9 Spectral optimization problems for Schrödinger operators

where Φ is a given function. The cases when D is unbounded or V takes on negative values may provide in general a continuous spectrum and are more delicate to treat; some examples in this framework are considered in [171] and in the references therein. The largest framework in which Schrödinger operators can be considered is the one where the potentials are capacitary measures; these ones are nonnegative Borel measures on D, possibly taking on the value +∞ and vanishing on all sets of capacity zero (we refer to Section 2.2 for the de nition of capacity). This framework will be considered in Section 9.1 together with the related optimization problems. We want to stress here that the class of capacitary measures μ is very large and contains both the case of standard potentials V(x), in which μ = V dx, as well as the case of classical domains Ω, in which μ = +∞D\Ω. By this notation, we intend to reference themeasure de ned in (9.3). Optimization problems for domains, usually called shape optimization problems, are often considered in the literature; the other chapters in the present volume deal with this kind of problem and in particular with spectral optimization problems, in

In this chapter we consider Schrödinger operators of the form −∆+V(x) on the Sobolev space H (D), where D is an open subset of R d . We are interested in nding optimal potentials for some suitable criteria; the optimization problems we deal with are then written as where F is a suitable cost functional and V is a suitable class of admissible potentials. For simplicity, we consider the case when D is bounded and V ≥ ; under these conditions the resolvent operator of −∆ + V(x) is compact and the spectrum λ(V) of the Schrödinger operator is discrete and consists of an increasing sequence of positive eigenvalues This allows us to consider as cost functions the so-called spectral functionals, of the form where Φ is a given function. The cases when D is unbounded or V takes on negative values may provide in general a continuous spectrum and are more delicate to treat; some examples in this framework are considered in [171] and in the references therein. The largest framework in which Schrödinger operators can be considered is the one where the potentials are capacitary measures; these ones are nonnegative Borel measures on D, possibly taking on the value +∞ and vanishing on all sets of capacity zero (we refer to Section 2.2 for the de nition of capacity). This framework will be considered in Section 9.1 together with the related optimization problems. We want to stress here that the class of capacitary measures µ is very large and contains both the case of standard potentials V(x), in which µ = V dx, as well as the case of classical domains Ω, in which µ = +∞ D\Ω . By this notation, we intend to reference the measure de ned in (9.3).
Optimization problems for domains, usually called shape optimization problems, are often considered in the literature; the other chapters in the present volume deal with this kind of problem and in particular with spectral optimization problems, in which the cost functional depends on the spectrum of the Laplace operator −∆ on H (Ω): being Ω a domain which varies in the admissible class. For further details on shape optimization problems we refer the reader to the other chapters of this book and to [207], [505], [510]; here we simply recall some key facts. The existence of optimal domains for a problem of the form min F(Ω) : Ω ⊂ D, |Ω| ≤ m (9.1) has been obtained under some additional assumptions, that we resume below.
-On the admissible domains Ω, some additional geometrical constraints are imposed, including convexity, uniform Lipschitz condition, uniform exterior cone properties, capacitary conditions, Wiener properties, . . . ; a detailed analysis of these conditions can be found in the book [207]. -No geometrical conditions are required on the admissible domains Ω but the functional F is assumed to satisfy some monotonicity conditions; in particular it is supposed to be decreasing with respect to set inclusion. The rst result in this direction has been obtained in [238] and several generalizations, mainly to the cases where the set D is not bounded, have been made in [206] and in [700].
Without the extra assumptions above, the existence of an optimal shape may fail, in general, as several counterexamples show (see for instance [207]); in these cases the minimizing sequences (Ωn) for the problem (9.1) converge in the γ-convergence sense (see De nition 9.1) to capacitary measures µ. In Section 9.1 we will see that many problems admit a capacitary measure as an optimal solution; this class is very large and only mild assumptions on the cost functional are required to provide the existence of a solution. In Section 9.2 we restrict our attention to the subclass of Schrödinger potentials V(x) that belong to some space L p (D); we call them integrable potentials and we will see that suitable assumptions on the cost functional still imply the existence of an optimal potential. Finally, in Section 9.3 we consider the case of con ning potentials V(x) that are very large out of a bounded set, or more generally ful ll some integral inequalities of the formˆD ψ V(x) dx ≤ for some suitable integrand ψ. The key ingredient we need is the notion of γ-convergence. For a given measure µ ∈ Mcap(D) we consider the Schrödinger-like operator −∆ + µ de ned on H (D) and its resolvent operator Rµ which associates to every f ∈ L (D) the unique solution u =

. Existence results for capacitary measures
The PDE above has to be de ned in the weak sense In the de nition above one can equivalently require that the resolvent operators Rµ n converge to the resolvent operator Rµ in the norm of the space of operators L L (D); L (D) . We summarize here below the main properties of the class Mcap(D); we refer for the details to [207].
-Every domain Ω can be seen as a capacitary measure, by taking µ = ∞ D\Ω , or more precisely -Every capacitary measure is the γ-limit of a suitable sequence (Ωn) of (smooth) domains; in other words, the class Mcap(D) is the closure with respect to the γconvergence, of the class of (smooth) domains D. -For every sequence (µn) of capacitary measures there exists a subsequence (µn k ) which γ-converges to a capacitary measure µ; in other words the class Mcap(D) is compact with respect to the γ-convergence. -If µ is a capacitary measure, we may consider the PDE formally written as The meaning of the equation above, as speci ed in (9.2), is in a weak sense, by considering the Hilbert space H µ (D) = H (D) ∩ L µ (D) with the norm u H µ (D) = u H (D) + u L µ (D) and de ning the solution in the weak sense (9.2). By Lax-Milgram theory, for every µ ∈ Mcap(D) and f ∈ L (D) (actually it would be enough to have f in the dual space of H µ (D)) there exists a unique solution u µ,f of the PDE above. Moreover, if µn → µ in the γ-convergence, we have u µn ,f → u µ,f weakly in in H (D), hence strongly in L (D). -In order to have the γ-convergence of µn to µ it is enough to have the weak convergence in H (D) of Rµ n ( ) to Rµ( ); in other words, we need to test the convergence of solutions of the PDEs related to the operators −∆ + µn only with f = . -The space Mcap(D), endowed with the γ-convergence, is metrizable; more precisely, the γ-convergence on Mcap(D) is equivalent to the distance where wµ and wν are the solutions of the problems Remark 9.2. We notice that the de nition of γ-convergence of a sequence of capacitary measures µn to µ can be equivalently expressed in terms of the Γ-convergence in L (D) of the corresponding energy functionals Jn(u) =ˆD |∇u| dx +ˆD u dµn to the limit energy For all details about Γ-convergence theory we refer to [313].
The γ-convergence is very strong, and so many functionals are γ-lower semicontinuous, or even continuous (see below some important examples). The classes of functionals we are interested in are the following.
Integral functionals. Given a function f ∈ L (D), for every µ ∈ Mcap(D) we consider the solution u µ,f = Rµ(f ) to the elliptic PDE (9.4). The integral cost functionals we consider are of the form where j(x, s, z) is a suitable integrand that we assume measurable in the x variable, lower semicontinuous in the s, z variables, and convex in the z variable. Moreover, the function j is assumed to ful ll bounds from below of the form with a ∈ L (D) and c smaller than the rst Dirichlet eigenvalue of the Laplace operator −∆ in D. In particular, the energy E f (µ) de ned by (9.6) belongs to this class since, integrating its Euler-Lagrange equation by parts, we have which corresponds to the integral functional above with Thanks to the assumptions above and to the strong-weak lower semicontinuity theorem for integral functionals (see for instance [235]) all functionals of the form (9.5) are γ-lower semicontinuous on Mcap(D).
Spectral functionals. For every capacitary measure µ ∈ Mcap(D) we consider the spectrum λ(µ) of the Schrödinger operator −∆+µ on H (D)∩L µ (D). Since D is bounded (it is enough to consider D to be of nite measure), then the operator −∆ + µ has a compact resolvent and so its spectrum λ(µ) is discrete: where λ k (µ) are the eigenvalues of −∆ + µ, counted with their multiplicity. The same occurs if D is unbounded, and the measure µ satis es some suitable con nement integrability properties (see for instance [208]). The spectral cost functionals we may allow are of the form for suitable functions Φ : R N → (−∞, +∞]. For instance, taking Φ(λ) = λ k we obtain Since a sequence (µn) γ-converges to µ if and only if the sequence of resolvent operators (Rµ n ) converges in the operator norm convergence of linear operators on L (D) to the resolvent operator Rµ, the spectrum λ(µ) is continuous with respect to the γconvergence, that is Therefore, the spectral functionals above are γ-lower semicontinuous, provided that the function Φ is lower semicontinuous, in the sense that where λn → λ in R N is intended in the componentwise convergence. The relation between γ-convergence and weak*-convergence of measures is given in the proposition below. Proof. It is enough to show that µ(K) ≤ ν(K) whenever K is a compact subset of D. Let u be a nonnegative smooth function with compact support in D such that u ≤ in D and u = on K; we have Since u is arbitrary, the conclusion follows from the de nition of Borel regularity of the measure ν.
Remark 9.4. When d = , as a consequence of the compact embedding of H (D) into the space of continuous functions on D, we obtain that any sequence (µn) weakly* converging to µ is also γ-converging to µ.
In several shape optimization problems the class of admissible domains Ω is slightly larger than the class of open sets.

for a suitable function u ∈ H (D). Since Sobolev functions are de ned only up to sets of capacity zero, a quasi-open set is de ned up to capacity zero sets too.
In many problems the admissible domains Ω are constrained to verify a measure constraint of the form |Ω| ≤ m; in order to relax this constraint to capacitary measures we have to introduce, for every µ ∈ Mcap(D), the set of niteness Ωµ. A precise denition would require the notion of ne topology and nely open sets (see for instance [207]); however, a simpler equivalent de nition can be given in terms of the solution wµ = Rµ( ) of the elliptic PDE −∆u + µu = , u ∈ H µ (D).
By de nition, the set Ωµ is quasi-open, being the set where a Sobolev function is positive. Of course, since the function wµ is de ned only up to sets of capacity zero, the set Ωµ is de ned up to sets of capacity zero too.
Proof. This follows from the de nition of Ωµ and from the fact that the γ-convergence µn →γ µ is equivalent to the convergence of the solutions wµ n = Rµ n ( ) to wµ = Rµ( ) in L (D). The conclusion then follows by the Fatou's lemma.
In summary, thanks to the γ-compactness of the class Mcap(D), the following general existence result holds.
Theorem 9.9. Let F : Mcap(D) → R be a γ-lower semicontinuous functional (for instance one of the classes above); then the minimization problem In general, the optimal measure µ opt is not unique; however, in the situation described below, the uniqueness occurs. Consider the optimization problem for the integral functional where f ≥ is a given function in L (D). We can write the problem as a double minimization, in µ and in u: Since f ≥ , by the maximum principle we know that u ≥ and, at least formally (the rigorous justi cation can be found in [269]), so that we can eliminate the variable µ from the minimization and the optimization problem can be reformulated in terms of the function u only, as where K is the subset of H (D) given by The inequality f + ∆u ≥ has to be formulated in a weak sense, aŝ The set K is clearly convex and it is easy to see that it is also closed. Hence, as a consequence, if the function j(x, s, z) is strictly convex with respect to the pair (s, z), the solution of (9.7) is unique. Thus the solution µ opt , that exists thanks to Theorem 9.9 is also unique. Note that in this case, no measure constraint of the form |Ωµ| ≤ m is imposed.
In several situations the optimal measure µ opt given by Theorem 9.9 has more regularity or summability properties than a general element of Mcap(D).This happens in the cases below: -If the functional F is monotonically increasing with respect to the usual order of measures, and a constraint |Ωµ| ≤ m is added, then an optimal measure µ opt that is actually a domain exists, that is µ opt = ∞ D\Ω for some quasi-open subset Ω of D. This fact should be rigorously justi ed (see [238]), but the argument consists in the fact that the measure ∞ D\Ω is smaller than µ and has the same set of niteness; then it provides an optimum for the minimization problem due to the monotonicity of F and to the constraint on the measure of the set of niteness.
-In [241] the optimization of the elastic compliance for a membrane is considered, with the additional constraint that the measure µ has a prescribed total mass. In this case it is shown that µ opt is actually an L (D) function, that is no singular parts with respect to the Lebesgue measure occur.
In general, we should not expect that µ opt is a domain or a function with any summability; the following example shows that even in simple and natural problems this does not occur.
Example 9.10. Let D be a ball of radius R and let f = ; consider the optimization problem for the integral functional where c is a given constant and u µ, denotes as before the solution of the PDE By the argument described above the problem can be reformulated in terms of the function u only, as where K is the convex closed subset of H (D) given by As we have seen, this auxiliary problem has a unique solution which is radially symmetric. Thus we can write the problem in polar coordinates as The minimum problem above can be fully analyzed and its solution is characterized as follows (see [207] for the details).
-If c is large enough, above a certain thresholdc that can be computed explicitly, we have for the optimal solution (u, µ) -Below the thresholdc the optimal measure µ is given by where L d denotes the Lebesgue measure in R d , αc > is a suitable constant, and Rc < R is a suitable radius. The solution u is computed correspondingly, through the equation A plot of the behavior of an optimal state function u is given in Figure 9.1. Note that the functional in (9.8) is not monotonically increasing with respect to µ.

. Existence results for integrable potentials
In this section we consider optimization problems of the form where p > and F(V) is a cost functional acting on Schrödinger potentials, or more generally on capacitary measures. We assume that F is γ-lower semicontinuous, an assumption that, as we have seen in the previous section, is very mild and veri ed for most of the functionals of integral or spectral type. When p > a general existence result follows from the following proposition, where we show that the weak L (D) convergence (that is the one having L ∞ (D) as the space of test functions) of potentials implies the γ-convergence.
Proposition 9.11. Let Vn ∈ L (D) converge weakly in L (D) to a function V. Then the capacitary measures Vn dx γ-converge to V dx.
Proof. We have to prove that the solutions un = R Vn ( ) of the PDE Equivalently, as noticed in Remark 9.2, we may prove that the functionals Let us prove the Γ-liminf inequality: by the strong-weak lower semicontinuity theorem for integral functionals (see for instance [235]).
Let us now prove the Γ-limsup inequality: there exists un → u in L (D) such that then, by the weak L (D) convergence of Vn to V, for every t xed we have Moreover, letting t → ∞ we have by the monotone convergence theorem Then, by a diagonal argument, we can nd a sequence tn → +∞ such that Taking now un = u tn , and noticing that for every t > ˆD |∇u t | dx ≤ˆD |∇u| dx, we obtain (9.10) and so the proof is complete.
The existence of an optimal potential for problems of the form (9.9) is now straightforward.
Theorem 9.12. Let F(V) be a functional de ned for V ∈ L + (D) the set of nonnegative functions in L (D) , lower semicontinuous with respect to the γ-convergence, and let V be a subset of L + (D), compact for the weak L -convergence. Then the problem admits a solution.
Proof. Let (Vn) be a minimizing sequence in V. By the compactness assumption on V, we may assume that Vn tends to some V ∈ V weakly in L (D). By Proposition 9.11, we have that Vn γ-converges to V and so, by the semicontinuity of F, which gives the conclusion.
In some cases the optimal potential can be explicitly determined through the solution of a partial di erential equation, as for instance in the examples below.
Example 9.13. Take F = −E f , where E f is the energy functional de ned in (9.6), with f a xed function in L (D), and Then, the problem we are dealing with is As we have already seen above, the energy functional can be written, by an integration by parts, as where R V is the resolvent operator of −∆ + V(x). Therefore, the functional F is γcontinuous and the existence Theorem 9.12 applies. In order to compute the optimal potential, interchanging the min and the max in (9.12) we obtain the inequality The maximization with respect to V is very easy to compute; in fact, for a xed u, the maximal value is reached at In order to nd the optimal potential V opt we have then to solve the auxiliary variational problem and then, by means of its solutionū, recovering V opt from (9.13). The auxiliary variational problem above can be written, via its Euler-Lagrange equation, as the nonlinear PDE with the constant C(p,ū) given by The fact that V opt actually solves our optimization problem (9.12) follows from the fact thatū = R Vopt (f ), hence we have We notice that, replacing −E f by E f transforms the maximization problem in (9.12) into the minimization of E f on V, which has the only trivial solution V ≡ .
Example 9.14. More generally, we may consider the optimization problem ˆD |∇u| + Vu dx :ˆD u dx = . (9.14) We are then dealing with the optimization problem where the constraint V is as in (9.11). Arguing as before, we interchange the max and the min above and we end up with the auxiliary problem In the same way as before, the optimal potential V opt can be recovered through the solutionū of the auxiliary problem above, by taking Remark 9.16. In the case p < problem (9.12) with the admissible class (9.11) does not admit any solution. Indeed, for a xed real number α > , take Vn(x) = nχ Ωn (x), where χ E denotes the characteristic function of the set E (with value on E and outside E) and Ωn ⊂ D are such that the sequence (Vn) converges weakly in L (D) to the constant function α. In particular, we have n|Ωn| → α as n → ∞ and so, since p < , we havê D V p n dx = n p |Ωn| → as n → ∞.
Therefore, for n large enough, the potentials Vn belong to the admissible class V. By Proposition 9.11 we have E f (Vn) → E f (α) and, since α was arbitrary, we obtain The limit on the right-hand side above is zero; on the other hand we have E f (V) ≤ for any V. Thus, if a maximal potential V opt exists, it should verify E f (V opt ) = which is impossible.
It remains to consider the maximization problem (9.12) when p = . In this case the result of Proposition 9.11 cannot be applied because the unit ball of L (D) is not weakly compact. However, the existence of an optimal potential still holds, as we show below. It is convenient to introduce the functionals Proposition 9 If v L ∞ = +∞, then setting ω k = {v > k}, for any k ≥ , and arguing as above, we obtain (9.16). Now, let un → u in L (D). Then, by the semicontinuity of the L norm of the gradient, by (9.16), and by the continuity of the term´D uf dx, we have J (u) ≤ lim inf n→∞ Jp n (un), for any decreasing sequence pn → . On the other hand, for any u ∈ L (D), we have Jp n (u) → J (u) as n → ∞ and so, we have the conclusion. In order to prove (b) we use an argument similar to that of the classical elliptic regularity theorem. For h ∈ R and k = , . . . , d, we use the notation and we consider a function ϕ ∈ C ∞ c (D) such that ϕ ≡ on Ω. Then we have that for h small enough ∂ h k u satis es the following equation on the support of ϕ : Multiplying (9.19) by ϕ ∂ h k u and taking into account the inequality By a change of variables, the Cauchy-Schwartz and the Poincaré inequalities we get thatˆD (9.20) where C ϕ is a constant depending on ϕ. On the other hand we havê Thus, there is a constant C D,ϕ depending on D and ϕ such that which nally gives that and since this last ineaqulity is true for every k = , . . . , d and every h small enough we get that u ∈ H (Ω) and for an appropriate constant C Ω depending on the function ϕ associated to Ω.
Now since for t ∈ R such that |t| ≤ n φ L ∞ we have that u + tφvn ≤ M, the and taking the derivative with respect to t at t = , we get that and passing to the limit as n → ∞ we obtain (9.24). We can now obtain (9.22) by (9.23) and the fact that u = M on ω+ and u = −M on ω−. (iii) Since u is the minimizer of J , we have Taking the derivative of this di erence at ε = , we obtain On the other hand, by (9.23), we havê For any non-negative φ ∈ C ∞ c (D) we have that for t small enough u + tφ L ∞ < M. Therefore, the optimality of u gives ≤ lim Since the last inequality holds for any φ ≥ and any ε > we get that On the other hand, ∆u = almost everywhere on ω− = {u = −M}, and so we obtain that f ≤ on ω−. Arguing in the same way, and considering test functions supported on {u ≥ −M + ε}, we can prove that f ≥ on ω+.
Theorem 9.20. Let D ⊂ R d be a bounded open set, let p = , and let f ∈ L (D). Then there is a unique solution to problem (9.25) given by Proof. For any u ∈ H (D) and any V ≥ withˆD V dx ≤ we havê Thus we obtain the inequality and taking the minimum with respect to u we get which nally gives where u is the minimizer of J . By Proposition 9.19 we have that u satis es the equa- By Proposition 9.19 (iii) we have thatˆD V u dx = M and so Moreover, again by (iii) and (iv) we obtain that V ≥ andˆD V dx = , which concludes the proof.
By the results of Section 9.1 the maximization problem admits a solution µ opt which is a capacitary measure. Repeating the proof of Theorem 9.20 we obtain the auxiliary variational problem

Denoting by u its unique solution and by M the maximum of u, we obtain that the optimal capacitary measure µ opt is supported by the set {u = M}, this is contained in S (since the function u is subharmonic on D \ S) and so µ opt is singular with respect to the Lebesgue measure. Moreover µ opt has the form
The result in the following Theorem was proved in [356] (see also [505,Theorem 8.2.4]). We present it in a slightly di erent form as a simple consequence of Proposition 9.19. We recall the notation λ (V) introduced in (9.14) for the rst eigenvalue related to the potential V. (9.28) Proof. We rst notice that due to the compact inclusion H (D) ⊂ L (D) and the semicontinuity of the norm of the gradient there is a solution u λ ∈ H (D) of the problem (9.28). We now set f = λu λ . Since for every u ∈ H (D) \ { } we have that we obtain that the minimizer of the functional J corresponding to the function f is also the minimizer of the functional On the other hand, for every u ∈ H (D) we havê which proves that u λ is the minimizer of J . Thus u λ satis es the equation where V is such that Thus we have that On the other hand for every V ≥ such thatˆD V dx = we havê D |∇u| dx +ˆD u V dx ≤ˆD |∇u| dx + u L ∞ , for every u ∈ H (D), which after taking the minimum with respect to u gives which proves that V is a solution of (9.27).
In order to prove the uniqueness of the solution it is su cient to check that there is a unique solution to the problem (9.28). In fact suppose that u and u are two distinct solutions of (9.28) and denote M i = u i L ∞ , ω i = {u i = M i } and V i = λχω i , for i = , .
We consider now the potential V = V + V . Since the function V → λ (V) is the in mum of a family of linear functions we know that it is concave and so, V is also a solution of (9.27). Now since V is optimal, we have that for every A, Since the rst eigenvalue is simple and the family of operators −∆ + V + ε(χ A − χ B ) is analytic with respect to ε, we have that the functions ε → λ V + ε(χ A − χ B ) and ε → uε, where uε is the solution of are analytic. Taking the derivatives in ε at ε = we obtain Multiplying both sides by u and integrating by parts we get Since A and B are arbitrary we get that u is a (positive, by the maximum principle) constant on ω ∪ ω and since u ∈ H loc (D) we obtain that and as a consequence V = λ on ω ∪ ω which gives that ω = ω , V = V and u = u .
Remark 9.23. The proof above is constructed for the maximization of the rst eigenvalue λ (V) on the class It would be interesting to consider the analogous maximization problem for λ k (V) on the same class of potentials

. Existence results for con ning potentials
In this section we consider the potential optimization problem where the functional F is as in the sections above and the admissible class V is given by The assumptions above on the function Ψ are for instance satis ed by the following functions: -Ψ(s) = s −p , for any p > ; -Ψ(s) = e −αs , for any α > . and justify the terminology "con ning potentials" we used. Indeed, large potentials turn out to be admissible.
The result showing the existence of an optimal potential in this case is as follows.
Then the optimization problem (9.29) has a solution, where the admissible class V is given by (9.30).
Proof. Let Vn ∈ V be a minimizing sequence for problem (9.29). Then the functions vn := Ψ(Vn) /p are bounded in L p (D) and so, up to a subsequence, we may assume that vn converges weakly in L p (D) to some function v. We will prove that the potential V := Ψ − (v p ) is optimal for the problem (9.29). Since vn converges to v weakly in L p (D) we haveˆD which shows that V ∈ V. It remains to prove that By the compactness of the γ-convergence on the class Mcap(D), we can suppose that, up to a subsequence, Vn γ-converges to some capacitary measure µ ∈ Mcap(D). Since F is assumed γ-lower semicontinuous, we have We will show that F(V) ≤ F(µ), which, together with (9.31) will conclude the proof. By the de nition of γ-convergence, we have that for any u ∈ H (D), there is a sequence un ∈ H (D) which converges to u in L (D) and is such that The inequality in (9.32) is due to the L (D) lower semicontinuity of the Dirichlet integral and to the strong-weak lower semicontinuity of integral functionals (see for instance [235]), which follows by the assumption b) on the function Ψ. Thus, for any u ∈ H (D), we haveˆD u dµ ≥ˆD u V dx, which implies V ≤ µ. Since F was assumed to increase monotonically, we obtain F(V) ≤ F(µ), which concludes the proof.
Just like in the previous section, in some special cases, the solution to the optimization problem (9.29) can be computed explicitly through the solution to some auxiliary variational problem. This occurs for instance when with f ∈ L (D). In fact, by the variational formulation we can rewrite the optimization problem (9.29) for F(V) = λ (V) as The minimization with respect to V is easy to compute; in fact, if Ψ is di erentiable with Ψ′ invertible, then the minimum with respect to V in (9.33) is achieved for where Λu is a constant such that Thus, the solution to the problem on the right hand side of (9.33) is given by the solution to the auxiliary variational problem Example 9.26. Consider the case Ψ(x) = e −αx with α > . Again, the same argument we used above shows that the optimal potentials for the functionals F(V) = λ (V) and F(V) = E f (V) are given by where u is the minimizer of the auxiliary variational problems (9.35) and (9.36) respectively. We also note that, in this casê D u (Ψ′) − (Λu u ) dx = α ˆD u dxˆD log u dx −ˆD u log u dx and so the auxiliary variational problems (9.35) and (9.36) give rise to the nonlinear PDEs respectively, where the constants C (u) and C (u) are given by The function Ψ(s) = e −αs in the constraint (9.30) can be used to simulate and approximate a volume constraint in a shape optimization problem of the form We note that on the set where u ≥ Λα we necessarily have that V = . On the other hand, if u < Λα, then by the optimality of V, we have that V > . Finally, the optimal potential V can be identi ed in terms of u by By the properties of the Γ-convergence this implies the convergence of the solutions uα of (9.40) and hence, thanks to the relation (9.39), of the optimal potentials Vα for (9.37) to a limit potential of the form where u is a solution to the limit problem min ˆD |∇u| dx −ˆD fu dx + Λ|{u = }| : u ∈ H (D) .
This limit problem is indeed a shape optimization problem written in terms of the state function u; several results on the regularity of the optimal domains are known (see for instance [25], [187], [189], as well as Chapter 3 of the present book).