that allows us to introduce a restricted subclass: FΨ (X) = {F : X → R, F lower semicontinuous, F ≥ Ψ} where Ψ is a given functional on X. The following result concerning metrizability of the Γ-convergence will be used (see [52], Theorem 10.22). Theorem 2.5. If Ψ : X → R is coercive, i.e. its sublevels {Ψ ≤ t} are relatively compact, then the class FΨ (X), endowed with the Γ-convergence structure, is a compact metric space, in the sense that there exists a compact distance δΓ on FΨ (X) such that δΓ (Fn , F ) → 0 2.3 ⇐⇒ Γ lim Fn = F. n→∞ γ-convergence for sequences of quasi-open sets Given a bounded domain D ⊂ Rd and a right-hand side f ∈ L2 (D) we are interested to study how the solution uΩ of the elliptic problem −∆u = f in Ω, u ∈ H01 (Ω) depends on the domain Ω ∈ A(D). The solutions are extended by zero on D \ Ω. Since the equation above can be also written as the minimum problem n Z h1 i o min |∇u|2 − f (x)u dx : u ∈ H01 (Ω) , D 2 by the definition of Γ-convergence introduced in Section 2.2 the study of the limits of (uΩn ) for a given sequence (Ωn ) of domains can be reduced to the study of the Γ-limit of the sequence of functionals Z h i 1 |∇u|2 − f (x)u dx + IΩn (u) D 2 where  IΩ (u) = 0 if u ∈ H01 (Ω) +∞ otherwise. 1 All the functionals above are defined on the R separable metric space H0 (D) endowed with 2 the L (D) convergence. Since the term D f (x)u dx is continuous, by Theorem 2.4 it is enough to study the Γ-limit of the functionals Z Fn (u) = |∇u|2 dx + IΩn (u). (2.1) D Of course, when (Ωn ) converges to Ω in a rather strong way, for instance when Ωn = (Id + εn V )(Ω) with εn → 0 and V ∈ Cc1 (Rd ; Rd ), the Γ-limit of the sequence Fn above is the functional Z F (u) = |∇u|2 dx + IΩ (u). D However, in shape optimization problems, we are interested to work with weaker convergences, in order to ensure, via the direct methods of the calculus of variations, the existence 5 of an optimal domain. Therefore we look for a kind of equilibrium between a convergence strong enough to provide continuity, or at least lower semicontinuity, of the cost functionals, and on the other hand weak enough to provide the compactness of minimizing sequences. When this equilibrium is achieved one obtains the existence of an optimal solution. A very natural definition is the following one. Definition 2.6. We say that a sequence of quasi-open setsR (Ωn ) in A(D) γ-converges to a 2 quasi-open set Ω ∈ R A(D)2if the corresponding functionals D1|∇u| dx + IΩn (u) Γ-converge to the functional D |∇u| dx + IΩ (u), in the metric space H0 (D) endowed with the L2 (D) convergence. By the properties of the Γ-convergence seen in Section 2.2 we have Ωn → Ω in the γ-convergence if and only if for every right-hand side f ∈ L2 (D) the solutions uΩn ,f of −∆u = f in Ωn , u ∈ H01 (Ωn ) (extended by zero on D \ Ωn ) converge in L2 (D) to the solution uΩ,f of −∆u = f in Ω, u ∈ H01 (Ω). Note that the sequence (uΩ,f ) is bounded in H01 (D), hence compact in L2 (D). Remark 2.7. The following facts for the γ-convergence hold. 1. By its definition, and by the equicoercivity of the functionals (2.1), the γ-convergence is metrizable on A(D). 2. It can be proven (see [26]) that the L2 (D) convergence of uΩn ,f to uΩ,f for every f holds if and only if it holds for f ≡ 1. Therefore a distance on A(D) equivalent to the γ-convergence is dγ (Ω1 , Ω2 ) = kuΩ1 ,1 − uΩ2 ,1 kL2 (D) . 3. For every Ω ∈ A(D) we may define the resolvent operator RΩ which associates to every f ∈ L2 (D) the solution uΩ,f ; in this way the γ-convergence of (Ωn ) to Ω coincides with the pointwise L2 (D) convergence of resolvent operators RΩn to RΩ . 4. It can actually be proven (see [26]) that if Ωn → Ω in the γ-convergence, the convergence of uΩn ,f to uΩ,f is in fact strong in H01
(D) and the resolvent operators RΩn converge to RΩ in the operator norm L L2 (D) . In particular, the spectrum of RΩn converges (componentwise) to the spectrum of RΩ , hence the spectrum of −∆ on H01 (Ωn ) converges (componentwise) to the spectrum of −∆ on H01 (Ω). 2.4 Capacitary measures Unfortunately, the γ-convergence introduced in Section 2.3 is not compact; in other words it is possible to construct a sequence (Ωn ) of domains (even smooth) such that the corresponding solutions uΩn ,1 do not converge to a function of the form uΩ,1 . The first example of a sequence (Ωn ) of this form was provided by Cioranescu and Murat in [47]: they constructed the sequence (Ωn ) by removing from D a periodic array of balls of equal radius rn → 0. If the radius rn is suitably chosen they proved (see also [26] for an interpretation in terms of γ-convergence) that the weak H01 (D) limit of the sequence of solutions uΩn ,1 satisfies the PDE −∆u + cu = 1 in D, 6 u ∈ H01 (D) Figure 1: The sets Ωn in the Cioranescu and Murat example. for a suitable constant c > 0, and thus the sequence of domains (Ωn ) cannot γ-converge to any domain Ω. The sets Ωn are represented in Figure 1 If we indicate by εn the distance between two adjacent centers, then the critical size of the radius rn is rn = exp(−cε−2 n ) if d = 2 rn = cεd/(d−2) if d > 2, n with c > 0. In this way the capacity of the array of removed balls is of order O(1), while the corresponding total volume vanishes as εn → 0. It is interesting to notice that, for any sequence (Ωn ) of domains in D and for any f ∈ L2 (D), the solutions uΩn ,f are bounded in H01 (D), so up to extracting a subsequence, they weakly converge to some function w ∈ H01 (D). In order to characterize this function w as the solution of some limit PDE we have to identify the compactification of the metric space A(D) of all quasi-open subsets of D, endowed with the γ-convergence. This identification has been obtained by Dal Maso and Mosco in [53], where it is proven that the compactification of the metric space A(D) endowed with the γ-convergence is the set M0 (D) of all nonnegative regular Borel measures µ on D, possibly +∞ valued, such that µ(B) = 0 for every Borel set B ⊂ D with Cap(B) = 0. We stress the fact that the measures µ ∈ M0 (D) do not need to be finite, and may take the value +∞ even on large parts of D. Example 2.8. 1. The Dirac measure δx0 does not belong to M0 (D) because it does not vanish on the set {x0 } which has capacity zero when d ≥ 2. Similarly, the Hausdorff measures Hk bS, with S manifold of dimension k, do not belong to M0 (D) when d ≥ k + 2. 2. If d − 2 < α ≤ d the α-dimensional Hausdorff measure Hα belongs to M0 (D), and consequently every µ absolutely continuous with respect to Hα as well. In fact every Borel set with capacity zero has an Hausdorff dimension which is less than or equal to d − 2. 3. For every S ⊂ D, the measure ∞S defined by  0 if Cap(B ∩ S) = 0, ∞S (B) = +∞ otherwise. 7 (2.2) belongs to the class M0 (D). Given µ ∈ M0 (D) we can consider the relaxed form of the Dirichlet problem by introducing the space Xµ (D) as the vector space of all functions u ∈ H01 (D) such that R 2 D u dµ < ∞. Note that, since µ vanishes on all sets with capacity zero and since Sobolev functions are defined up to sets of capacity zero, the definition of Xµ (D) is well posed. We may think to Xµ (D) as to the Hilbert space (see [38]) H01 (D) ∩ L2 (D, µ), endowed with the norm Z 1/2 Z 2 |∇u| dx + u2 dµ kukXµ (D) = D D which comes from the scalar product Z (u, v)Xµ (D) = D ∇u∇v dx + Z uv dµ. D Given f ∈ L2 (D) we can now consider the relaxed Dirichlet problem −∆u + µu = f in D, u ∈ Xµ (D) whose precise meaning has to be given in the weak form Z Z Z u ∈ Xµ (D), ∇u∇v dx + uv dµ = f v dx D D D ∀v ∈ Xµ (D). The usual Lax-Milgram method in the Hilbert space Xµ (D) provides, for every µ ∈ M0 (D) and every f ∈ L2 (D), a unique solution uµ,f . In this way we can construct the resolvent operator Rµ which associates to every f ∈ L2 (D) the solution uµ,f . Example 2.9. 1. Take µ = a(x) dx where a ∈ Lp (D) with p ≥ d/2 (any p > 1 if d = 2). Then, by the Sobolev embedding theorem and Hölder inequality, we have that Xµ (D) = H01 (D) with equivalent norms. 2. If Ω is a quasi-open subset of D and µ = ∞D\Ω then the space Xµ (D) coincides with the Sobolev space H01 (Ω) and the solution uµ,f defined above coincides with the solution uΩ,f of the Dirichlet problem in Ω. Thus we can identify the domain Ω to the capacitary measure µ = ∞D\Ω . 3. If µ ∈ M0 (D) and f ≥ 0, then by maximum principle the solution u = Rµ (f ) is nonnegative, and then also f + ∆u = µu is nonnegative. On the other hand, if f > 0 we can formally write f + ∆u µ= u which gives µ once u is known; of course it turns out that µ = +∞ whenever u = 0. This argument can be made rigorous (see [46, 26]) and therefore, working with the class M0 (D) is in this case equivalent to work with the class of functions {u ∈ H01 (D), u ≥ 0, ∆u + f ≥ 0}, which is a closed convex subset of the Sobolev space H01 (D). The γ-convergence can be extended to the space M0 (D): we have µn → µ in the γconvergence if the solutions Rµn (f ) converge to Rµ (f ) weakly in H01 (D) for every f ∈ L2 (D). Again, it can be proven that this is equivalent to require the convergence only for f ≡ 1. Proposition 2.10. The properties below for the γ-convergence on the space M0 (D) hold. 8 1. The space M0 (D) endowed with the γ-convergence is a compact metric space; a distance equivalent to the γ-convergence is dγ (µ1 , µ2 ) = kRµ1 (1) − Rµ2 (1)kL2 (D) . 2. The class A(D) is included in M0 (D) via the identification Ω 7→ ∞D\Ω and A(D) is dense in M0 (D) for the γ-convergence. Actually also the class of all smooth domains Ω is dense in M0 (D). 3. The measures of the form a(x) dx with a ∈ L1 (D) belong to M0 (D) and are dense in M0 (D) for the γ-convergence. Actually also the class of measures a(x) dx with a smooth is dense in M0 (D). 4. If µn → µ for the γ-convergence, then the spectrum of the compact resolvent operator Rµn converges to the spectrum of Rµ ; in other words, the eigenvalues of the Schrödinger-like operator −∆ + µn defined on H01 (D) converge to the corresponding eigenvalues of the operator −∆ + µ. Remark 2.11. Even if in this paper we limit ourselves to consider only the case of the Laplace operator, the same construction of relaxed domains can be performed for the p-Laplacian −∆p in W01,p (D) for 1 < p ≤ d, and for more general classes of nonlinear operators (see [54]). Given f ∈ Lp (D) and a sequence (Ωn ) of p-quasi-open subsets of D (i.e. defined in a similar way by means of the p-capacity) we denote by un the solution of the following equation on Ωn : −∆p un = f in Ωn , un ∈ W01,p (Ωn ), which has to be understood in the weak sense Z Z 1,p p−2 |∇un | ∇un ∇v dx = un ∈ W0 (Ωn ), f v dx Ωn Ωn ∀v ∈ W01,p (Ωn ). A suitable subsequence (still denoted by the same indices) of (un ) weakly converges in W01,p (D) to the solution of the equation u ∈ W01,p (D) ∩ Lpµ (D), −∆p u + µ|u|p−2 u = f, µ being the Borel measure defined by  Z dν  p−1 µ(A) = A w  +∞ if Capp (A ∩ {w = 0}) = 0 if Capp (A ∩ {w = 0}) > 0. Here w is the weak limit in W01,p (D) of the solutions above with f = 1 and ν = 1 + ∆p w. 2.5 Geometrical restrictions As we have seen, the γ-convergence makes the space M0 (D) a compact metric space, while the subspace of shapes A(D) is not compact (it is even dense in M0 (D)). However, if we consider smaller classes than the whole A(D), introducing geometrical constraints on the class of admissible shapes, we can provide some extra compactness properties. We refer to [26] for all details concerning the restricted classes below. • A very strong geometrical constraint is convexity: the class Aconvex of all convex open subsets of D 9 is compact with respect to many kinds of convergences, among which the γ-convergence. • A weaker geometrical constraint is given by the so-called uniform exterior cone condition: an open set Ω satisfies the uniform exterior cone condition if for every point x0 ∈ ∂Ω there exists a closed cone, with uniform height and opening, and with vertex in x0 , contained in the complement of Ω. The class Aunif cone of all open subsets of D with the uniform exterior cone condition is still compact for the γ-convergence. • The class Aunif f lat cone is the class of domains satisfying a uniform flat cone condition (see [35]), i.e. as above, but with the weaker requirement that the cone may be flat, that is of dimension d − 1. The class Aunif f lat cone is larger than the previous ones and is still compact for the γ-convergence. • The previous geometrical constraints can be further weakened by considering the classes: Acap density of domains Ω satisfying a uniform capacitary density condition, i.e. such that there exist c, r > 0 with
cap Bt (x) \ Ω, B2t (x)
≥ c, ∀t ∈ (0, r), ∀x ∈ ∂Ω cap Bt (x), B2t (x) where Bs (x) denotes the ball of radius s centered at x. Aunif W iener of domains Ω satisfying a uniform Wiener condition, i.e. such that for every point x ∈ ∂Ω
Z R cap Bt (x) \ Ω, B2t (x) dt
≥ g(r, R, x) ∀0 < r < R < 1 t cap Bt (x), B2t (x) r where g : (0, 1) × (0, 1) × D → R+ is a fixed function such that for every R ∈ (0, 1) lim g(r, R, x) = +∞ locally uniformly on x. r→0 For the classes above the following inclusions can be established: Aconvex ⊂ Aunif cone ⊂ Aunif f lat cone ⊂ Acap density ⊂ Aunif W iener and on each of the classes the γ-convergence is equivalent to the Hausdorff complementary convergence Hc (i.e. the Hausdorff convergence of the sets D \ Ωn ) which is then compact. • Another interesting class, which is only of topological type and is not contained in any of the previous ones, was found by Šverák in [83]; it is concerned only with the case d = 2 and consists of all open subsets Ω of D for which the number of “holes”, i.e. connected components of D \ Ω, is uniformly bounded. More precisely, the classes  Om (D) = Ω ⊂ D, Ω open, # connected components of D \ Ω ≤ m are such that on them the γ-convergence is equivalent to the Hausdorff complementary convergence Hc , which is then compact. For higher dimensions d ≥ 3 the same result is no more true but it can be recovered again if the Laplace operator is replaced by the p-Laplace operator, with p ∈]d − 1, d] (see [34]). For p > d the compactness result is trivial, due to the fact that points have a strictly positive p-capacity and that the Sobolev embedding theorem gives uniform continuity of Sobolev maps. 10 3 3.1 Optimization problems for integral functionals Nonexistence of optimal domains We present an explicit example of optimization problem for an integral functional where the existence of an optimal domain does not occur. We fix a bounded open subset D in Rd (d ≥ 2), a function f ∈ L2 (D), and we consider the minimization problem Z n o
min J(Ω) = j x, uΩ (x) dx : Ω ∈ A(D) (3.1) D where uΩ is the solution of the Dirichlet problem in Ω −∆u = f in Ω, u ∈ H01 (Ω), (3.2) extended by zero on D \ Ω. For instance, taking j(x, s) = |s − u0 (x)|2 , the problem amounts to find a state function u as close as possible, in the L2 (D) norm, to the desired state u0 , only acting on the shape of the Dirichlet region D \ Ω, which can then be seen as a control variable, with (3.2) as the corresponding state equation. Considering the state function u as the temperature of a conducting medium D with prescribed heat sources f in a stationary configuration, the optimization problem (3.1) consists in finding an optimal distribution of the Dirichlet region D \ Ω in order to achieve a temperature as close as possible to the desired temperature u0 . We shall see that for a large class of integrands j(x, s) an optimal solution does not exist; the reason is that a Dirichlet region D \ Ω composed of many small pieces is more efficient, in terms of the given cost, than a domain composed by a single piece. Therefore a minimizing sequence (Ωn ) of domains is such that D \ Ωn splits more and more, then leading to the nonexistence of the optimum. We argue by contradiction and assume that the optimization problem (3.1) admits a solution Ω that does not coincide with the entire set D; assume also for simplicity some mild regularity on Ω, as that Ω is an open set whose boundary has zero Lebesgue measure and whose closure Ω does not fill all the set D. All these extra assumptions could be actually removed (see [37, 38, 46]) only requiring that Ω is a quasi-open set. For every point x0 ∈ D \ Ω we perform the so-called topological derivative on the cost functional, which consists in adding to Ω a small ball Bε of radius ε centered at x0 , then obtaining a competing domain Ωε = Ω ∪ Bε . Since for small ε the ball Bε is disjoint from Ω we obtain Z Z Z

J(Ωε ) = j x, uΩ (x) dx + j x, uε (x) dx + j(x, 0) dx Ω Bε D\Ωε where we denoted by uε the solution of −∆u = f in Bε , The same computation for J(Ω) gives Z Z
J(Ω) = j x, uΩ (x) dx + Ω u ∈ H01 (Bε ). Z j(x, 0) dx + Bε so that J(Ωε ) − J(Ω) = Z Bε j(x, 0) dx, D\Ωε
j x, uε (x) − j(x, 0) dx. Assuming that j(x, s) is continuous in x and continuously differentiable in s, and that the source f is continuous (these conditions can be easily weakened), we obtain uε (x) = f (x0 ) ε2 − |x − x0 |2 + o(ε2 ) 2d 11 ∀x ∈ Bε so that j x, uε (x)
= j(x, 0) + uε (x)js (x, 0) + o(ε2 )
= j(x, 0) + f (x0 )js (x, 0) ε2 − |x − x0 |2
/(2d) + o(ε2 ) = j(x, 0) + f (x0 )js (x0 , 0) ε2 − |x − x0 |2 /(2d) + o(ε2 ). Therefore f (x0 )js (x0 , 0) J(Ωε ) − J(Ω) = 2d Z
ε2 − |x − x0 |2 dx + o(εd+2 ) Bε and, using the optimality of Ω we end up with the necessary condition of optimality: f (x0 )js (x0 , 0) ≥ 0 ∀x0 ∈ D \ Ω. (3.3) For instance, if j(x, s) = |s − u0 (x)|2 and f (x) > 0 on D, the necessary condition above becomes u0 (x) ≤ 0 on D \ Ω. Taking the desired state u0 (x) > 0 on D, we deduce that the only possibility for a domain Ω to be optimal is that Ω = D. However, also this possibility can be excluded if the function u0 is small enough; in fact, in this case the empty set gives a better value of the cost functional, since for small u0 we have Z Z 2 |u0 | dx = J(∅) < J(D) = |uD − u0 |2 dx. D 3.2 D Necessary conditions of optimality Following the argument illustrated in the previous section we can obtain the result below (see [37, 38] for a more detailed proof). Theorem 3.1. Let f ∈ L2 (D) and assume j : D × R → R satisfies the conditions: 1. j(x, s) is measurable in x and continuous in s; 2. |j(x, s)| ≤ a(x) + b|s|2 for suitable a ∈ L1 (D) and b ∈ R; 3. j(x, ·) is continuously differentiable and |js (x, s)| ≤ a1 (x) + b1 |s| for suitable a1 ∈ L2 (Ω) and b1 ∈ R. Then, if a smooth domain Ω solves the shape optimization problem (3.1), then the following necessary condition of optimality holds: f (x)js (x, 0) ≥ 0 a.e. on D \ Ω. (3.4) Remark 3.2. When the optimal set Ω is not smooth but only quasi-open, a necessary condition of optimality of the form of (3.4) has to be given in terms of the fine topology. We recall that the fine topology on D is the weakest topology on D for which all superharmonic functions are continuous (see for instance [57] for a detailed study of properties of the fine topology); by ∂ ∗ Ω and cl∗ Ω we denote respectively the fine boundary and the fine closure of Ω in D. Under the same conditions on f and j as in Theorem 3.1 the topological derivative argument gives (see [37, 38]) that an optimal quasi-open set Ω must fulfill the necessary condition of optimality: f (x)js (x, 0) ≥ 0 a.e. on D \ cl∗ Ω. 12 Other kinds of necessary conditions of optimality can be obtained for the optimization problem (3.1). Assuming again that a smooth optimal domain Ωopt exists, it is convenient to introduce the adjoint state equation, that for any admissible Ω reads −∆v = js (x, uΩ ) in Ω, v ∈ H01 (Ω). In [38] the following optimality conditions have been obtained:  q.e. on Ωopt ;  uv ≤ 0 f (x)js (x, 0) ≥ 0 a.e. on D \ Ωopt ;  (∂u/∂n)(∂v/∂n) = 0 HN −1 -a.e. on D ∩ ∂Ωopt . 3.3 (3.5) (3.6) Relaxed Dirichlet problems We have seen that in general the minimization problem (3.1) may have no solution and its relaxed formulation on M0 (D) Z n o
min J(µ) = j x, uµ (x) dx : µ ∈ M0 (D) , (3.7) D where uµ is the solution of the relaxed Dirichlet problem −∆u + µu = f in D, u ∈ Xµ (D), has to be considered, in order to obtain a relaxed solution, which is a measure of the class M0 (D). The previous analysis leading to necessary conditions of optimality can be performed for relaxed solutions too, considering the adjoint relaxed state equation −∆v + µv = js (x, uµ ) in D, v ∈ Xµ (D). (3.8) Let now µopt an optimal relaxed solution in the space M0 (D) (which always exists thanks to the compactness of the γ-convergence in M0 (D)). The optimality conditions (3.6) can be written for a general µopt as  q.e. in D;  uv ≤ 0 uv = 0 µ-a.e. in D; (3.9)  ∗ f (x)js (x, 0) ≥ 0 a.e. on int {uµopt = 0}, where int* denotes the interior with respect to the fine topology. The last condition in (3.6), involving normal derivatives, has a more involved translation in terms of capacitary measures, and we refer to [38, 26]. Consider now again the problem of finding the temperature closest to the desired one u0 , acting on the Dirichlet region D \ Ω. We have seen that when u0 is small enough and the heat source f is positive, no optimal domain exists and the problem has to be written in the relaxed formulation nZ o min |uµ − u0 (x)|2 dx : µ ∈ M0 (D) D in order to obtain a solution in M0 (D). Thanks to the fact that f > 0 and to the identification seen in Example 2.9 (iii), the relaxed formulation above can be rewritten as nZ o min |u − u0 (x)|2 dx : u ∈ H01 (D), u ≥ 0, ∆u + f ≥ 0 D 13 which is now a strictly convex problem on the closed convex subset {u ∈ H01 (D), u ≥ 0, ∆u + f ≥ 0} of the Sobolev space H01 (D). Therefore there is a unique solution u from which we may deduce the optimal measure µ through the formula 1 2x − 1 µ = f + ∆u . 1 2x − 1  1 − 0,0532617 + 1 + u     − 1u and In particular, 2 if D is a ball2x and−u01 , f are radially symmetric,2the optimal2x solution so the optimal measure µ are radially 1 symmetric too and an explicit 3 computation can be lnunit (1 − x) 2 Let usconsiderthe 0,25    made in polar coordinates. particular case when d = 2, D is the ( 1 − x ) − 0,0532617 − ln x  After   some computations we find  that for  small disk, f≡ 1, and u0 ≡ c is a constant. ln 2 16 ln 1 − x values of the constant c the optimal measure µ has the form y = 1 µ = H2 bBRc + Kc H1 b∂BRc c where the values of Rc and of Kc can be computed explicitely by one-dimensional optimization problems. Note that u = c in BRc . A plot of the corresponding optimal temperature u(r) is given below. Figure 2: Plot of the optimal temperature u(r) for c small. 4 4.1 Spectral optimization problems A general existence result We have seen that in general for a shape optimization problem of the form (1.1), either we consider on the admissible class A some additional geometrical constraints of the kind listed in Section 2.5, or in general the problem does not admit any optimal domain solution and the problem has to be considered in the relaxed class of capacitary measures M0 (D). However, for some particular type of cost functionals, the existence of optimal domains can be obtained (in this case we say that a classical solution exists) without the addition of any geometrical constraint but under the only volume constraint |Ω| ≤ m. In this section we show some interesting classes of cost functionals for which the existence of classical solutions holds. The problems we consider have the general form  min F (Ω) : Ω ∈ A(D), |Ω| ≤ m (4.1) 14 where the admissible class A(D) is as before the class of quasi-open subsets of D. A general existence result for optimal domains has been obtained in [39] under the following conditions on the cost functional F . Theorem 4.1. Let F be a γ-lower semicontinuous functional which is decreasing with respect to the set inclusion. Then the optimization problem (4.1) admits at least one classical solution, i.e. a domain Ω ∈ A(D). Remark 4.2. Since the γ is a rather strong convergence (for instance it implies the convergence of the spectrum, as we have seen in Proposition 2.10) the assumption that F is γ-lower semicontinuous is very mild and is always fulfilled in spectral optimization problems. On the contrary, the monotonicity assumption on F is quite severe; for instance it does not hold in the case Z F (Ω) = |uΩ − u0 (x)|2 dx D considered in Section 3. Remark 4.3. By the monotonicity assumption on the cost functional F in Theorem 4.1, it is clear that the inequality constraint |Ω| ≤ m in (4.1) can be replaced by the equality constraint |Ω| = m. Example 4.4. (Optimal domains for integral functionals). Consider a given f ∈ L2 (D), with f ≥ 0, and let g : D × R →] − ∞, +∞] be a Borel function such that g(x, ·) is lower semicontinuous and decreasing on R for a.e. x ∈ D, and g(x, s) ≥ −a(x) − bs2 for a suitable a ∈ L1 (D) and b ∈ R. For every Ω ∈ A(D) let uΩ = RΩ (f ) be the unique solution of −∆u = f on H01 (Ω), and let Z
F (Ω) = g x, uΩ (x) dx . D Then F is lower semicontinuous with respect to the γ-convergence and, since we assumed f ≥ 0, the maximum principle and the monotonicity properties of g imply that F is decreasing with respect to set inclusion. Therefore, by Theorem 4.1 (and Remark 4.3) the minimum problem Z 
min g x, uΩ (x) dx : Ω ∈ A(D), |Ω| = m D admits at least a solution. Example 4.5. (Optimal domains for spectral problems). For every Ω ∈ A(D) let λk (Ω) be the k th eigenvalue of the Dirichlet Laplacian on H01 (Ω), with the convention λk (Ω) = +∞ if Cap(A) = 0. It is well known that all the mappings Ω 7→ λk (Ω) are decreasing with respect to set inclusion; moreover they are continuous with respect to the γ-convergence (see Proposition 2.10 (iv) and Remark 2.7 (iv)), so that Theorem 4.1 (and Remark 4.3) applies and for every k ∈ N and 0 ≤ m ≤ |D| we obtain that the minimization problem  min λk (Ω) : Ω ∈ A(D), |Ω| = m admits a (classical) solution. More generally, arguing in the same way, the minimum 
min Φ λ(Ω) : Ω ∈ A(D), |Ω| = m is achieved too, where λ(Ω) denotes the spectrum of −∆ on H01 (Ω), that is the sequence
λk (Ω) , and the function Φ : [0, +∞]N → [0, +∞] is lower semicontinuous and increasing, in the sense that λhk → λk ∀k ∈ N ⇒ Φ(λ) ≤ lim inf Φ(λh ) , λk ≤ µk ∀k ∈ N ⇒ Φ(λ) ≤ Φ(µ) . h→∞ 15 Example 4.6. (Domains with minimal capacity). Since Cap(E) is a increasing set function, the mapping Ω 7→ F (Ω) = Cap(D \ Ω) is decreasing with respect to the set inclusion. It is not difficult to verify that the mapping F is also γ-continuous, so that the existence Theorem 4.1 applies, and the minimum  min F (Ω) : Ω ∈ A(D), |Ω| ≤ m is achieved. If F denotes the class of all quasi-closed subsets of D, passing to complements and taking m = |D| − k in the previous problem, we have that the minimum  min Cap(E) : E ∈ F, |E| = k (4.2) is achieved. If we denote by E0 a solution to (4.2) we show that  Cap(E0 ) = min Cap(E) : E ⊂ D, |E| = k . (4.3) In fact, for every subset E of D there exists a quasi-closed set E 0 such that E ⊂ E 0 and Cap(E) = Cap(E 0 ) (see for instance Section 2 of [60], or Proposition 1.9 of [51]). If |E| = k, then |E 0 | ≥ k, so that there exists E 00 ∈ F with E 00 ⊂ E 0 and |E 00 | = k. By (4.2)) we then have Cap(E0 ) ≤ Cap(E 00 ) ≤ Cap(E 0 ) = Cap(E), which proves (4.3). The idea of the proof of Theorem 4.1 is to consider a relaxed solution, that always exists, which is a measure µ of the class M0 (D). If we denote by F the relaxed functional, defined on M0 (D) we then have  F (µ) = inf F (Ω) : Ω ∈ A(D), |Ω| ≤ m . We associate to the measure µ the solution uµ of the relaxed Dirichlet problem −∆u + µu = 1 in D, u ∈ Xµ (D) and the domain Ω = {uµ > 0} which is quasi-open, since uµ ∈ H01 (D). The new measure ν = ∞D\Ω fulfills the inequality ν ≤ µ and the monotonicity assumption on the cost functional gives F (Ω) = F (ν) ≤ F (µ), then showing the optimality of the domain Ω. 4.2 The weak γ-convergence To give the proof of Theorem 4.1 in a more rigorous way we introduce a new convergence, much weaker than γ, that makes the class A(D) compact. We call weak γ this new convergence and we denote it by wγ. Definition 4.7. We say that a sequence (Ωn ) of domains in A(D) weakly γ-converges to a domain Ω ∈ A(D) if the solutions wΩn = RΩn (1) converges weakly in H01 (D) to a function w ∈ H01 (D) (that we may take quasi-continuous) such that Ω = {w > 0}. Remark 4.8. We stress the fact that, in general, the function w in Definition 4.7 does not coincide with the solution wΩ = RΩ (1); this happens only if Ωn γ-converges to Ω, which in general does not occur, because γ-convergence is not compact on A(D). We only have 16 that Ωn γ converges to some µ ∈ M0 (D), so that the function w in Definition 4.7 coincides with the solution wµ = Rµ (1) of the relaxed Dirichlet problem −∆u + µu = 1 in D, u ∈ Xµ (D). Also, we notice that, by its definition, the wγ-convergence is compact, since the sequence wΩn = RΩn (1) is bounded in H01 (D) so it always has a subsequence (Ωnk ) weakly converging to some function w ∈ H01 (D), and the set of positivity Ω = {w > 0} (which is quasi-open since w ∈ H01 (D)) is then the wγ-limit of (Ωnk ). Finally, by Definitions 2.6 and 4.7 we obtain that the wγ-convergence is weaker than the γ-convergence. Since the wγ-convergence is rather weak, the class of wγ-lower semicontinuous functionals is much smaller than the class of γ-lower semicontinuous functionals. However, the proposition below shows that some relevant examples are still valid. Proposition 4.9. Let f ∈ L1 (D) be a nonnegative function. Then the mapping Ω 7→ R Ω f dx is wγ-lower semicontinuous on A(D). Proof. Let (Ωn ) be a sequence in A(D) that wγ-converges to some Ω ∈ A(D); this means that wΩn → w in L2 (RN ) and that Ω = {w > 0}. Passing to a subsequence we may assume that wΩn → w a.e. on D. Suppose x ∈ Ω is a point where wΩn (x) → w(x). Then w(x) > 0, and for n large enough we have that wΩn (x) > 0. Hence x ∈ Ωn . So we have shown that 1Ω (x) ≤ lim inf 1Ωn (x) n→+∞ for a.e. x ∈ D. Fatou’s lemma now completes the proof. In order to show the wγ-lower semicontinuity of other shape functionals, the following lemma is needed. Lemma 4.10. Let (Ωn ) be a sequence of quasi-open sets wγ-converging to a quasi-open set Ω. Then there exists a subsequence (still denoted by the same indices) and a sequence of quasi-open sets Gn ⊂ D with Ωn ⊂ Gn such that Gn γ-converges to Ω. Proof. Let us denote by wΩ the solution RΩ (1) of the Dirichlet problem −∆u = 1 in Ω, u ∈ H01 (Ω); we have w = wΩ = 0 on D \ Ω and (see for instance [54] for a detailed proof) w ≤ wΩ on Ω, and hence w ≤ wΩ on the entire set D. For each ε > 0 we define the quasi-open set Ωε = {wΩ > ε}. For a subsequence, we may suppose that wΩn ∪Ωε converge to some wε weakly in H01 (D) and, since by a comparison principle we have wΩn ∪Ωε ≥ wΩε , passing to the limit as n → +∞ we have that wε ≥ wΩε . Let us show that wε ∈ H01 (Ω). Indeed, defining v ε = 1 − 1ε min{wΩ , ε} we get 0 ≤ v ε ≤ 1, v ε = 0 on Ωε , and v ε = 1 on D \ Ω. Taking un = min{v ε , wΩn ∪Ωε } we get un = 0 on Ωε ∪ (D \ (Ωn ∪ Ωε )), and in particular on D \ Ωn . Moreover un converges to min{v ε , wε } weakly in H01 (D) and hence min{v ε , wε } vanishes q.e. on {w = 0}. Since v ε = 1 on D \ Ω we get that wε = 0 q.e. on D \ Ω. Using Theorem 5.1 of [54], from the fact that −∆wΩn ∪Ωε ≤ 1 in D we get −∆wε ≤ 1 and hence wε ≤ wΩ . Finally wΩε ≤ wε ≤ wΩ , and by a diagonal extraction procedure we get that wΩn ∪Ωεn converge to wΩ weakly in H01 (D). Therefore the quasi-open sets Gn = Ωn ∪Ωεn γ-converge to Ω, which concludes the proof. 17 For monotone decreasing shape functionals F the γ-lower semicontinuity and the wγlower semicontinuity coincide, as the following result shows. Proposition 4.11. Let F : A(D) → [−∞, +∞] be a γ-lower semicontinuous shape functional which is monotone decreasing with respect to the set inclusion. Then F is wγ-lower semicontinuous. Proof. Let us consider a sequence of quasi-open sets (Ω)n in A(D) wγ-converging to some quasi-open set Ω ∈ A(D), and let Ωnk be a subsequence such that lim F (Ωnk ) = lim inf F (Ωn ). n→+∞ k→+∞ By Lemma 4.10 there exists a subsequence (which we still denote by {Ωnk }) and Gnk ⊃ Ωnk such that Gnk γ-converge to Ω. The γ-lower semicontinuity of F gives F (Ω) ≤ lim inf F (Gnk ) k→+∞ and the monotonicity of F gives F (Gnk ) ≤ F (Ωnk ). Therefore F (Ω) ≤ lim inf F (Gnk ) ≤ lim inf F (Ωnk ) = lim inf F (Ωn ) k→+∞ k→+∞ n→+∞ which proves the required wγ-lower semicontinuity. Proof of Theorem 4.1. The proof of Theorem 4.1 is now straightforward. In fact, if (Ωn ) is a minimizing sequence of quasi-open sets for the optimization problem (4.1), by the compactness of the wγ-convergence we may extract a subsequence (still denoted by the same indices) that wγ-converges to some quasi-open set Ω ∈ A(D). By Proposition 4.9 we have |Ω| ≤ lim inf |Ωn | ≤ m n→+∞ and by Proposition 4.11 we have F (Ω) ≤ lim inf F (Ωn ). n→+∞ Therefore Ω is an optimal set for the minimum problem (4.1). Example 4.12. The minimization  min λ1 (Ω) : Ω ∈ A(D), |Ω| ≤ m is one of the first examples of spectral optimization problems. By Theorem 4.1 we know that an optimal solution exists and, it was conjectured by lord Rayleigh that, for D large enough (to contain at least a ball of measure m) the optimal domain is a ball of measure m. The first proof of this fact was obtained by Faber and Krahn by symmetrization techniques (see [59, 74, 75]). On the contrary, if D does not contain a ball of measure m, the optimal sets Ωopt have to touch the boundary ∂D; it has been shown (see [64, 23]) that such sets Ωopt are actually open sets and that their free boundary (i.e. the part of ∂Ωopt included in D) is smooth. However, the free boundary D ∩ ∂Ωopt does not contain any part of spherical surface (see [68, 69] and also [67] Theorem 3.4.1), in the sense that no part of D ∩ ∂Ωopt locally coincides with a sphere. When the bounding box D is convex we expect that the optimal sets Ωopt for λ1 are convex too; however, this result, even if very natural and strongly expected, is not yet available, and the question is still open. 18 Example 4.13. The minimization problem  min λ2 (Ω) : Ω ∈ A(D), |Ω| ≤ m also verifies the assumptions of the existence Theorem 4.1; as proved in [75, 81], for D large enough (to contain at least two disjoint balls of measure m/2 each) the optimal domain is the union of two disjoint balls of measure m/2 each. As before, if D is not large enough in the sense above, the optimal sets Ωopt have to touch the boundary ∂D, but in this case, even if it seems reasonable to conjecture that the free boundary of optimal sets is regular, the regularity question is still open. Actually, the proof that Ωopt are open sets is not yet available.  Minimizing λ2 (Ω) in the more restricted class Ω ∈ A(D), |Ω| ≤ m, Ω convex also admits an optimal solution Ωopt , as it is easy to prove. When D is large enough, since without the convexity assumption the solution is given by two equal disjoint balls, a reasonable expectation, also supported in [85] by some numerical computations, is that Ωopt is a stadium, i.e. the convex hull of two equal disjoint balls tangent each other. This conjecture has been disproved in [68, 69] where again it has been shown that, even if Ωopt is very close to a stadium, the nonflat parts do not locally coincide with a sphere. When k ≥ 3 the optimal shapes for the minimization problem  min λk (Ω) : Ω ∈ A(D), |Ω| ≤ m , that exist again thanks to the existence Theorem 4.1, are not known. For k = 3 and D large enough the conjecture, still open, is that: the optimal domains for λ3 are balls if the dimension d is 2 or 3 and the union of three equal disjoint balls when d ≥ 4. For k = 4, d = 2 and D large enough the conjecture, also still open is: the optimal domains for λ4 in dimension d = 2 are the union of two disjoint balls of different radius, whose radii are in the ratio (j0,1 /j1,1 )1/2 ∼ 0.79, where j0,1 and j1,1 are the first zeroes of the Bessel functions J0 and J1 respectively. For k ≥ 5 one could expect that optimal sets are also made by a suitable array of balls. This is false, as shown by Keller and Wolf for d = 2 and k = 13; in fact in [77] numerical computation are provided for d = 2 showing that for k ≥ 5 optimal arrays of disks are not optimal (see Figure 3). Remark 4.14. In Theorem 4.1 the assumption that admissible domains are all contained in a given bounded domain D is crucial. A similar existence result for D = Rd is not known. For instance, the eigenvalue optimization problem  (Pk ) min λk (Ω) : Ω ⊂ Rd , |Ω| ≤ m is known to have a solution for k = 1 (a ball of measure m), for k = 2 (two disjoint balls of measure m/2 each) and for k = 3 (a ball conjectured if d = 2, 3 and three equal disjoint balls for d ≥ 4). This last existence result was proved in [33], where more generally it is shown that if the minimizers of problem (Pj ) exist and are bounded for j = 1, . . . , k −1, then the minimizers of problem (Pk ) exist. For instance, a proof of boundedness of minimizers of λ3 (Ω) with Ω ⊂ Rd would imply the existence of optimal domains for λ4 (Ω) with Ω ⊂ Rd . Several other explicit solutions for minimization problems involving eigenvalues are known or conjectured; we refer the interested reader to [10, 26, 66, 67, 70] for a wider presentation of this subject. 19 No Optimal union of discs Computed shapes 3 46.125 46.125 4 64.293 64.293 5 82.462 78.47 6 92.250 88.96 7 110.42 107.47 8 127.88 119.9 9 138.37 133.52 10 154.62 143.45 Figure 3: Optimal domains for λk and optimall arrays of disks, 3 ≤ k ≤ 10. 4.3 Spectral problems with perimeter constraint We consider in this section the case when a perimeter constraint is imposed on the admissible domains. Here we use the De Giorgi definition of perimeter (see for instance [7]): a measurable set Ω is said to have a finite perimeter if its characteristic function  1 if x ∈ Ω 1Ω (x) = 0 otherwise belongs to the space BV (Rd ) of functions with bounded variation on Rd , i.e. functions in L1 (Rd ) whose distributional gradient is a measure with finite total variation. In this case the perimeter of Ω is defined by Z Per(Ω) = |∇1Ω |. Rd Since for a sequence of domains equi-bounded perimeter implies strong L1 compactness of the characteristic functions, the situation with perimeter constraint on the admissible class could seem a priori better than the one with volume constraint. However the following difficulties arise. • The relaxation of Dirichlet problems, and hence the appearance of capacitary measures as limits of minimizing sequences, may occur even with a perimeter constraint on the admissible domains. It is enough to look at the Cioranescu and Murat example presented in Section 2.4: the domains γ-converging to the Lebesgue measure can be taken with perimeter arbitrarily small. • The perimeter is not wγ-lower semicontinuous. Indeed, again by a construction like the one of Cioranescu and Murat, we may produce a sequence (Ωn ) of subsets of the 20 ball B2 centered at the origin and of radius 2 made by removing n small holes from the ball B1 of radius 1. If rn denotes the critical radius in Cioranescu and Murat example, it is enough to take the radius sn of the small holes such that rn  sn  n1/(1−d) . In this way, the limit set is Ω = B2 \ B1 and Per(B2 ) + Per(B1 ) = Per(Ω) > lim Per(Ωn ) = Per(B2 ). n→+∞ • The perimeter constraint gives the compactness of minimizing sequences in the L1 convergence of characteristic functions, but the γ-limit can be considerably smaller. Indeed, the same example as above, with removed balls radius sn such that rn  sn  n−1/d shows that Ωn → B2 in the L1 sense, while Ωn → B2 \ B1 in the γ-convergence sense. The link between wγ-convergence and L1 -convergence is given by the following proposition. Proposition 4.15. Let (An ) be a sequence of quasi-open sets which wγ-converges to a quasi-open set A, and assume that there exist measurable sets Ωn such that An ⊂ Ωn , and that (Ωn ) converges in L1 to a measurable set Ω. Then we have |A \ Ω| = 0. Proof. By applying Proposition 4.9 with f = 1D\Ω we obtain |A \ Ω| ≤ lim inf |An \ Ω| = lim inf |An \ Ωn | = 0, n→∞ n→∞ which concludes the proof. In order to consider shape optimization problems with perimeter constraints it is convenient to extend the definition of a monotone decreasing functional F defined on A(D) also to measurable sets by setting  F (A) = inf F (Ω) : Ω ⊂ A a.e., Ω ∈ A(D) . (4.4) We notice that, since F is monotone decreasing, we have that its value on A(D) is not modified by this extension. The shape optimization problems we are interested in are of the form  inf F (Ω) : Ω ⊂ D, Per(Ω) ≤ L , (4.5) where L is a given positive real number. Theorem 4.16. If F is γ-lower semicontinuous and decreasing with respect to the set inclusion, then there exists a finite perimeter set Ωopt which solves the variational problem (4.5). Proof. Let (Ωn ) be a minimizing sequence for problem (4.5). Since Per(Ωn ) ≤ L we may extract a subsequence (still denoted by (Ωn )) that converges in L1 to a set Ω with Per(Ω) ≤ L. By the construction of the extension of the functional F to measurable sets there are quasi-open sets ωn ⊂ Ωn a.e. such that F (ωn ) = F (Ωn ). 21 By the compactness of wγ-convergence we may assume that (ωn ) is wγ-converging to some quasi-open set ω, and by Proposition 4.15 we have |ω \ Ω| = 0. Therefore, we have that F (Ω) ≤ F (ω) ≤ lim inf F (ωn ) = lim inf F (Ωn ). n→∞ n→∞ Hence Ω solves the variational problem (4.5). Of course, if F (Ω) = λ1 (Ω) and D is large enough to contain a ball of perimeter L, the balls solve the shape optimization problem (4.5) and are the unique minimizers. Indeed, symmetrizing Ω both reduces the cost λ1 (Ω) as well as the perimeter Per(Ω). The situation is different for the cost F (Ω) = λ2 (Ω), which was first considered in [61]. We summarize here the results obtained in [29], to which we refer for the detailed proofs. • In the case d = 2 it is easy to see that the optimal set Ωopt has to be convex, since convexification both reduces the cost λ2 as well as the perimeter. • In the case d = 2, if Ωopt is a minimizer for λ2 with perimeter constraint, then λ2 (Ω) is simple. • Using the facts above, in the case d = 2 it is possible to prove the C ∞ regularity of Ωopt . Moreover, if u2 denotes the second eigenfunction, with unitary L2 norm, we have the necessary condition of optimality |∇u2 (x)|2 = 2λ2 (Ωopt ) k(x) Per(Ωopt ) ∀x ∈ ∂Ωopt where k(x) is the curvature of ∂Ωopt at x. • Always in the case d = 2 additional necessary conditions of optimality can be proved, as: - the boundary of Ωopt does not contain any segment; - the boundary of Ωopt does not contain any arc of circle; - the boundary of Ωopt contains exactly two points where the curvature vanishes. A numerical plot of Ωopt in the two-dimensional case is given in Figure 4. The picture shows that minimizers in two dimensions should have two axes of symmetry (one of these containing the nodal line), but the proof of this fact is not yet available. • If d ≥ 3 no regularity results for the optimal domains are available; actually, at present we do not even know if optimal domains are open sets. For a similar problem with perimeter penalization the regularity of optimal domains has been proved in [12]. • If d ≥ 3 optimal domains are not convex (see [71]); they should have a cylindrical symmetry, even if this fact has not yet been proved. 5 An example of nonmonotone cost functional In this section we consider as a particular cost functional the case of a function which only depends on the first two eigenvalues of the Dirichlet Laplacian:
F (Ω) = Φ λ1 (Ω), λ2 (Ω) . By the results of Section 4.1 (see Theorem 4.1) we know that the optimization problem n o
min Φ λ1 (Ω), λ2 (Ω) : Ω ∈ A(D), |Ω| ≤ m , (5.1) 22 Figure 4: Plot of the optimal set for λ2 with perimeter constraint, in the case d = 2. without additional geometrical constraints on the admissible domains Ω, admits a classical solution whenever the function Φ is increasing in each of its variables. The following very natural question then arises: what happens when Φ does not satisfy the monotonicity condition above? In the rest of this section we suppose the bounding box D is large enough to allow all the constructions that will be made. It is convenient to introduce the attainable set  E = (x, y) ∈ R2 : x = λ1 (Ω), y = λ2 (Ω) for some Ω ∈ A(D), |Ω| ≤ m , so that the optimization problem (5.1) can be rewritten as  min Φ(x, y) : (x, y) ∈ E . (5.2) The set E is a subset of R2 which verifies the following properties. 1. Being the coordinates of points in E the first and second eigenvalues of the Dirichlet Laplacian in some domain Ω, we have that x > 0 and y > 0 for every (x, y) ∈ E. 2. Since λ2 (Ω) ≥ λ1 (Ω) we have that x ≤ y for every (x, y) ∈ E. 3. Since the ball B of measure m makes λ1 (Ω) minimal, we have x ≥ λ1 (B) for every (x, y) ∈ E. 4. Since the union A of two disjoint balls of measure m/2 each makes λ2 (Ω) minimal, we have y ≥ λ2 (A) for every (x, y) ∈ E. 5. Taking the domain Ω/t with t ≥ 1 and using the fact that λk (Ω/t) = t2 λk (Ω), we have that the set E is conical, i.e. if (x, y) ∈ E then (tx, ty) ∈ E for all t ≥ 1. 6. By a result obtained in [11] (proving a conjecture by Payne, Pólya and Weinberger stated in [78]) the balls minimize the ratio λ1 (Ω)/λ2 (Ω) among all domains Ω, therefore y ≤ xλ2 (B)/λ1 (B) for every (x, y) ∈ E. 23 A numerical output of the set E in the two-dimensional case has been obtained in [87] and 6.4 The first two eigenvalues 155 is reported below. Figure set E for N = 2ofand = 1. Figure6.1. 5: The A numerical plot thecset E. If the function Φ is lower semicontinuous on R2 and satisfies the coercivity condition Theorem 6.4.1 The set E is closed in R2 . lim Φ(x, y) = +∞ |(x,y)|→+∞ The proof of the theorem above is based on the following lemma. (5.3) then the existence of a solution to problem (5.2), and then to problem (5.1), follows straightLemma If the set Etheorem is convex on the and horizontal directions, then forward by6.4.2 the Weierstrass as soon as vertical we can prove that the set E is closed in R2 . 2 E is closed in R . Proposition 5.1. If the set E is convex, then it has to be closed. ProofLetConsider y) consider ∈ Ē. There exists a sequence of sets (An )n∈N ⊆ Ac (B) Proof. (x̄, ȳ) ∈ E(x, and the minimization problem such that s(An ) → (x, y).From the weak γ -compactness of the set Ac (B), for a + minby(xthe − x̄) + indices (y − ȳ)+we: can (x, y)write ∈E A , n → A in the weak subsequence still denoted same γ -sense. Then A ∈ Ac (B) and since the eigenvalues of the Laplacian are weakly which is equivalent to γ -lower semicontinuous we get 
+
+ min λ1 (Ω) − x̄ + λ2 (Ω) − ȳ : Ω ∈ A(D), |Ω| ≤ m . λ1 (A) ≤ liminf λ1 (An ) = x n→∞ and (x − x̄)+ + (y − ȳ)+ λ2 (A) ≤ liminf λ2 (An ) = y. n→∞ Since the function is increasing in each of its variables, by the existence Theorem the minimum problems above admitsegment optimaljoining solutions (xopt , yoptthe ) ∈half E and From the4.1vertical convexity of E, the vertical s(A) with E it is also clear that the minimum value has to Ωline ∈ A(D) respectively. Since (x̄, ȳ) ∈ {ts(B ) : t ≥ 1} is contained in E. If y < λ (B ) we can find the point opt 1 2 1 be(λzero, so that 1 (A), y) on this segment and using now the horizontal convexity, the segment yoptBut ≤ ȳ. joining (λ1 (A), y) to {ts(B2 ) : xt opt ≥≤ 1}x̄,is in E. this segment contains the point λ1 (A)that ≤ x.the set E is convex, the segment joining (xopt , yopt ) to the point of If(x, we y) aresince assuming
If yby≥ λ12(B), (B1 ),λ2then horizontal convexity gives directly (x,contained y) ∈ E. in E, as well E given (B) the , where B is the ball of measure m, is all
as the segment joining (xopt , yopt ) to the point of E given by λ1 (B̃), λ2 (B̃) , where B̃ is the Lemma it suffices proveBythe of E on (v) vertical unionFollowing of two disjoint balls6.4.2 of measure m/2toeach. theconvexity conicity property aboveand of the horizontal directions. For this purpose, we split the proof in two steps: set E the point (x̄, ȳ) has to belong to E thus proving that E is closed. 24 Unfortunately, in spite of the numerical evidence of the convexity of E provided by Figure 5, a proof of the convexity of E is still missing, so the result of Proposition 5.1 above is useless to deduce the closedness of E and then the existence of optimal domains for the minimization problem (5.1). However, a result stronger than the one of Proposition 5.1 holds. Definition 5.2. We say that the set E is: • horizontally convex if for every point (x0 , y0 ) ∈ E the horizontal segment joining (x0 , y0 ) to the straight line y = x is all contained in E; • vertically convex if for every point (x0 , y0 ) ∈ E the vertical segment joining (x0 , y0 ) to the straight line y = kx, with k = λ2 (B)/λ1 (B) being B any ball of measure m, is all contained in E. Of course, if E is convex it is also horizontally convex and vertically convex, while the converse could not in principle be true. It happens that horizontal and vertical convexity are still enough to prove that E is closed. Proposition 5.3. If the set E is both horizontally convex and vertically convex, then it has to be closed. Proof. It is enough to repeat the proof of Proposition 5.1 and to conclude that (x̄, ȳ) ∈ E by using horizontal and vertical convexity instead of convexity. While the convexity of the set E is still unproved, the horizontal and vertical convexity can be showed; we give a sketch of the proof omitting some technical details. A complete proof can be found in [27]. Proof of horizontal convexity. We have to show that, given a domain Ω ∈ A(D) with |Ω| ≤ m, we can construct domains Ω(t) ∈ A(D) with |Ω(t)| ≤ m such that λ2 (Ω(t)) = λ2 (Ω), λ1 (Ω(t)) = tλ2 (Ω) + (1 − t)λ1 (Ω), ∀t ∈ [0, 1]. (5.4) Assume that Ω is a regular set; if we denote by u a second eigenfunction for Ω and by S the corresponding nodal set, i.e. the set {u = 0}, we may continuously shrink S, obtaining then for t ∈ [0, 1] sets S(t) ⊂ S continuously shrinking from S to ∅. The sets Ω(t) = Ω \ S(1 − t) then γ-continuously move from Ω to Ω \ S. Since S is a nodal set we have
λ2 Ω(t) = λ2 (Ω) ∀t ∈ [0, 1]. On the
other hand, since Ω(t) γ-continuously decreases from Ω to Ω \ S, we have that λ1 Ω(t) continuously increases from λ1 (Ω) to λ1 (Ω \ S) = λ2 (Ω). A reparametrization of the map t 7→ Ω(t) now gives the required property (5.4). A more careful analysis is made in [27] where the following result is shown. The application to horizontal convexity is then as above, with Ω0 = Ω and Ω1 = Ω \ S. Proposition 5.4. Let Ω1 ⊂ Ω0 be two quasi-open sets. Then there exists a decreasing homotopy from Ω0 to Ω1 which is γ-continuous, namely there exists a γ-continuous mapping h : [0, 1] → A(D) such that h(t) ⊂ h(s) for s < t, h(0) = Ω0 , 25 h(1) = Ω1 . An important tool that we shall use in the proof of horizontal convexity is the so-called continuous Steiner symmetrization (see [24, 25]). Roughly speaking it consists in a path t 7→ Ωt starting at t = 0 from any domain Ω ∈ A(D) and symmetrizing it more and more, finishing at t = 1 in a ball of the same measure. The crucial fact is that during the evolution the first eigenvalue λ1 (Ωt ) decreases, as a consequence of the fact that Ωt is more and more symmetric. Proof of horizontal convexity. We have to show that, given a domain Ω ∈ A(D) with |Ω| ≤ m, we can construct domains Ωt ∈ A(D) with |Ωt | ≤ m such that λ1 (Ωt ) = λ1 (Ω), λ1 (Ωt ) = tλ2 (B) + (1 − t)λ2 (Ω), ∀t ∈ [0, 1] (5.5) where, again, B denotes the ball of measure m. If the continuous Steiner symmetrization Ωt was γ-continuous, the proof could be easily achieved because the path x(t) = λ1 (Ωt ), y(y) = λ2 (Ωt )

would join continuously the point λ1 (Ω), λ2 (Ω) to the point λ1 (B), λ2 (B) with x(t) descreasing, and this, together with the conicity property (v) of the set E would allow us to conclude that for every t ∈ [0, 1] there is a set Ωt verifying the property (5.5). Actually, the continuous Steiner symmetrization path Ωt provides a mapping t 7→ λ2 (Ωt ) which is only lower semicontinuous on the left and upper semicontinuous on the right, but this is still enough to achieve the proof, using again the conicity property (v) of the set E. Summarizing, we have obtained the following existence result Theorem 5.5. If D is large enough, and Φ is lower semicontinuous and verifies the coercivity condition (5.3), then the shape optimization problem (5.1) admits a solution. In the proof, several properties of the first two eigenvalues of the Laplace operator have been used; we do not know if more general problems of the form n o
min Φ λi1 (Ω), . . . , λik (Ω) : Ω ∈ A(D), |Ω| ≤ m still admit an optimal domain solution, without monotonicity assumptions on the function Φ, for different choices of the indices i1 , . . . , ik . 6 6.1 Optimal partition problems Existence of optimal partitions In this section we consider the minimization problem for shape cost functionals of the form F (Ω1 , . . . ΩN ) where N is a fixed integer, the unknown domains Ωi ∈ A(D), the cost is a map F : A(D)N → [0, +∞] and the domains Ωi fulfill the condition Ωi ∩ Ωj = ∅ for i 6= j. A family {Ω1 , . . . , ΩN } of pairwise disjoint subsets of D will be called a partition. The minimization problem we consider is then  min F (Ω1 , . . . ΩN ) : Ωj ∈ A(D), Ωi ∩ Ωj = ∅ for i 6= j . (6.1) As in the case of a single domain Ω we may consider several interesting cases of partition cost functionals F : 26 1. integral costs given by Z F (Ω1 , . . . ΩN ) = D
j x, RΩ1 (f1 ), . . . , RΩN (fN ) dx where fi ∈ L2 (D) are given and as usual the solutions RΩi (fi ) of −∆u = fi on Ωi u ∈ H01 (Ωi ) are intended as extended by zero outside Ωi 2. spectral costs given by
F (Ω1 , . . . ΩN ) = Φ λ(Ω1 ), . . . , λ(ΩN ) where we denoted by λ(Ω) the spectrum of the Dirichlet Laplacian in Ω. By the same methods used in Section 3.2 we may obtain some necessary conditions of optimality for the case of integral costs (i) above. Assuming that an optimal partition (Ω1 , . . . , ΩN ) exists and is made of smooth domains we have  q.e. on Ωi ;  ui vi ≤ 0 fi (x)jsi (x, 0, . . . , 0) ≥ 0 a.e. on D \ Ωi ; (6.2)
S  (∂ui /∂n)(∂vi /∂n) = 0 HN −1 -a.e. on D ∩ ∂Ωi \ j6=i ∂Ωj for all i = 1, . . . , N , where we denoted by ui and vi the solutions of the state and adjoint state equation −∆u = fi on Ωi u ∈ H01 (Ωi ) and −∆v = jsi (x, u1 , . . . , uN ) on Ωi v ∈ H01 (Ωi ) respectively. In general, without extra conditions either on the admissible partitions or on the cost functional F , we could reproduce counterexamples to the existence of optimal partitions similar to the ones seen in Section 3.1, and optimal solutions only exist in a relaxed sense. For optimal partition problems the relaxed formulation will be considered later in Section 6.2; here we focus our attention on some monotonicity assumptions on the cost F that will imply the existence of classical solutions to problem (6.1). Definition 6.1. We say that F : A(D)N → [0, +∞] is: 1. γ-lower semicontinuous if F (Ω1 , . . . , ΩN ) ≤ lim inf F (Ωn1 , . . . , ΩnN ) n→+∞ whenever Ωni → Ωi in the γ-convergence, for i = 1, . . . , N ; 2. wγ-lower semicontinuous if F (Ω1 , . . . , ΩN ) ≤ lim inf F (Ωn1 , . . . , ΩnN ) n→+∞ whenever Ωni → Ωi in the wγ-convergence, for i = 1, . . . , N ; 27 3. monotonically decreasing in the sense of the set inclusion if for all (A1 , . . . , AN ), (B1 , . . . , BN ) ∈ A(D)N such that Ai ⊂ Bi for all i = 1, . . . , N in the sense of capacity, i.e. Cap(Ai \ Bi ) = 0, then F (B1 , . . . , BN ) ≤ F (A1 , . . . , AN ). The following existence result for optimal partition problems with monotonically decreasing shape cost functional has been obtained in [28]. Theorem 6.2. Let F : A(D)N → [0, +∞] be a shape cost functional which is wγ-lower semicontinuous. Then the optimal partition problem (6.1) admits a classical solution. This happens for instance if F is γ-lower semicontinuous and monotonically decreasing in the sense of the set inclusion. For instance Theorem 6.2 applies to the case
F (Ω1 , . . . , ΩN ) = Φ λk1 (Ω1 ), . . . , λkN (ΩN ) where ki ≥ 1 are integers and Φ is increasing in each variable, i.e. Φ(s1 , . . . , sN ) ≤ Φ(t1 , . . . , tN ) whenever si ≤ ti , i = 1, . . . , N. Note that no measure constraints on the sets Ωi are imposed; however, thanks to the wγ-lower semicontinuity result of Proposition 4.9, if F : A(D)N → [0, +∞] is wγ-lower semicontinuous, the measure constrained minimization problem  min F (Ω1 , . . . ΩN ) : Ωj ∈ A(D), Ωi ∩ Ωj = ∅ for i 6= j, |Ωi | ≤ mi still admits a classical solution. A particularly interesting example is given by the shape cost functional F (Ω1 , . . . , ΩN ) = λ1 (Ω1 ) + · · · + λ1 (ΩN ). In [43, 44] Caffarelli and Lin considered the equivalent variational formulation of the problem Z nZ o 2 1 N 2 min |∇u| dx : u ∈ H0 (D; R ), ui dx = 1 for all i, G(u) = 0 D D where G(u) = X u2i u2j , i6=j and showed the regularity of the free boundary of the optimal partition, up to a singular set of Hausdorff dimension less than or equal to d − 2. They also formulated a very interesting conjecture about the behaviour of the optimal partitions when the number of parts N tends to +∞: in the two-dimensional case d = 2 the optimal partitions {Ωopt → +∞, to be i } tend, as NP opt made by a regular exagonal tiling; in particular, for the optimal value PN = N i=1 λ1 (Ωi ) the estimate λ1 (H) N −2 PN = + o(1) |D| holds, where H is a regular exagon of area equal to 1. The numerical computations made in [19] confirm the conjecture, as Figure 6 shows. 28 simple and orthogonal projection. The leftmost lgorithm using simple projection. Note how the figure corresponds to the final result obtained g. 3.5. Optimization of the sum of the first eigenvalue of the Dirichlet Laplacian on 384 cells with C = 105 . First Compare value of the grids objective at127), (253 × 253), and (505 × 505). Second row: sum of the first ell shape on the recursively refined (64 ×function 64), (127 × alues on the same grids. Figure 6: Optimal partitions for 6.2 PN i=1 λ1 (Ωi ) with N = 16, N = 384, N = 512. Relaxed formulation of optimal partition problems For a general partition functional F (Ω1 , . . . , ΩN ) which does not satisfy any monotonicity assumption we cannnot expect the existence of a classical solution to the optimization problem (6.1). Similarly to what happens in the case of a single domain, a relaxation procedure is needed to describe the behaviour of minimizing sequences, and this will be done through the use of the capacitary measures described in Sections 2.4 and 3.3. 5 . First sum of the first eigenvalue ofsum the Dirichlet onof384 = 105 . problem First g. 3.6. Optimization of of the firstLaplacian eigenvalue thecells Dirichlet of 512 cells with C =to10(6.1) Inthe order to characterize the expression ofwith theCLaplacian relaxed associated we on increasingly refined grids (4 leftmost figures) d grids (64 × 64), (127 × 127), (253 × 253), and (505 × 505). Second row: sum of the first ell shape on recursively refined grids (64 × 64), (127 × 127), (253 × 253), and (505 × 505). Second row: sum of n n e (from left to right) 25 × 25, × 50, 100 × 100, consider, for 50 every sequence {Ω1 , . . . , ΩN } of pairwise disjoint quasi-open subsets ofthe Dfirst the alues on the same grids. 1,902.1, 2,033.8,associated 2,095.7, 2,124.6, 2,048.8 measuresand µni = ∞D\Ωni (i = 1, . . . , N ) of the class M0 (D) introduced in Section 2.4. Since M0 (D) is compact with respect to the γ-convergence, up to a subsequence, there N of measures (D) (i = 1, Again, . . . , N ) modulo such thatthe µi flattening is the γ-limit of µni . to achieve i ∈ M0hexagons. ned by a partitionexist made pairs of µregular necessary n that the original Ωi were pairwise disjoint imply some conditions on the of large parallel superdicity on a computations unit cell, The this fact is on the configuration that we observe. For k =will 3 (Figure 3.7-right), we obtain limit measures µ , that are not “independent”, and characterize the relaxed optimization 3.2, domain is againhexagons, the iunit which square. odic the tiling by non-regular can be proven to be a sub-optimal solution, as a tiling problem associated to (6.1). For instance, it is is not possible to obtain N -tuple madeour by gular wouldenough lead to that a lower this most certainly due toanthe fact that (n =hexagons 384) is large weenergy. expect Again, all the measures µ equal to the Lebesgue measure on D. tive function admits a great deal ofi local minima, are difficult to avoid in optimization problems he domain. The For computations were run µwhich every capacitary measure ∈ M (D)eigenvalue we denote of byan Ωµoptimal the set “of of to µ, 0 s size. difficulty when k ≥ 2 and is that the k-th cell finiteness” is expected (64 × An 64),additional (127more × 127), (253 × 253), precisely defined as  multiplicity greater than 1 hence and may not be differentiable. 106 , the bounds on the admissible steps Ωµ = Rµ (1) > 0 . oticing that the analysis and algorithm are not restricted to the two–dimensional case, we ported our perator, and the final objective functions The following result to has been proved in [41]. results. We believe that the convergence am to the 3D case, but were unable obtain any meaningful observe that the solution corresponds to Theorem 6.3. An N -tuple (µ111 , . . . , µN ) of capacitary measures is made by the γ-limit m a “good” local minimizer. of pairwise disjoint quasi-open sets {Ωn1 , . . . , ΩnN } if and only if it satisfies the following ion of 512 processors, property: for 512 cells. The 6 ∀i 6= j. he final energies are 2.342 106 , 2.243 10Cap(Ω , µi ∩ Ωµj ) = 0 edges of the domain is that of For instance, if a network Z F (Ω1 , . . . ΩN ) =
j x, RΩ1 (f1 ), . . . , RΩN (fN ) dx, D y be adapted to objective function involvthen theofrelaxed problem associated to cells with C = 105 . First sum of theorder. first eigenvalue thenumerical Dirichlet Laplacian different A classical issue of 512 d grids (64 × 64), (127 × 127), (253 × 253), and (505 × 505). Second row: sum of the first iple eigenvalues with respectminto Fchanges (Ω1 , . . . ΩN ) : Ωi ∈ A(D), Ωi ∩ Ωj = ∅ for i 6= j obtained interesting results nevertheless. is given by 2 and k = 3, respectively, using periodic  modulo f pairs of regular hexagons. Again, the flattening necessary to achieve min for F (µk . . µ2N )is: µi ∈ M0 (D), Cap(Ωµi ∩ Ωµj ) = 0 for i 6= j 1 , .= olds, the optimal partition is the configuration that we observe. For k = 3 (Figure 3.7-right), we obtain ar hexagons, which can be proven to be a sub-optimal solution, as a tiling where Z ad to a lower energy. Again, this isFmost fact that our
(µ1 , . .certainly . µN ) = due j x,toRthe µ1 (f1 ), . . . , RµN (fN ) dx. D optimization problems eat deal of local minima, which are difficult to avoid in culty when k ≥ 2 is that the k-th eigenvalue of an optimal cell is expected to 1 hence and may not be differentiable. 29 and algorithm are not restricted to the two–dimensional case, we ported our ere unable to obtain any meaningful results. We believe that the convergence 11 7 Further problems 7.1 Cheeger-type problems We consider here minimization problems that we may call of Cheeger type, for the similarity with the well-known Cheeger problem min n Per(Ω) |Ω| o : Ω⊂D . It is possible to prove that for every bounded domain D there exists an optimal Cheeger set Ωopt and (see [4, 45]) this set is unique and convex whenever D is convex. Moreover, in this case the boundary ∂Ωopt does not contain the points of ∂D with too large mean curvature; more precisely, ∂Ωopt coincides with ∂D if and only if kHkL∞ (∂D) ≤ λ(D) d−1 where H(x) is the mean curvature of ∂D at x and λ(D) is the minimal value of the Cheeger minimization problem above. Note that the two quantities Per(Ω) and |Ω| scale in a different way and, due to the isoperimetric inequality 1/d Per(Ω)|Ω|(1−d)/d ≥ Per(B)|B|(1−d)/d = dωd (where ωd denotes the Lebesgue measure of the unit ball in Rd ), we have Per(Ω) = +∞. |Ω| |Ω|→0 lim The same analysis occurs for the cost |Ω|−α Per(Ω) whenever α > d−1 . d In this section we consider rescaled minimization problems of the form  min M (Ω)J(Ω) : Ω ∈ A(D) (7.1) where the mappings M and J fulfill some rather general assumptions related to the variational γ-convergence on the family A(D) of quasi-open subsets of D. These problems have been studied in [42] to which we refer for all the details. On the mappings M and J, from A(D) into [0, +∞] we assume 1. M and J are nonnegative, and J(D) > 0; 2. J is γ-l.s.c. and decreasing with respect to the set inclusion; 3. M is wγ-lower semicontinuous; 4. the Cheeger scaling condition is verified: lim M (Ω)→0 M (Ω)J(Ω) = +∞. With this last assumption we may define the cost M (Ω)J(Ω) = +∞ whenever M (Ω) = 0 and obtain the following existence result. 30 Theorem 7.1. Under the assumptions above the minimum problem (7.1) admits a solution Ωopt and M (Ωopt ) > 0. Proof. If (Ωn ) is a minimizing sequence, by the compactness of the wγ-convergence (see Section 4.2) we may assume that Ωn tend in wγ to some Ω ∈ A(D), with M (Ω) > 0 by the assumptions above. By the properties of wγ-convergence and by Proposition 4.11 the functional J is wγ-lower semicontinuous, as well as the product M J, which allows to conclude that Ω is an optimal domain for the minimum problem (7.1). Example 7.2. The following cases fulfill the assumptions of Theorem 7.1.
• Let J(Ω) = Φ λ(Ω) where λ(Ω) denotes the spectrum of the Dirichlet Laplacian in Ω and Φ is a lower semicontinuous function increasing in each variable and positively p-homogeneous. Taking M (Ω) = |Ω|α we have that the assumptions of Theorem 7.1 are fulfilled whenever α < 2p/d, so that in this case the minimization problem 
min |Ω|α Φ λ(Ω) : Ω ∈ A(D) admits a solution. • The case J(Ω) as above with M (Ω) = | Per(Ω)|α does not fall into the framework of Theorem 7.1 because the shape functional Per(Ω) is not wγ-lower semicontinuous (see Section 4.3). However, using the arguments of Section 4.3, we obtain that the minimization problem 
min | Per(Ω)|α Φ λ(Ω) : Ω ∈ A(D) admits a solution whenever α < 2p/(d − 1). • Consider the integral functional Z a(x)|uΩ (x)|p dx I(Ω) = D where uΩ is the solution of the Dirichlet problem in Ω −∆u = f in Ω, u ∈ H01 (Ω), extended by zero on D \ Ω. Assume that a ∈ Lq is nonnegative, with q ≥ p/(p − 1), f ∈ L2 is nonnegative, and 0 < p < 2d/(d − 2). Then the shape functional J(Ω) = 1/I(Ω) fulfills the assumptions of Theorem 7.1 and, taking M (Ω) = |Ω|α we have that the minimization problem n |Ω|α o min : Ω ∈ A(D) I(Ω) admits a solution whenever α < (2p + d)/d. A similar analysis holds for the shape functional Z J(Ω) = a(x)|uΩ (x)|−p dx. D 31 • The case of the minimization problem n | Per(Ω)|α o min : Ω ∈ A(D) I(Ω) where I(Ω) is as above, does not fall into the framework of Theorem 7.1 but the arguments of Section 4.3 apply, and we can conclude that an optimal domain exists, provided that α < (2p + d)/(d − 1). As it happens in the Cheeger problem, due to the scaling condition we assumed, the optimal domains Ωopt for the examples above have to touch the boundary ∂D. Here below we describe some necessary conditions of optimality, obtained in [42] for the minimization problem o −1 n Z α : Ω ∈ A(D) uΩ dx min |Ω| D where −∆uΩ = 1 in Ω, with Dirichlet boundary conditions. Theorem 7.3. Assuming that Ωopt is smooth, and that α < (2 + d)/d, the following necessary conditions of optimality hold:  Z 1 |∇uΩopt |2 = αK on ∂Ωopt ∩ D where K = uΩ dx. |∇uΩopt |2 ≥ αK on ∂Ωopt ∩ ∂D |Ωopt | D opt Remark 7.4. Assume that ∂D has isolated conical points, but is smooth otherwise. The gradient of the solution uΩopt then vanishes in the conical points and a pointwise decay estimate for the gradient holds. Theorem 7.3 then shows, that the optimal domain Ωopt cannot fill the entire set D. In a similar way we can obtain necessary conditions of optimality for the minimum problem o  min |Ω|α λk (Ω) : Ω ∈ A(D) where α < 2/d. If we denote by u a normalized k-th eigenfunction of Ωopt we obtain (see [42]):  λk (Ωopt ) |∇u|2 = αH on ∂Ωopt ∩ D; where H = . 2 |∇u| ≥ αH on ∂Ωopt ∩ ∂D. |Ωopt | We conclude this section by pointing out some shape optimization problems for which the existence of a solution is still unavailable. For a fixed k ≥ 1 consider the minimization problem Z  α uΩ dx : Ω ∈ A(D) min λk (Ω) D with α > 1 + d/2. By the results of [72, 73] we have Z Z 1+d/2 1+d/2 λ1 (Ω) uΩ dx ≥ λ1 (B) uB dx D D for any ball B, so that the scaling condition lim M (Ω)→0 M (Ω)J(Ω) = +∞ is fulfilled whenever α > 1 + d/2, taking J(Ω) = λαk (Ω) Z and M (Ω) = uΩ dx. D However, the wγ-lower semicontinuity condition fails for M (Ω), as it can be easily seen by a sequence of finely perforated domains like the Cioranescu and Murat ones (see Section 2.4), and so the existence Theorem 7.1 cannot be applied. It would be interesting to prove (or disprove) that an optimal domain Ωopt for the problem above exists. 32 7.2 Spectral flows In [56] De Giorgi introduced a very general theory to study evolution problems with an underlying variational structure. The theory was called minimizing movement theory and its framework is so flexible to be applied both to quasistatic evolutions as well as to gradient flows, under rather mild assumptions. Here we limit ourselves to recall the part of the scheme which is interesting for our purposes, referring to [8] for further details. Let (X, d) be a complete metric space, let u0 ∈ X be an initial condition, and let F : X →] − ∞, +∞] be a functional defined on X. For every fixed ε > 0 the implicit Euler scheme of time step ε and initial condition u0 consists in constructing a function uε (t) = w([t/ε]), where [·] stands for the integer part function, in the following way: w(0) = u0 , n d2 (v, w(n)) o w(n + 1) ∈ argmin F (v) + . 2ε Definition 7.5. If σ is another topology on X, we say that u : [0, T ] → X is a generalized minimizing movement (or a variational flow) associated to F and σ, with initial condition u0 , and we write u ∈ GM M (F, σ, u0 ), if there exist a sequence εn → 0 such that uεn (t) → u(t) in σ ∀t ∈ [0, T ]. The simplest situation occurs when σ is the topology of the metric d; in this case the following existence result for variational flows holds (see Proposition 2.2.3 of [8]). Theorem 7.6. If F is d-lower semicontinuous and d-coercive, in the sense that its sublevels {F ≤ k} are d-compact in X, then for every initial condition u
0 ∈ X there exists u ∈ GM M (F, d, u0 ). Moreover, u belongs to the space AC 2 (0, T ); X of mappings such that, for a suitable m ∈ L2 (0, T ) Z t
d u(s), u(t) ≤ m(r) dr ∀s, t ∈ [0, T ]. s The result above applies straightforward, thanks to Proposition 2.10 (i), to the case X = M0 (D) endowed with the compact distance dγ . By Theorem 7.6 for every initial condition
µ0 ∈ M0 (D) there exists a GM M (F, dγ , µ0 ) flow µ(t) which is of class AC 2 (0, T ); M0 (D) . There is a natural one-to-one map between the class of capacitary measures M0 (D) and the closed convex set of L2 (D) K = {w ∈ H01 (D) : w ≥ 0, 1 + ∆w ≥ 0}, given by 1 + ∆w . w Moreover, the metric structures on M0 (D) and K are the same, since µ 7→ Rµ (1) := wµ , with inverse w 7→ µw = dγ (µ1 , µ2 ) = kwµ1 − wµ2 kL2 (D) . Therefore, every functional F : M0 (D) → R can be identified with a functional J : K → R by F (µ) = J(wµ ) or equivalently J(w) = F (µw ). The variational flow for F in M0 (D) can be then obtained through the variational flow for J in the metric space K endowed with the L2 (D) distance, generated by the implicit Euler scheme Z n o 1 n+1 wε ∈ argmin J(w) + |w − wεn |2 dx . (7.2) 2ε D 33 For instance, we may consider a functional J of integral type, of the form Z J(w) = j(x, w(x)) dx, D where j : D × R → R is a suitable integrand. In particular, we may take j(x, w) = −w which leads to the energy of the system for the constant force f ≡ 1, or j(x, w) = w which gives the torsional rigidity. If j(x, s) is convex and lower semicontinuous in the second variable the variational flow for capacitary measures is reduced to the gradient flow of a convex lower semicontinuous map on L2 (D). An interesting question is the following: if we start from an initial condition which is a quasi-open set, i.e. µ0 = ∞D\Ω0 , in which cases the flow remains in the family of quasiopen sets? It can be shown that in general this does not happen, at least at the discrete level. This kind of phenomenon was numerically observed in the framework of quasi-static debonding membranes [30]. We may consider, instead of capacitary measures, flows of shapes by endowing the class A(D) of quasi-open subsets of D with a suitable distance. There are no “standard” distances on A(D) and several choices can be made. A first possibility is to consider the Lebesgue measure of the symmetric difference set dchar (Ω1 , Ω2 ) = |Ω1 ∆Ω2 |. Since two quasi-open sets may differ for a negligible set (as for instance in R2 a disk and a disk minus a segment), this is not a proper metric in A(D), so that one should consider equivalence classes in the family of shapes. The distance dchar is not compact on A(D); nevertheless, for γ-lower semicontinuous functionals which are monotone decreasing for the set inclusion, we have the following result. Theorem 7.7. Let F : A(D) → R be a γ-lower semicontinuous functional monotone decreasing for the set inclusion, and let Ω0 ∈ A(D). There exists a GMM map t 7→ Ω(t) associated to F and to the distance dchar , with initial condition Ω0 . Moreover, the flow Ω(t) is increasing for the set inclusion. For instance, spectral functionals of the form
F (Ω) = Φ λ(Ω) , where λ(Ω) is the spectrum of the Dirichlet Laplacian in Ω and Φ : Rk → R is increasing in each variable and lower semicontinuous, fulfill the assumptions above. Several questions about the flow Ω(t) arise. • The flow Ω(t) is not obtained through Theorem 7.6 so the functional F could be discontinuous on the curve Ω(t); it would be interesting to analyze this issue and to identify some cases where this continuity occurs. • In R2 , assume that Ω0 is simply connected. Prove or disprove that the GMM associated to λ1 in the framework of Theorem 7.7 consists only on simply connected open sets. • Assume that Ω0 is convex. Prove or disprove that a GMM associated to λ1 in the framework of Theorem 7.7 consists only on convex sets. 34 • Assume that Ω0 is convex. Is it true that any minimizer of min λ1 (Ω) + |Ω|, Ω0 ⊂Ω is convex? • Is it true that the GMM associated to λ1 in the framework of Theorem 7.7 converges, up to a suitable rescaling, to a ball? • Prove or disprove that the metric derivative of λ1 , computed at a bounded smooth set Ω is given by (u1 denotes the normalized first eigenfunction) |λ01 (Ω)| = lim sup |Ωn \Ω|→0, Ω⊂Ωn λ1 (Ω) − λ1 (Ωn ) ∂u1 2 . = max ∂Ω |Ωn \ Ω| ∂n Other possibilities for defining spectral flows are possible, as for instance the frameworks below. ◦ Work in the class of sets with prescribed measure {Ω ∈ A(D) : |Ω| = m}. ◦ Introduce a penalization on the perimeter, that is work with the functional F (Ω) + Per(Ω). ◦ Work with the Hausdorff complementary distance dH c in the family of open subsets of D: dH c (Ω1 , Ω2 ) = max |d(x, D \ Ω1 ) − d(x, D \ Ω2 )|. x∈D ◦ Work - The - The - The 7.3 in the class of convex open subsets Ω metrized by one of the distances: Hausdorff distance; L1 distance dchar of the characteristic functions; L2 distance of the oriented distance functions. Other kinds of boundary conditions All the results presented in the previous sections are concerned with the framework of Dirichlet boundary conditions. The main reason why this framework allows to treat the various questions we have seen, is the monotonicity relation f ≥ 0, Ω1 ⊂ Ω2 ⇒ RΩ1 (f ) ≤ RΩ2 (f ) and λ(Ω2 ) ≤ λ(Ω1 ). When working with different boundary conditions, the monotonicity properties above are no more true, which creates several difficulties if one wants to obtain similar kinds of results. In addition, while a function u ∈ H01 (Ω) is easily extended to H01 (D) by putting u = 0 outside Ω, similar extension operators do not exist if the Dirichlet boundary condition is replaced by Neumann or Robin type conditions. Finally, in order to work with solutions of elliptic PDEs with Neumann or Robin boundary conditions, some mild regularity on the domains Ω has to be required, which could be lost when passing to the limit on sequences (Ωn ). Nevertheless, some shape optimization results are known in these more difficult frameworks; we mention few of them together with some open problems, referring for instance to [67] for more details. • Neumann boundary conditions. If we denote by µ(Ω) the spectrum of the NeumannLaplacian on the domain Ω (notice that µ1 (Ω) = 0 for all Ω), the minimization problem 35 for µk (Ω) under the volume constraint |Ω| = m has a trivial solution. Indeed, we have µk (Ω) = 0 for any domain Ω which has at least k connected components. If we impose that the domains Ω must be convex and with a given diameter, then the infimum is not zero, but it is not achieved, as shown in [79]. A more interesting problem is the maximization of µk (Ω) under a volume constraint. It is known (see [84, 86]) that for µ2 (Ω) the maximum is attained for Ω a ball, which are the only maximizers. An analogous result for k ≥ 3 (i.e. existence of optimal domains and possibly their characterization) is not known. Similarly, the existence of a maximizing domain for µ2 (Ω), with the constraint |Ω| = m, is not known if we impose the constraint Ω ⊂ D with D narrow enough to not contain a ball of measure m. Some more issues on spectral optimization problems for the Neumann-Laplacian, together with related numerical computations, can be found in [15]. One could also consider mixed boundary conditions: Dirichlet on one part and Neumann on the remaining part. We mention the paper [49] where this problem has been studied, and [9] where the behaviour of the spectrum of the Neumann Laplacian under boundary perturbations is analyzed. Finally, some results are available for minimization problems of functions of reciprocals of µk under volume constraint (see [67]) as: X 1 X 1 1 1 . + , , µ2 (Ω) µ3 (Ω) µk (Ω) µ2k (Ω) k≥2 k≥2 • Robin boundary conditions. If Ω is a sufficiently regular open subset of Rd the Robin eigenvalue problem for the Laplace operator is given by ∂u + βu = 0 on ∂Ω, ∂n where β > 0 is a given real number. It is known (see for instance [55, 31, 17, 18]) that, denoting by η1 (Ω) the smallest Robin eigenvalue, the minimum of η1 (Ω) among the class of bounded Lipschitz sets Ω with prescribed volume is achieved on balls, which are also the unique minimizers. However, the problem of setting spectral optimization problems with Robin boundary conditions for general nonsmooth domains, faces some difficulties dues to the presence of boundary terms. In [32] a new approach has been proposed, which, using the features of SBV functions, introduced by De Giorgi for treating free discontinuity problems (see [7]), allows to define η1 (Ω) also for nonsmooth domains. Denoting by SBV 1/2 (Rd ) the space  SBV 1/2 (Rd ) = u : Rd → [0, +∞[ measurable and u2 ∈ SBV (Rd ) −∆u = ηu in Ω, we may define η1 (Ω) for a general open set Ω through the Rayleigh type formula Z Z nZ o  + 2  η1 (Ω) = min |∇u|2 dx + β (u ) + (u− )2 dHd−1 : u2 dx = 1 Rd Rd Ju where the minimum above is taken over the class of functions u ∈ SBV 1/2 (Rd ) with | spt u \ Ω| = Hd−1 (Ju \ ∂Ω) = 0, and ∇u is the approximate gradient of u, Ju is the set of discontinuity points of u, u± are the traces of u on Ju from the two sides. In this way in [32] it is proved that for every open set Ω the isoperimetric inequality η1 (Ω) ≥ η1 (B) holds, where B is a ball with |B| = |Ω|. Moreover, equality holds if and only if Ω is a ball up to a negligible set. However, all the questions of general spectral optimization problems still remain open, as: 36 - existence results under the constraint Ω ⊂ D;
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