Optimal Potentials For Schrodinger Operators
Giuseppe Buttazzo, Augusto Gerolin, Berardo Ruffini, Bozhidar Velichkov
OOPTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS
G. BUTTAZZO, A. GEROLIN, B. RUFFINI, AND B. VELICHKOV
Abstract.
We consider the Schr¨odinger operator − ∆ + V ( x ) on H (Ω), where Ω is a givendomain of R d . Our goal is to study some optimization problems where an optimal potential V ≥ Keywords:
Schr¨odinger operators, optimal potentials, spectral optimization, capacity.
Introduction sintro
In this paper we consider the Schr¨odinger operator − ∆ + V ( x ) on H (Ω), where Ω is a givendomain of R d . Our goal is to study some optimization problems where an optimal potential V ≥ (cid:8) F ( V ) : V ∈ V (cid:9) , where F denotes the cost functional and V the admissible class. The cost functionals we aim toinclude in our framework are for instance the following. Integral functionals.
Given a right-hand side f ∈ L (Ω) we consider the solution u V of theelliptic PDE − ∆ u + V u = f in Ω , u ∈ H (Ω) . The integral cost functionals we may consider are of the form F ( V ) = (cid:90) Ω j (cid:0) x, u V ( x ) , ∇ u V ( x ) (cid:1) dx, where j is a suitable integrand that we assume convex in the gradient variable and boundedfrom below. One may take, for example, j ( x, s, z ) ≥ − a ( x ) − c | s | , with a ∈ L (Ω) and c smaller than the first Dirichlet eigenvalue of the Laplace operator − ∆ inΩ. In particular, the energy E f ( V ) defined by E f ( V ) = inf (cid:26)(cid:90) Ω (cid:16) |∇ u | + 12 V ( x ) u − f ( x ) u (cid:17) dx : u ∈ H (Ω) (cid:27) , (1.1) energy belongs to this class since, integrating by parts its Euler-Lagrange equation, we have E f ( V ) = − (cid:90) Ω f ( x ) u V dx, which corresponds to the integral functional above with j ( x, s, z ) = − f ( x ) s. a r X i v : . [ m a t h . A P ] M a y G. BUTTAZZO, A. GEROLIN, B. RUFFINI, AND B. VELICHKOV
Spectral functionals.
For every admissible potential V ≥ V )of the Schr¨odinger operator − ∆+ V ( x ) on H (Ω). If Ω is bounded or has finite measure, or if thepotential V satisfies some suitable integral properties, the operator − ∆ + V ( x ) has a compactresolvent and so its spectrum Λ( V ) is discrete:Λ( V ) = (cid:0) λ ( V ) , λ ( V ) , . . . (cid:1) , where λ k ( V ) are the eigenvalues counted with their multiplicity. The spectral cost functionalswe may consider are of the form F ( V ) = Φ (cid:0) Λ( V ) (cid:1) , for a suitable function Φ : R N → R . For instance, taking Φ(Λ) = λ k we obtain F ( V ) = λ k ( V ) . Concerning the admissible classes we deal with, we consider mainly the cases V = (cid:26) V ≥ (cid:90) Ω V p dx ≤ (cid:27) and V = (cid:26) V ≥ (cid:90) Ω V − p dx ≤ (cid:27) ;in some situations more general admissible classes V will be considered, see Theorem 3.1 andTheorem 4.1.In Section 3.1 our assumptions allow to take F ( V ) = −E f ( V ) and thus the optimizationproblem becomes the maximization of E f under the constraint (cid:82) Ω V p dx ≤
1. We prove that for p ≥
1, there exists an optimal potential for the problemmax (cid:26) E f ( V ) : (cid:90) Ω V p dx ≤ (cid:27) . (1.2)The existence result is sharp in the sense that for p < p = 1 is particularly interesting and we show that in this case the optimalpotentials are of the form V opt = fM (cid:0) χ ω + − χ ω − (cid:1) , where χ U indicates the characteristic function of the set U , f ∈ L (Ω), M = (cid:107) u V (cid:107) L ∞ (Ω) , and ω ± = { u = ± M } . In Section 4 we deal with minimization problems of the formmin (cid:8) F ( V ) : (cid:90) Ω V − p dx ≤ (cid:9) . (1.3) We prove a general result (Theorem 4.1) establishing the existence of an optimal potential undersome mild conditions on the functional F . In particular, we obtain the existence of optimalpotentials for a large class of spectral and energy functionals (see Corollary 4.3).In Section 5 we deal with the case of unbounded domains Ω. precisely, we prove that in thecase Ω = R d and F = E f or F = λ , the solutions of problem (1.3) exist and are such that 1 /V is compactly supported, provided f is compactly supported. Finally, in Section 6 we make somefurther remarks and present some open questions. PTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS 3 Capacitary measures and γ -convergence s2 For a subset E ⊂ R d its capacity is defined bycap( E ) = inf (cid:26)(cid:90) R d |∇ u | dx + (cid:90) R d u dx : u ∈ H ( R d ) , u ≥ E (cid:27) . If a property P ( x ) holds for all x ∈ Ω, except for the elements of a set E ⊂ Ω of capacity zero,we say that P ( x ) holds quasi-everywhere (shortly q.e. ) in Ω, whereas the expression almosteverywhere (shortly a.e. ) refers, as usual, to the Lebesgue measure, which we often denote by | · | . A subset A of R d is said to be quasi-open if for every ε > A ε of R d , with A ⊂ A ε , such that cap( A ε \ A ) < ε . Similarly, a function u : R d → R is said to be quasi-continuous (respectively quasi-lower semicontinuous ) if there exists a decreasing sequenceof open sets ( A n ) n such that cap( A n ) → u n of u to the set A cn is continuous(respectively lower semicontinuous). It is well known (see for instance [18]) that every function u ∈ H ( R d ) has a quasi-continuous representative (cid:101) u , which is uniquely defined up to a set ofcapacity zero, and given by (cid:101) u ( x ) = lim ε → | B ε ( x ) | (cid:90) B ε ( x ) u ( y ) dy , where B ε ( x ) denotes the ball of radius ε centered at x . We identify the (a.e.) equivalence class u ∈ H ( R d ) with the (q.e.) equivalence class of quasi-continuous representatives (cid:101) u .We denote by M + ( R d ) the set of positive Borel measures on R d (not necessarily finiteor Radon) and by M +cap ( R d ) ⊂ M + ( R d ) the set of capacitary measures , i.e. the measures µ ∈ M + ( R d ) such that µ ( E ) = 0 for any set E ⊂ R d of capacity zero. We note that when µ isa capacitary measure, the integral (cid:82) R d | u | dµ is well-defined for each u ∈ H ( R d ), i.e. if (cid:101) u and (cid:101) u are two quasi-continuous representatives of u , then (cid:82) R d | (cid:101) u | dµ = (cid:82) R d | (cid:101) u | dµ .For a subset Ω ⊂ R d , we define the Sobolev space H (Ω) as H (Ω) = (cid:110) u ∈ H ( R d ) : u = 0 q.e. on Ω c (cid:111) . (2.1)Alternatively, by using the capacitary measure I Ω defined as I Ω ( E ) = (cid:40) E \ Ω) = 0+ ∞ if cap( E \ Ω) > E ⊂ R d , (2.2) Iomega the Sobolev space H (Ω) can be defined as H (Ω) = (cid:26) u ∈ H ( R d ) : (cid:90) R d | u | dI Ω < + ∞ (cid:27) . More generally, for any capacitary measure µ ∈ M +cap ( R d ), we define the space H µ = (cid:26) u ∈ H ( R d ) : (cid:90) R d | u | dµ < + ∞ (cid:27) , which is a Hilbert space when endowed with the norm (cid:107) u (cid:107) ,µ , where (cid:107) u (cid:107) ,µ = (cid:90) R d |∇ u | dx + (cid:90) R d u dx + (cid:90) R d u dµ. If u / ∈ H µ , then we set (cid:107) u (cid:107) ,µ = + ∞ . G. BUTTAZZO, A. GEROLIN, B. RUFFINI, AND B. VELICHKOV
For Ω ⊂ R d , we define M +cap (Ω) as the space of capacitary measures µ ∈ M +cap ( R d ) suchthat µ ( E ) = + ∞ for any set E ⊂ R d such that cap( E \ Ω) >
0. For µ ∈ M +cap ( R d ), we denotewith H µ (Ω) the space H µ ∨ I Ω = H µ ∩ H (Ω). Gamma
Definition 2.1.
Given a metric space ( X, d ) and sequence of functionals J n : X → R ∪ { + ∞} ,we say that J n Γ -converges to the functional J : X → R ∪ { + ∞} , if the following two conditionsare satisfied:(a) for every sequence x n converging in to x ∈ X , we have J ( x ) ≤ lim inf n →∞ J n ( x n ); (b) for every x ∈ X , there exists a sequence x n converging to x , such that J ( x ) = lim n →∞ J n ( x n ) . For all details and properties of Γ-convergence we refer to [8]; here we simply recall that,whenever J n Γ-converges to J , min x ∈ X J ( x ) ≤ lim inf n →∞ min x ∈ X J n ( x ) . (2.3) gamma Definition 2.2.
We say that the sequence of capacitary measures µ n ∈ M +cap (Ω) , γ -convergesto the capacitary measure µ ∈ M +cap (Ω) if the sequence of functionals (cid:107) · (cid:107) ,µ n Γ -converges to thefunctional (cid:107) · (cid:107) ,µ in L (Ω) , i.e. if the following two conditions are satisfied: • for every sequence u n → u in L (Ω) we have (cid:90) R d |∇ u | dx + (cid:90) R d u dµ ≤ lim inf n →∞ (cid:26)(cid:90) R d |∇ u n | dx + (cid:90) R d u n dµ n (cid:27) ; • for every u ∈ L (Ω) , there exists u n → u in L (Ω) such that (cid:90) R d |∇ u | dx + (cid:90) R d u dµ = lim n →∞ (cid:26)(cid:90) R d |∇ u n | dx + (cid:90) R d u n dµ n (cid:27) . If µ ∈ M +cap (Ω) and f ∈ L (Ω) we define the functional J µ ( f, · ) : L (Ω) → R ∪ { + ∞} by J µ ( f, u ) = 12 (cid:90) Ω |∇ u | dx + 12 (cid:90) Ω u dµ − (cid:90) Ω f u dx. (2.4) F If Ω ⊂ R d is a bounded open set, µ ∈ M +cap (Ω) and f ∈ L (Ω), then the functional J µ ( f, · )has a unique minimizer u ∈ H µ that verifies the PDE formally written as − ∆ u + µu = f, u ∈ H µ (Ω) , (2.5) dumuf and whose precise meaning is given in the weak form (cid:90) Ω ∇ u · ∇ ϕ dx + (cid:90) Ω uϕ dµ = (cid:90) Ω f ϕ dx, ∀ ϕ ∈ H µ (Ω) ,u ∈ H µ (Ω) . The resolvent operator of − ∆ + µ , that is the map R µ that associates to every f ∈ L (Ω) thesolution u ∈ H µ (Ω) ⊂ L (Ω), is a compact linear operator in L (Ω) and so, it has a discretespectrum 0 < · · · ≤ Λ k ≤ · · · ≤ Λ ≤ Λ . Their inverses 1 / Λ k are denoted by λ k ( µ ) and are the eigenvalues of the operator − ∆ + µ . PTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS 5
In the case f = 1 the solution will be denoted by w µ and when µ = I Ω we will use thenotation w Ω instead of w I Ω . We also recall (see [2]) that if Ω is bounded, then the strong L -convergence of the minimizers w µ n to w µ is equivalent to the γ -convergence of Definition 2.2. convres Remark . An important well known characterization of the γ -convergence is the following: asequence µ n γ -converges to µ , if and only if, the sequence of resolvent operators R µ n associatedto − ∆ + µ n , converges (in the strong convergence of linear operators on L ) to the resolvent R µ of the operator − ∆ + µ . A consequence of this fact is that the spectrum of the operator − ∆ + µ n converges (pointwise) to the one of − ∆ + µ . s2r2 Remark . The space M +cap (Ω) endowed with the γ -convergence is metrizable. If Ω is bounded,one may take d γ ( µ, ν ) = (cid:107) w µ − w ν (cid:107) L . Moreover, in this case, in [10] it is proved that the space M +cap (Ω) endowed with the metric d γ is compact. wgamma Proposition 2.5.
Let Ω ⊂ R d and let V n ∈ L (Ω) be a sequence weakly converging in L (Ω) toa function V . Then the capacitary measures V n dx γ -converge to V dx .Proof.
We have to prove that the solutions u n = R V n (1) of (cid:40) − ∆ u n + V n ( x ) u n = 1 u ∈ H (Ω)weakly converge in H (Ω) to the solution u = R V (1) of (cid:40) − ∆ u + V ( x ) u = 1 u ∈ H (Ω) , or equivalently that the functionals J n ( u ) = (cid:90) Ω |∇ u | dx + (cid:90) Ω V n ( x ) u dx Γ (cid:0) L (Ω) (cid:1) -converge to the functional J ( u ) = (cid:90) Ω |∇ u | dx + (cid:90) Ω V ( x ) u dx. The Γ-liminf inequality (Definition 2.1 (a)) is immediate since, if u n → u in L (Ω), we have (cid:90) Ω |∇ u | dx ≤ lim inf n →∞ (cid:90) Ω |∇ u n | dx by the lower semicontinuity of the H (Ω) norm with respect to the L (Ω)-convergence, and (cid:90) Ω V ( x ) u dx ≤ lim inf n →∞ (cid:90) Ω V n ( x ) u n dx by the strong-weak lower semicontinuity theorem for integral functionals (see for instance [4]).Let us now prove the Γ-limsup inequality (Definition 2.1 (b)) which consists, given u ∈ H (Ω), in constructing a sequence u n → u in L (Ω) such thatlim sup n →∞ (cid:90) Ω |∇ u n | dx + (cid:90) Ω V n ( x ) u n dx ≤ (cid:90) Ω |∇ u | dx + (cid:90) Ω V ( x ) u dx. (2.6) potgls For every t > u t = ( u ∧ t ) ∨ ( − t ); then, by the weak convergence of V n , for t fixed we havelim n →∞ (cid:90) Ω V n ( x ) | u t | dx = (cid:90) Ω V ( x ) | u t | dx, G. BUTTAZZO, A. GEROLIN, B. RUFFINI, AND B. VELICHKOV and lim t → + ∞ (cid:90) Ω V ( x ) | u t | dx = (cid:90) Ω V ( x ) | u | dx. Then, by a diagonal argument, we can find a sequence t n → + ∞ such thatlim n →∞ (cid:90) Ω V n ( x ) | u t n | dx = (cid:90) Ω V ( x ) | u | dx. Taking now u n = u t n , and noticing that for every t > (cid:90) Ω |∇ u t | dx ≤ (cid:90) Ω |∇ u | dx, we obtain (2.6) and so the proof is complete. (cid:3) In the case of weak* convergence of measures the statement of Proposition 2.5 is no longertrue, as the following proposition shows.
VgeW
Proposition 2.6.
Let Ω ⊂ R d ( d ≥ ) be a bounded open set and let V, W ∈ L (Ω) be twofunctions such that V ≥ W . Then, there is a sequence V n ∈ L (Ω) , uniformly bounded in L (Ω) ,such that the sequence of measures V n ( x ) dx converges weakly* to V ( x ) dx and γ -converges to W ( x ) dx .Proof. Without loss of generality we can suppose (cid:82) Ω ( V − W ) dx = 1. Let µ n be a sequence ofprobability measures on Ω weakly* converging to ( V − W ) dx and such that each µ n is a finitesum of Dirac masses. For each n ∈ N consider a sequence of positive functions V n,m ∈ L (Ω)such that (cid:82) Ω V n,m dx = 1 and V n,m dx converges weakly* to µ n as m → ∞ . Moreover, we choose V n,m as a convex combination of functions of the form | B /m | − χ B /m ( x j ) .We now prove that for fixed n ∈ N , ( V n,m + W ) dx γ -converges, as m → ∞ , to W dx or,equivalently, that the sequence w W + V n,m converges in L to w W , as m → ∞ . Indeed, by theweak maximum principle, we have w W + I Ω m,n ≤ w W + V n,m ≤ w W , where Ω m,n = Ω \ ∪ j B /m ( x j ) and I Ω m,n is as in (2.2).Since a point has zero capacity in R d ( d ≥
2) there exists a sequence φ m → H ( R d ) with φ m = 1 on B /m (0) and φ m = 0 outside B / √ m (0). We have (cid:90) Ω | w W − w W + I Ω m,n | dx ≤ (cid:107) w W (cid:107) L ∞ (cid:90) Ω ( w W − w W + I Ω m,n ) dx = 4 (cid:107) w W (cid:107) L ∞ (cid:0) E ( W + I Ω m,n ) − E ( W ) (cid:1) (2.7) VgeWeq1 ≤ (cid:107) w W (cid:107) L ∞ (cid:18)(cid:90) Ω |∇ w m | + 12 W w m − w m dx − (cid:90) Ω |∇ w W | + 12 W w W − w W dx (cid:19) , where w m is any function in ∈ H (Ω m,n ). Taking w m ( x ) = w W ( x ) (cid:89) j (1 − φ m ( x − x j )) , since φ m → H ( R d ), it is easy to see that w m → w W strongly in H (Ω) and so,by (2.7), w W + I Ω m,n → w W in L (Ω) as m → ∞ . Since the weak convergence of probability the idea of this proof was suggested by Dorin Bucur PTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS 7 measures and the γ -convergence are both induced by metrics, a diagonal sequence argumentbrings to the conclusion. (cid:3) Remark . When d = 1, a result analogous to Proposition 2.5 is that any sequence ( µ n )weakly* converging to µ is also γ -converging to µ . This is an easy consequence of the compactembedding of H (Ω) into the space of continuous functions on Ω.We note that the hypothesis V ≥ W in Proposition 2.6 is necessary. Indeed, we have thefollowing proposition, whose proof is contained in [9, Theorem 3.1] and we report it here for thesake of completeness. gamma Let µ n ∈ M + cap (Ω) be a sequence of capacitary Radon measures weakly* con-verging to the measure ν and γ -converging to the capacitary measure µ ∈ M + cap (Ω) . Then µ ≤ ν in Ω .Proof. We note that it is enough to show that µ ( K ) ≤ ν ( K ) whenever K ⊂⊂ Ω is a compactset. Let u be a nonnegative smooth function with compact support in Ω such that u ≤ u = 1 on K ; we have µ ( K ) ≤ (cid:90) Ω u dµ ≤ lim inf n →∞ (cid:90) Ω u dµ n = (cid:90) Ω u dν ≤ ν ( { u > } ) . Since u is arbitrary, we have the conclusion by the Borel regularity of ν . (cid:3) Existence of optimal potentials in L p (Ω) s3 In this section we consider the optimization problemmin (cid:26) F ( V ) : V : Ω → [0 , + ∞ ] , (cid:90) Ω V p dx ≤ (cid:27) , (3.1) pop where p > F ( V ) is a cost functional depending on the solution of some partial differentialequation on Ω. Typically, F ( V ) is the minimum of some functional J V : H (Ω) → R dependingon V . A natural assumption in this case is the lower semicontinuity of the functional F withrespect to the γ -convergence, that is F ( µ ) ≤ lim inf n →∞ F ( µ n ) , whenever µ n → γ µ. (3.2) lscmain Theorem 3.1. Let F : L (Ω) → R be a functional, lower semicontinuous with respect to the γ -convergence, and let V be a weakly L (Ω) compact set. Then the problem min { F ( V ) : V ∈ V} , (3.3) popK admits a solution.Proof. Let ( V n ) be a minimizing sequence in V . By the compactness assumption on V , we mayassume that V n tends weakly L (Ω) to some V ∈ V . By Proposition 2.5, we have that V n γ -converges to V and so, by the semicontinuity of F , F ( V ) ≤ lim inf n →∞ F ( V n ) , which gives the conclusion. (cid:3) Remark . Theorem 3.1 applies for instance to the integral functionals and to the spectralfunctionals considered in the introduction; it is not difficult to show that they are lower semi-continuous with respect to the γ -convergence. G. BUTTAZZO, A. GEROLIN, B. RUFFINI, AND B. VELICHKOV Remark . In some special cases the solution of (3.1) can be written explicitly in terms of thesolution of some partial differential equation on Ω. This is the case of the Dirichlet Energy, thatwe discuss in Subsection 3.1, and of the first eigenvalue of the Dirichlet Laplacian λ (see [12,Chapter 8]).The compactness assumption on the admissible class V for the weak L (Ω) convergencein Theorem 3.1 is for instance satisfied if Ω has finite measure and V is a convex closed andbounded subset of L p (Ω), with p ≥ 1. In the case of measures an analogous result holds. mainmeas Theorem 3.4. Let Ω ⊂ R d be a bounded open set and let F : M +cap (Ω) → R be a functionallower semicontinuous with respect to the γ -convergence. Then the problem min (cid:8) F ( µ ) : µ ∈ M +cap (Ω) , µ (Ω) ≤ (cid:9) , (3.4) popmeas admits a solution.Proof. Let ( µ n ) be a minimizing sequence. Then, up to a subsequence µ n converges weakly* tosome measure ν and γ -converges to some measure µ ∈ M +cap (Ω). By Proposition 2.8, we havethat µ (Ω) ≤ ν (Ω) ≤ µ is a solution of (3.4). (cid:3) The following example shows that the optimal solution of problem (3.4) is not, in general,a function V ( x ), even when the optimization criterion is the energy E f introduced in (1.1). Onthe other hand, an explicit form for the optimal potential V ( x ) will be provided in Proposition3.9 assuming that the right-hand side f is in L (Ω). examdelta Example . Let Ω = ( − , 1) and consider the functional F ( µ ) = − min (cid:26) (cid:90) Ω | u (cid:48) | dx + 12 (cid:90) Ω u dµ − u (0) : u ∈ H (Ω) (cid:27) . Then, for any µ such that µ (Ω) ≤ 1, we have F ( µ ) ≥ − min (cid:26) (cid:90) Ω | u (cid:48) | dx + 12 (cid:0) sup Ω u (cid:1) − u (0) : u ∈ H (Ω) , u ≥ (cid:27) . (3.5) examdelta1 By a symmetrization argument, the minimizer u of the right-hand side of (3.5) is radiallydecreasing; moreover, u is linear on the set u < M , where M = sup u , and so it is of the form u ( x ) = M − α x + M − α , x ∈ [ − , − α ] ,M, x ∈ [ − α, α ] , − M − α x + M − α , x ∈ [ α, , (3.6)for some α ∈ [0 , α = 0 and M = 1 / 3. Thus, u is alsothe minimizer of F ( δ ) = − min (cid:26) (cid:90) Ω | u (cid:48) | dx + 12 u (0) − u (0) : u ∈ H (Ω) (cid:27) , and so δ is the solution of min { F ( µ ) : µ (Ω) ≤ } . PTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS 9 s31 Minimization problems in L p concerning the Dirichlet Energy functional. LetΩ ⊂ R d be a bounded open set and let f ∈ L (Ω). By Theorem 3.1, the problemmin {−E f ( V ) : V ∈ V} with V = (cid:26) V ≥ , (cid:90) Ω V p dx ≤ (cid:27) , (3.7) maxpb admits a solution, where E f ( V ) is the energy functional defined in (1.1). We notice that, replacing −E f ( V ) by E f ( V ), makes problem (3.7) trivial, with the only solution V ≡ 0. Minimizationproblems for E f will be considered in Section 4 for admissible classes of the form V = (cid:26) V ≥ , (cid:90) Ω V − p dx ≤ (cid:27) . Analogous results for F ( V ) = − λ ( V ) were proved in [12, Theorem 8.2.3]. maxex Proposition 3.6. Let Ω ⊂ R d be a bounded open set, < p < ∞ and f ∈ L (Ω) . Then theproblem (3.7) has a unique solution V p = (cid:18)(cid:90) Ω | u p | p/ ( p − dx (cid:19) − /p | u p | − p +1) / ( p − , where u p ∈ H (Ω) ∩ L p/ ( p − (Ω) is the minimizer of the functional J p ( u ) := 12 (cid:90) Ω |∇ u | dx + 12 (cid:18)(cid:90) Ω | u | p/ ( p − dx (cid:19) ( p − /p − (cid:90) Ω uf dx. (3.8) Ja Moreover, we have E f ( V p ) = J p ( u p ) .Proof. We first show that we havemax V ∈V min u ∈ H (Ω) (cid:90) Ω (cid:18) |∇ u | + u V − uf (cid:19) dx ≤ min u ∈ H (Ω) max V ∈V (cid:90) Ω (cid:18) |∇ u | + u V − uf (cid:19) dx, (3.9) maxexfeb1 where the maximums are taken over all positive functions V ∈ L p (Ω). For a fixed u ∈ H (Ω), themaximum on the right-hand side (if finite) is achieved for a function V such that Λ pV p − = u ,where Λ is a Lagrange multiplier. By the condition (cid:82) Ω V p dx = 1 we obtain that the maximumis achieved for V = (cid:18)(cid:90) Ω | u | pp − dx (cid:19) /p | u | p − . Substituting in (3.9), we obtainmax {E f ( V ) : V ∈ V} ≤ min (cid:8) J p ( u ) : u ∈ H (Ω) (cid:9) . (3.10) E Let u n be a minimizing sequence for J p . Since inf J p ≤ 0, we can assume J p ( u n ) ≤ n ∈ N . Thus, we have12 (cid:90) Ω |∇ u n | dx + 12 (cid:18)(cid:90) Ω | u n | p/ ( p − dx (cid:19) ( p − /p ≤ (cid:90) Ω u n f dx ≤ C (cid:107) f (cid:107) L (Ω) (cid:107)∇ u n (cid:107) L , (3.11) apriori where C is a constant depending on Ω. Thus we obtain (cid:90) Ω |∇ u n | dx + (cid:18)(cid:90) Ω | u n | p/ ( p − dx (cid:19) ( p − /p ≤ C (cid:107) f (cid:107) L (Ω) , (3.12) apriori2 and so, up to subsequence u n converges weakly in H (Ω) and L p/ ( p − (Ω) to some u p ∈ H (Ω) ∩ L p/ ( p − (Ω). By the semicontinuity of the L -norm of the gradient and the L pp − -norm and the fact that (cid:82) Ω f u n dx → (cid:82) Ω f u p dx , as n → ∞ , we have that u p is a minimizer of J p . By thestrict convexity of J p , we have that u p is unique. Moreover, by (3.11) and (3.12), J p ( u p ) > −∞ .Writing down the Euler-Lagrange equation for u p , we obtain − ∆ u p + (cid:18)(cid:90) Ω | u p | p/ ( p − dx (cid:19) − /p | u p | / ( p − u p = f. Setting V p = (cid:18)(cid:90) Ω | u p | p/ ( p − dx (cid:19) − /p | u p | / ( p − , we have that (cid:82) Ω V pp dx = 1 and u p is the solution of − ∆ u p + V p u p = f. (3.13) eqalpha In particular, we have J p ( u p ) = E p ( V p ) and so V p solves (3.7). The uniqueness of V p follows bythe uniqueness of u p and the equality case in the H¨older inequality (cid:90) Ω u V dx ≤ (cid:18)(cid:90) Ω V p dx (cid:19) /p (cid:18)(cid:90) Ω | u | p/ ( p − dx (cid:19) ( p − /p ≤ (cid:18)(cid:90) Ω | u | p/ ( p − dx (cid:19) ( p − /p . (cid:3) When the functional F is the energy E f , the existence result holds also in the case p = 1.Before we give the proof of this fact in Proposition 3.9, we need some preliminary results. Wealso note that the analogous results were obtained in the case F = − λ (see [12, Theorem 8.2.4])and in the case F = −E f , where f is a positive function (see [9]). unifest Remark . Let u p be the minimizer of J p , defined in (3.8). By (3.12), we have the estimate (cid:107)∇ u p (cid:107) L (Ω) + (cid:107) u p (cid:107) L p/ ( p − (Ω) ≤ C (cid:107) f (cid:107) L (Ω) , (3.14) unifest1 where C is the constant from (3.11). Moreover, we have u p ∈ H loc (Ω) and for each open setΩ (cid:48) ⊂⊂ Ω, there is a constant C not depending on p such that (cid:107) u p (cid:107) H (Ω (cid:48) ) ≤ C ( f, Ω (cid:48) ) . Indeed, u p satisfies the PDE − ∆ u + c | u | α u = f, (3.15) unifest2 with c > α = 2 / ( p − u ∈ H loc (Ω). To show that (cid:107) u p (cid:107) H (Ω (cid:48) ) is bounded independently of p we applythe Nirenberg operator ∂ hk u = u ( x + he k ) − u ( x ) h on both sides of (3.15), and multiplying by φ ∂ hk u ,where φ is an appropriate cut-off function which equals 1 on Ω (cid:48) , we have (cid:90) Ω φ |∇ ∂ hk u | dx + (cid:90) Ω ∇ ( ∂ hk u ) · ∇ ( φ ) ∂ hk u dx + c ( α + 1) (cid:90) Ω φ | u | α | ∂ hk u | dx (3.16) unifest3 = − (cid:90) f ∂ hk ( φ ∂ hk u ) dx, for all k = 1 , . . . , d . Some straightforward manipulations now give (cid:107)∇ u (cid:107) L (Ω (cid:48) ) ≤ d (cid:88) k =1 (cid:90) Ω φ |∇ ∂ k u | dx ≤ C (Ω (cid:48) ) (cid:0) (cid:107) f (cid:107) L ( { φ > } ) + (cid:107)∇ u (cid:107) L (Ω) (cid:1) . (3.17) unifest4 PTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS 11 sc Lemma 3.8. Let Ω ⊂ R d be an open set and f ∈ L (Ω) . Consider the functional J : L (Ω) → R defined by J ( u ) := 12 (cid:90) Ω |∇ u | dx + 12 (cid:107) u (cid:107) ∞ − (cid:90) Ω uf dx, (3.18) J1 Then, J p Γ -converges in L (Ω) to J , as p → , where J p is defined in (3.8) .Proof. Let v n ∈ L (Ω) be a sequence of positive functions converging in L to v ∈ L (Ω) andlet α n → + ∞ . Then, we have that (cid:107) v (cid:107) L ∞ (Ω) ≤ lim inf n →∞ (cid:107) v n (cid:107) L αn (Ω) . (3.19) sc1 In fact, suppose first that (cid:107) v (cid:107) L ∞ = M < + ∞ and let ω ε = { v > M − ε } , for some ε > 0. Then,we have lim inf n →∞ (cid:107) v n (cid:107) L αn (Ω) ≥ lim n →∞ | ω ε | (1 − α n ) /α n (cid:90) ω ε v n dx = | ω ε | − (cid:90) ω ε v dx ≥ M − ε, and so, letting ε → 0, we have lim inf n →∞ (cid:107) v n (cid:107) L αn (Ω) ≤ M . If (cid:107) v (cid:107) L ∞ = + ∞ , then setting ω k = { v > k } , for any k ≥ 1, and arguing as above, we obtain (3.19).Let u n → u in L (Ω). Then, by the semicontinuity of the L norm of the gradient and (3.19)and the continuity of the term (cid:82) Ω uf dx , we have J ( u ) ≤ lim inf n →∞ J p n ( u n ) , (3.20)for any decreasing sequence p n → 1. On the other hand, for any u ∈ L , we have J p n ( u ) → J ( u )as n → ∞ and so, we have the conclusion. (cid:3) maxone Proposition 3.9. Let Ω ⊂ R d be a bounded open set and f ∈ L (Ω) . Then there is a uniquesolution of problem (3.7) with p = 1 , given by V = 1 M (cid:0) χ ω + f − χ ω − f (cid:1) , where M = (cid:107) u (cid:107) L ∞ (Ω) , ω + = { u = M } , ω − = { u = − M } , being u ∈ H (Ω) ∩ L ∞ (Ω) theunique minimizer of the functional J , defined in (3.18) . In particular, (cid:82) ω + f dx − (cid:82) ω − f dx = M , f ≥ on ω + and f ≤ on ω − .Proof. For any u ∈ H (Ω) and any V ≥ (cid:82) Ω V dx ≤ (cid:90) Ω u V dx ≤ (cid:107) u (cid:107) ∞ (cid:90) Ω V dx ≤ (cid:107) u (cid:107) ∞ , where for sake of simplicity, we write (cid:107) · (cid:107) ∞ instead of (cid:107) · (cid:107) L ∞ (Ω) . Arguing as in the proof ofProposition 3.6, we obtain the inequalities12 (cid:90) Ω |∇ u | dx + 12 (cid:90) Ω u V dx − (cid:90) Ω uf dx ≤ J ( u ) , max (cid:26) E f ( V ) : (cid:90) Ω V ≤ (cid:27) ≤ min (cid:8) J ( u ) : u ∈ H (Ω) (cid:9) . As in (3.11), we have that a minimizing sequence of J is bounded in H (Ω) ∩ L ∞ (Ω) and thusby semicontinuity there is a minimizer u ∈ H (Ω) ∩ L ∞ (Ω) of J , which is also unique, bythe strict convexity of J . Let u p denotes the minimizer of J p as in Proposition 3.6. Then, byRemark 3.7, we have that the family u p is bounded in H (Ω) and in H (Ω (cid:48) ) for each Ω (cid:48) ⊂⊂ Ω.Then, we have that each sequence u p n has a subsequence converging weakly in L (Ω) to some u ∈ H loc (Ω) ∩ H (Ω). By Lemma 3.8, we have u = u and so, u ∈ H loc (Ω) ∩ H (Ω). Thus u p n → u in L (Ω).Let us define M = (cid:107) u (cid:107) ∞ and ω = ω + ∪ ω − . We claim that u satisfies, on Ω the PDE − ∆ u + χ ω f = f. (3.21) maxone1 Indeed, setting Ω t = Ω ∩ {| u | < t } for t > 0, we compute the variation of J with respect to anyfunction ϕ ∈ H (Ω M − ε ). Namely we consider functions of the form ϕ = ψw ε where w ε is thesolution of − ∆ w ε = 1 on Ω M − ε , and w ε = 0 on ∂ Ω M − ε . Thus we obtain that − ∆ u = f onΩ M − ε and letting ε → − ∆ u = f on Ω M = Ω \ ω. Moreover, since u ∈ H loc (Ω), we have that ∆ u = 0 on ω and so, we obtain (3.21).Since u is the minimizer of J , we have that for each ε ∈ R , J ((1 + ε ) u ) − J ( u ) ≥ 0. Takingthe derivative of this difference at ε = 0, we obtain (cid:90) Ω |∇ u | dx + M = (cid:90) Ω f u dx. (3.22) maxone2 By (3.21), we have (cid:82) Ω |∇ u | dx = (cid:82) Ω \ ω f u dx and so M = (cid:90) ω + f dx − (cid:90) ω − f dx. (3.23) maxone2 Setting V := M (cid:0) χ ω + f − χ ω − f (cid:1) , we have that (cid:82) Ω V dx = 1, − ∆ u + V u = f in H − (Ω) and J ( u ) = 12 (cid:90) Ω |∇ u | dx + 12 (cid:90) Ω u V dx − (cid:90) Ω u f dx. We are left to prove that V is admissible, i.e. V ≥ 0. To do this, consider w ε the energyfunction of the quasi-open set { u < M − ε } and let ϕ = w ε ψ where ψ ∈ C ∞ c ( R d ), ψ ≥ 0. Since ϕ ≥ 0, we get that0 ≤ lim t → + J ( u + tϕ ) − J ( u ) t = (cid:90) Ω (cid:104)∇ u , ∇ ϕ (cid:105) dx − (cid:90) Ω f ϕ dx. This inequality holds for any ψ so that, integrating by parts, we obtain − ∆ u − f ≥ { u < M − ε } . In particular, since ∆ u = 0 almost everywhere on ω − = { u = − M } , we obtain that f ≤ ω − . Arguing in the same way, and considering testfunctions supported on { u ≥ − M + ε } , we can prove that f ≥ ω + . This implies V ≥ (cid:3) Remark . Under some additional assumptions on Ω and f one can obtain some more preciseregularity results for u . In fact, in [17, Theorem A1] it was proved that if ∂ Ω ∈ C and if f ∈ L ∞ (Ω) is positive, then u ∈ C , (Ω). controesempio Remark . In the case p < f and Ω. We give a counterexample in dimension one, which can be easily adapted tohigher dimensions.Let Ω = (0 , f = 1, and let x n,k = k/n for any n ∈ N and k = 1 , . . . , n − 1. We define the(capacitary) measures µ n = n − (cid:88) k =1 + ∞ δ x n,k , PTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS 13 where δ x is the Dirac measure at the point x . Let w n be the minimizer of the functional J µ n (1 , · ),defined in (2.4). Then w n vanishes at x n,k , for k = 1 , . . . , n − 1, and so we have E ( µ n ) = n min (cid:40) (cid:90) /n | u (cid:48) | dx − (cid:90) /n u dx : u ∈ H (0 , /n ) (cid:41) = − Cn , where C > n and j , let V nj be the sequence of positive functions such that (cid:82) | V nj | p dx = 1,defined by V nj = C n n − (cid:88) k =1 j /p χ (cid:104) kn − j , kn + j (cid:105) < n − (cid:88) k =1 I (cid:104) kn − j , kn + j (cid:105) , (3.24) esempioVjn where C n is a constant depending on n and I is as in (2.2). By the compactness of the γ -convergence, we have that, up to a subsequence, V nj dx γ -converges to some capacitary measure µ as j → ∞ . On the other hand it is easy to check that (cid:80) n − k =1 I (cid:104) kn − j , kn + j (cid:105) ( x ) γ -converges to µ n as j → ∞ . By (3.24), we have that µ ≤ µ n . In order to show that µ = µ n it is enough tocheck that each nonnegative function u ∈ H ((0 , (cid:82) u dµ < + ∞ , vanishes at x n,k for k = 1 , . . . , n − 1. Suppose that u ( k/n ) > 0. By the definition of the γ -convergence, there isa sequence u j ∈ H (Ω) = H V nj (Ω) such that u j → u weakly in H (Ω) and (cid:82) u j V nj dx ≤ C , forsome constant C not depending on j ∈ N . Since u j are uniformly 1 / u j ≥ ε > I containing k/n . But then for j large enough I contains [ k/n − /j, k/n + 1 /j ] so that C ≥ (cid:90) u j V nj dx ≥ (cid:90) k/n +1 /jk/n − /j u j V nj dx ≥ C n ε j /p − , which is a contradiction for p < 1. Thus, we have that µ = µ n and so V nj γ -converges to µ n as j → ∞ . In particular, E ( µ n ) = lim j →∞ E ( V nj ) and since the left-hand side converges to zero as n → ∞ , we can choose a diagonal sequence V nj n such that E ( V nj n ) → n → ∞ . Since there isno admissible functional V such that E ( V ) = 0, we have the conclusion.4. Existence of optimal potentials for unbounded constraints s4 In this section we consider the optimization problemmin { F ( V ) : V ∈ V} , (4.1) popPhi where V is an admissible class of nonnegative Borel functions on the bounded open set Ω ⊂ R d and F is a cost functional on the family of capacitary measures M +cap (Ω). The admissible classeswe study depend on a function Ψ : [0 , + ∞ ] → [0 , + ∞ ] V = (cid:26) V : Ω → [0 , + ∞ ] : V Lebesgue measurable, (cid:90) Ω Ψ( V ) dx ≤ (cid:27) . mainPhi Theorem 4.1. Let Ω ⊂ R d be a bounded open set and Ψ : [0 , + ∞ ] → [0 , + ∞ ] a strictly decreas-ing function with Ψ − convex. Then, for any functional F : M +cap (Ω) → R which is increasingand lower semicontinuous with respect to the γ -convergence, the problem (4.1) has a solution.Proof. Let V n ∈ V be a minimizing sequence for problem (4.1). Then, v n := Ψ( V n ) is a boundedsequence in L (Ω) and so, up to a subsequence, v n converges weakly* to some measure ν . Wewill prove that V := Ψ − ( ν a ) is a solution of (4.1), where ν a denotes the density of the absolutelycontinuous part of ν with respect to the Lebesgue measure. Clearly V ∈ V and so it remains to prove that F ( V ) ≤ lim inf n F ( V n ). In view of Remark 2.4, we can suppose that, up to asubsequence, V n γ -converges to a capacitary measure µ ∈ M +cap (Ω). We claim that the followinginequalities hold true: F ( V ) ≤ F ( µ ) ≤ lim inf n →∞ F ( V n ) . (4.2) th1 In fact, the second inequality in (4.2) is the lower semicontinuity of F with respect to the γ -convergence, while the first needs a more careful examination. By the definition of γ -convergence,we have that for any u ∈ H (Ω), there is a sequence u n ∈ H (Ω) which converges to u in L (Ω)and is such that (cid:90) Ω |∇ u | dx + (cid:90) Ω u dµ = lim n →∞ (cid:90) Ω |∇ u n | dx + (cid:90) Ω u n V n dx = lim n →∞ (cid:90) Ω |∇ u n | dx + (cid:90) Ω u n Ψ − ( v n ) dx (4.3) ineqth2 ≥ (cid:90) Ω |∇ u | dx + (cid:90) Ω u Ψ − ( ν a ) dx = (cid:90) Ω |∇ u | dx + (cid:90) Ω u V dx, where the inequality in (4.3) is due to strong-weak* lower semicontinuity of integral functionals(see for instance [4]). Thus, for any u ∈ H (Ω), we have (cid:90) Ω u dµ ≥ (cid:90) Ω u V dx, which gives V ≤ µ . Since F is increasing, we obtain the first inequality in (4.2) and so theconclusion. (cid:3) Remark . The condition on the function Ψ in Theorem 4.1 is satisfied for instance by thefollowing functions:(1) Ψ( x ) = x − p , for any p > x ) = e − αx , for any α > s41 Optimal potentials for the Dirichlet Energy and the first eigenvalue of theDirichlet Laplacian. In some special cases, the solution of the optimization problem (4.1)can be computed explicitly through the solution of some PDE, as in Subsection 3.1. This occursfor instance when F = λ or when F = E f , with f ∈ L (Ω). We note that, by the variationalformulation λ ( V ) = min (cid:26)(cid:90) Ω |∇ u | dx + (cid:90) Ω u V dx : u ∈ H (Ω) , (cid:90) Ω u dx = 1 (cid:27) , (4.4)we can rewrite problem (4.1) asmin (cid:26) min (cid:107) u (cid:107) =1 (cid:110) (cid:90) Ω |∇ u | dx + (cid:90) Ω u V dx (cid:111) : V ≥ , (cid:90) Ω Ψ( V ) dx ≤ (cid:27) = min (cid:107) u (cid:107) =1 (cid:26) min (cid:110) (cid:90) Ω |∇ u | dx + (cid:90) Ω u V dx : V ≥ , (cid:90) Ω Ψ( V ) dx ≤ (cid:111)(cid:27) . (4.5) op3 One can compute that, if Ψ is differentiable with Ψ (cid:48) invertible, then the second minimum in(4.5) is achieved for V = (Ψ (cid:48) ) − (Λ u u ) , (4.6) vopt PTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS 15 where Λ u is a constant such that (cid:82) Ω Ψ (cid:0) (Ψ (cid:48) ) − (Λ u u ) (cid:1) dx = 1. Thus, the solution of the problemon the right hand side of (4.5) is given through the solution ofmin (cid:26)(cid:90) Ω |∇ u | dx + (cid:90) Ω u (Ψ (cid:48) ) − (Λ u u ) dx : u ∈ H (Ω) , (cid:90) Ω u dx = 1 (cid:27) . (4.7) gs1 Analogously, we obtain that the optimal potential for the Dirichlet Energy E f is given by (4.6),where this time u is a solution ofmin (cid:26)(cid:90) Ω |∇ u | dx + (cid:90) Ω u (Ψ (cid:48) ) − (Λ u u ) dx − (cid:90) Ω f u dx : u ∈ H (Ω) (cid:27) . (4.8) gs2 Thus we obtain the following result. lb Corollary 4.3. Under the assumptions of Theorem 4.1, for the functionals F = λ and F = E f there exists a solution of (4.1) given by V = (Ψ (cid:48) ) − (Λ u u ) , where u ∈ H (Ω) is a minimizer of (4.7) , in the case F = λ , and of (4.8) , in the case F = E f .Example . If Ψ( x ) = x − p with p > 0, the optimal potentials for λ and E f are given by V = (cid:18)(cid:90) Ω | u | p/ ( p +1) dx (cid:19) /p u − / ( p +1) , (4.9) vopt2 where u is the minimizer of (4.7) and (4.8), respectively. We also note that, in this case (cid:90) Ω u (Ψ (cid:48) ) − (Λ u u ) dx = (cid:18)(cid:90) Ω | u | p/ ( p +1) dx (cid:19) (1+ p ) /p . Example . If Ψ( x ) = e − αx with α > 0, the optimal potentials for λ and E f are given by V = 1 α (cid:18) log (cid:18)(cid:90) Ω u dx (cid:19) − log (cid:0) u (cid:1)(cid:19) , (4.10) vopt2 where u is the minimizer of (4.7) and (4.8), respectively. We also note that, in this case (cid:90) Ω u (Ψ (cid:48) ) − (Λ u u ) dx = 1 α (cid:18)(cid:90) Ω u dx (cid:90) Ω log (cid:0) u (cid:1) dx − (cid:90) Ω u log (cid:0) u (cid:1) dx (cid:19) . Optimization problems in unbounded domains s5 In this section we consider optimization problems for which the domain region is the en-tire Euclidean space R d . General existence results, in the case when the design region Ω isunbounded, are hard to achieve since most of the cost functionals are not semicontinuous withrespect to the γ -convergence in these domains. For example, it is not hard to check that if µ isa capacitary measure, infinite outside the unit ball B , then, for every x n → ∞ , the sequence oftranslated measures µ n = µ ( · + x n ) γ -converges to the capacitary measure I ∅ ( E ) = (cid:40) , if cap( E ) = 0 , + ∞ , if cap( E ) > . Thus increasing and translation invariant functionals are never lower semicontinuous with respectto the γ -convergence. In some special cases, as the Dirichlet Energy or the first eigenvalue ofthe Dirichlet Laplacian, one can obtain existence results by more direct methods, as those inProposition 3.6.For a potential V ≥ f ∈ L q ( R d ), we define the Dirichlet energy as E f ( V ) = inf (cid:26)(cid:90) R d (cid:16) |∇ u | + 12 V ( x ) u − f ( x ) u (cid:17) dx : u ∈ C ∞ c ( R d ) (cid:27) . (5.1) energyrd In some cases it is convenient to work with the space . H ( R d ), obtained as the closure of C ∞ c ( R d )with respect to the L norm of the gradient, instead of the classical Sobolev space H ( R d ). Werecall that if d ≥ 3, the Gagliardo-Nirenberg-Sobolev inequality (cid:107) u (cid:107) L d/ ( d − ≤ C d (cid:107)∇ u (cid:107) L , ∀ u ∈ . H ( R d ) , (5.2) gnsd3 holds, while in the cases d ≤ 2, we have respectively (cid:107) u (cid:107) L ∞ ≤ (cid:18) r + 22 (cid:19) / ( r +2) (cid:107) u (cid:107) r/ ( r +2) L r (cid:107) u (cid:48) (cid:107) / ( r +2) L , ∀ r ≥ , ∀ u ∈ . H ( R ); (5.3) gnsd1 (cid:107) u (cid:107) L r +2 ≤ (cid:18) r + 22 (cid:19) / ( r +2) (cid:107) u (cid:107) r/ ( r +2) L r (cid:107)∇ u (cid:107) / ( r +2) L , ∀ r ≥ , ∀ u ∈ . H ( R ) . (5.4) gnsd2 Optimal potentials in L p ( R d ) . In this section we consider optimization problems for theDirichlet energy E f among potentials V ≥ (cid:107) V (cid:107) L p ≤ 1. Wenote that the results in this section hold in a generic unbounded domain Ω. Nevertheless, forsake of simplicity, we restrict our attention to the case Ω = R d . pinterv Proposition 5.1. Let p > and let q be in the interval with end-points a = 2 p/ ( p + 1) and b = max { , d/ ( d + 2) } (with a included for every d ≥ , and b included for every d (cid:54) = 2 ). Then,for every f ∈ L q ( R d ) , there is a unique solution of the problem max (cid:26) E f ( V ) : V ≥ , (cid:90) R d V p dx ≤ (cid:27) . (5.5) maxrd Proof. Arguing as in Proposition 3.6, we have that for p > V p is givenby V p = (cid:18)(cid:90) R d | u p | p/ ( p − dx (cid:19) − /p | u p | / ( p − , (5.6) Vprd where u p is the solution of the problemmin (cid:40) (cid:90) R d |∇ u | dx + 12 (cid:18)(cid:90) R d | u | p/ ( p − dx (cid:19) ( p − /p − (cid:90) R d uf dx : (5.7) Jard u ∈ . H ( R d ) ∩ L p/ ( p − ( R d ) (cid:41) . Thus, it is enough to prove that there exists a solution of (5.7). For a minimizing sequence u n we have12 (cid:90) R d |∇ u n | dx + 12 (cid:18)(cid:90) R d | u n | p/ ( p − dx (cid:19) ( p − /p ≤ (cid:90) R d u n f dx ≤ C (cid:107) f (cid:107) L q (cid:107) u n (cid:107) L q (cid:48) . (5.8) apriorird Suppose that d ≥ 3. Interpolating q (cid:48) between 2 p/ ( p − 1) and 2 d/ ( d − 2) and using the Gagliardo-Nirenberg-Sobolev inequality (5.2), we obtain that there is a constant C , depending only on p, d and f , such that 12 (cid:90) R d |∇ u n | dx + 12 (cid:18)(cid:90) R d | u n | p/ ( p − dx (cid:19) ( p − /p ≤ C. Thus we can suppose that u n converges weakly in . H ( R d ) and in L p/ ( p − ( R d ) and so, theproblem (5.7) has a solution. In the case d ≤ 2, the claim follows since, by using (5.3), (5.4)and interpolation, we can still estimate (cid:107) u n (cid:107) L q (cid:48) by means of (cid:107)∇ u n (cid:107) L and (cid:107) u n (cid:107) L p/ ( p − . (cid:3) PTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS 17 Repeating the argument of Subsection 3.1, one obtains an existence result for (5.5) in thecase p = 1, too. maxonerd Proposition 5.2. Let f ∈ L q ( R d ) , where q ∈ [1 , dd +2 ] , if d ≥ , and q = 1 , if d = 1 , . Thenthere is a unique solution V of problem (5.5) with p = 1 , which is given by V = fM (cid:0) χ ω + − χ ω − (cid:1) , where M = (cid:107) u (cid:107) L ∞ ( R d ) , ω + = { u = M } , ω − = { u = − M } , and u is the unique minimizer of min (cid:26) (cid:90) R d |∇ u | dx + 12 (cid:107) u (cid:107) L ∞ − (cid:90) R d uf dx : u ∈ . H ( R d ) ∩ L ∞ ( R d ) (cid:27) . (5.9) J1xx In particular, (cid:82) ω + f dx − (cid:82) ω − f dx = M , f ≥ on ω + and f ≤ on ω − . We note that, when p = 1, the support of the optimal potential V is contained in thesupport of the function f . This is not the case if p > 1, as the following example shows. exammaxrd Example . Let f = χ B (0 , and p > 1. By our previous analysis we know that there exist asolution u p for problem (5.7) and a solution V p for problem (5.5) given by (5.6). We note that u p is positive, radially decreasing and satisfies the equation − u (cid:48)(cid:48) ( r ) − d − r u (cid:48) ( r ) + Cu α = 0 , r ∈ (1 , + ∞ ) , where α = 2 p/ ( p − > C is a positive constant. Thus, we have that u p ( r ) = kr / (1 − α ) , where k is an explicit constant depending on C , d and α . In particular, we have that u p is notcompactly supported on R d (see Figure 1). y -3 -1 1 3 u p Figure 1. The solution u p of problem (5.7), with p > f = χ B (0 , doesnot have a compact support. fig1ss52 Optimal potentials with unbounded constraint. In this subsection we consider theproblems min (cid:26) E f ( V ) : V ≥ , (cid:90) R d V − p dx ≤ (cid:27) , (5.10) minrd min (cid:26) λ ( V ) : V ≥ , (cid:90) R d V − p dx ≤ (cid:27) , (5.11) lbrd for p > f ∈ L q ( R d ). We will see in Proposition 5.4 that in order to have existence for(5.10) the parameter q must satisfy some constraint, depending on the value of p and on thedimension d . Namely, we need q to satisfy the following conditions q ∈ [ 2 dd + 2 , pp − , if d ≥ p > ,q ∈ [ 2 dd + 2 , + ∞ ] , if d ≥ p ≤ ,q ∈ (1 , pp − , if d = 2 and p > , (5.12) admq q ∈ (1 , + ∞ ] , if d = 2 and p ≤ ,q ∈ [1 , pp − , if d = 1 and p > ,q ∈ [1 , + ∞ ] , if d = 1 and p ≤ . We say that q = q ( p, d ) ∈ [1 , + ∞ ] is admissible if it satisfy (5.12). Note that q = 2 is admissiblefor any d ≥ p > exErd Proposition 5.4. Let p > and f ∈ L q ( R d ) , where q is admissible in the sense of (5.12) .Then the minimization problem (5.10) has a solution V p given by V p = (cid:18)(cid:90) R d | u p | p/ ( p +1) dx (cid:19) /p | u p | − / (1+ p ) , (5.13) V-prd where u p is a minimizer of min (cid:40) (cid:90) R d |∇ u | dx + 12 (cid:18)(cid:90) R d | u | p/ ( p +1) dx (cid:19) ( p +1) /p − (cid:90) R d uf dx : (5.14) J-ard u ∈ . H ( R d ) , | u | p/ ( p +1) ∈ L ( R d ) (cid:41) . Moreover, if p ≥ , then the functional in (5.14) is convex, its minimizer is unique and so is thesolution of (5.10) .Proof. By means of (5.2), (5.3) and (5.4), and thanks to the admissibility of q , we get theexistence of a solution of (5.14) through an interpolation argument similar to the one used inthe proof of Proposition 5.1. The existence of an optimal potential follows by the same argumentas in Subsection 4.1. (cid:3) In Example 5.3, we showed that the optimal potentials for (5.5), may be supported on thewhole R d . The analogous question for the problem (5.10) is whether the optimal potentials givenby (5.13) have a bounded set of finiteness { V p < + ∞} . In order to answer this question, it issufficient to study the support of the solutions u p of (5.14), which solve the equation − ∆ u + C p | u | − / ( p +1) u = f, (5.15) eulagrd where C p > p . cs1 Proposition 5.5. Let p > and let f ∈ L q ( R d ) , for q > d/ , be a nonnegative function with acompact support. Then every solution u p of problem (5.14) has a compact support. PTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS 19 Proof. With no loss of generality we may assume that f is supported in the unit ball of R d . Wefirst prove the result when f is radially decreasing. In this case u p is also radially decreasingand nonnegative. Let v be the function defined by v ( | x | ) = u p ( x ). Thus v satisfies the equation − v (cid:48)(cid:48) − d − r v (cid:48) + C p v s = 0 r ∈ (1 , + ∞ ) ,v (1) = u p (1) , (5.16) ode1 where s = ( p − / ( p + 1) and C p > p . Since v ≥ v (cid:48) ≤ 0, wehave that v is convex. Moreover, since (cid:90) + ∞ v r d − dr < + ∞ , (cid:90) + ∞ | v (cid:48) | r d − dr < + ∞ , we have that v , v (cid:48) and v (cid:48)(cid:48) vanish at infinity. Multiplying (5.16) by v (cid:48) we obtain (cid:18) v (cid:48) ( r ) − C p v ( r ) s +1 s + 1 (cid:19) (cid:48) = − d − r v (cid:48) ( r ) ≤ . Thus the function v (cid:48) ( r ) / − C p v ( r ) s +1 / ( s + 1) is decreasing and vanishing at infinity and thusnonnegative. Thus we have − v (cid:48) ( r ) ≥ Cv ( r ) ( s +1) / , r ∈ (1 , + ∞ ) , (5.17) ode2 where C = (cid:0) C p / ( s + 1) (cid:1) / . Arguing by contradiction, suppose that v is strictly positive on(1 , + ∞ ). Dividing both sides of (5.17) and integrating, we have − v ( r ) (1 − s ) / ≥ Ar + B, where A = 2 C/ (1 − s ) and B is determined by the initial datum v (1). This cannot occur, sincethe left hand side is negative, while the right hand side goes to + ∞ , as r → + ∞ .We now prove the result for a generic compactly supported and nonnegative f ∈ L q ( R d ).Since the solution u p of (5.14) is nonnegative and is a weak solution of (5.15), we have that oneach ball B R ⊂ R d , u p ≤ u , where u ∈ H ( B R ) is the solution of − ∆ u = f in B R , u = u p on ∂B R . Since f ∈ L d/ ( R d ), by [19, Theorem 9.11] and a standard bootstrap argument on the integra-bility of u , we have that u is continuous on B R/ . As a consequence, u p is locally bounded in R d . In particular, it is bounded since u p ∧ M , where M = (cid:107) u p (cid:107) L ∞ ( B ) , is a better competitorthan u p in (5.14). Let w be a radially decreasing minimizer of (5.14) with f = χ B . Thus w isa solution of the PDE − ∆ w + C p w s = χ B , in R d , where C p is as in (5.16). Then, the function w t ( x ) = t / (1 − s ) w ( x/t ) is a solution of theequation − ∆ w t + C p w st = t s/ (1 − s ) χ B t . Since u p is bounded, there exists some t ≥ w t ≥ u p on the ball B t .Moreover, w t minimizes (5.14) with f = t s/ (1 − s ) χ B t and so w t ≥ u p on R d (otherwise w t ∧ u p would be a better competitor in (5.14) than w p ). The conclusion follows since, by the first stepof the proof, w t has compact support. (cid:3) The problems (5.11) and (5.10) are similar both in the questions of existence and the qual-itative properties of the solutions. lbrdex Proposition 5.6. For every p > there is a solution of the problem (5.11) given by V p = (cid:18)(cid:90) R d | u p | p/ ( p +1) dx (cid:19) /p | u p | − / (1+ p ) , (5.18) V-prd2 where u p is a radially decreasing minimizer of min (cid:40) (cid:90) R d |∇ u | dx + (cid:18)(cid:90) R d | u | p/ ( p +1) dx (cid:19) ( p +1) /p : u ∈ H ( R d ) , (cid:90) R d u dx = 1 (cid:41) . (5.19) J-ard2 Moreover, u p has a compact support, hence the set { V p < + ∞} is a ball of finite radius in R d .Proof. Let us first show that the minimum in (5.19) is achieved. Let u n ∈ H ( R d ) be a mini-mizing sequence of positive functions normalized in L . Note that by the P´olya-Szeg¨o inequalitywe may assume that each of these functions is radially decreasing in R d and so we will use theidentification u n = u n ( r ). In order to prove that the minimum is achieved it is enough to showthat the sequence u n converges in L ( R d ). Indeed, since u n is a radially decreasing minimizingsequence, there exists C > r > u n ( r ) p/ ( p +1) ≤ | B r | (cid:90) B r u p/ ( p +1) n dx ≤ Cr d . Thus, for each R > 0, we obtain (cid:90) B cR u n dx ≤ C (cid:90) + ∞ R r − d ( p +1) /p r d − dr = C R − /p , (5.20) lbrdex1 where C and C do not depend on n and R . Since the sequence u n is bounded in H ( R d ), itconverges locally in L ( R d ) and, by (5.20), this convergence is also strong in L ( R d ). Thus, weobtain the existence of a radially symmetric and decreasing solution u p of (5.19) and so, of anoptimal potential V p given by (5.18).We now prove that the support of u p is a ball of finite radius. By the radial symmetry of u p we can write it in the form u p ( x ) = u p ( | x | ) = u p ( r ), where r = | x | . With this notation, u p satisfies the equation: − u (cid:48)(cid:48) p − d − r u (cid:48) p + C p u sp = λu p , where s = ( p − / ( p +1) < C p > p . Arguing as in Proposition5.5, we obtain that, for r large enough, − u (cid:48) p ( r ) ≥ (cid:18) C p s + 1 u p ( r ) s +1 − λ u p ( r ) (cid:19) / ≥ (cid:18) C p s + 1) u p ( r ) s +1 (cid:19) / , where, in the last inequality, we used the fact that u p ( r ) → 0, as r → ∞ , and s + 1 < u p has a compact support. InFigure 2 we show the case d = 1 and f = χ ( − , . (cid:3) igns Remark . We note that the solution u p ∈ H ( R d ) of (5.19) is the function for which the bestconstant C in the interpolated Gagliardo-Nirenberg-Sobolev inequality (cid:107) u (cid:107) L ( R d ) ≤ C (cid:107)∇ u (cid:107) d/ ( d +2 p ) L ( R d ) (cid:107) u (cid:107) p/ ( d +2 p ) L p/ ( p +1) ( R d ) (5.21) igns1 PTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS 21 y -3 -1 1 3 u p Figure 2. The solution u p of problem (5.14), with p > f = χ ( − , . fig2 is achieved. Indeed, for any u ∈ H ( R d ) and any t > 0, we define u t ( x ) := t d/ u ( tx ). Thus, wehave that (cid:107) u (cid:107) L ( R d ) = (cid:107) u t (cid:107) L ( R d ) , for any t > 0. Moreover, up to a rescaling, we may assumethat the function g : (0 , + ∞ ) → R , defined by g ( t ) = (cid:90) R d |∇ u t | dx + (cid:18)(cid:90) R d | u t | p/ ( p +1) dx (cid:19) ( p +1) /p = t (cid:90) R d |∇ u | dx + t − d/p (cid:18)(cid:90) R d | u | p/ ( p +1) dx (cid:19) ( p +1) /p , achieves its minimum in the interval (0 , + ∞ ) and, moreover, we havemin t ∈ (0 , + ∞ ) g ( t ) = C (cid:18)(cid:90) R d |∇ u | dx (cid:19) d/ ( d +2 p ) (cid:18)(cid:90) R d | u | pp +1 dx (cid:19) p +1) / ( d +2 p ) , where C is a constant depending on p and d . In the case u = u p , the minimum of g is achievedfor t = 1 and so, we have that u p is a solution also ofmin (cid:40) (cid:18)(cid:90) R d |∇ u | dx (cid:19) d/ ( d +2 p ) (cid:18)(cid:90) R d | u | p/ ( p +1) dx (cid:19) p +1) / ( d +2 p ) : u ∈ H ( R d ) , (cid:90) R d u dx = 1 (cid:41) , which is just another form of (5.21).6. Further remarks and open questions s6 We recall (see [3]) that the injection H V ( R d ) (cid:44) → L ( R d ) is compact whenever the potential V satisfies (cid:82) R d V − p dx < + ∞ for some 0 < p ≤ 1. In this case the spectrum of the Schr¨odingeroperator − ∆+ V is discrete and we denote by λ k ( V ) its eigenvalues. The existence of an optimalpotential for spectral optimization problems of the formmin (cid:26) λ k ( V ) : V ≥ , (cid:90) R d V − p dx ≤ (cid:27) , (6.1) lbrdk for general k ∈ N , cannot be deduced by the direct methods used in Subsection 5.2. In this lastsection we make the following conjectures: Conjecture 1) For every k ≥ 1, there is a solution V k of the problem (6.1). Conjecture 2) The set of finiteness { V k < + ∞} , of the optimal potential V k , is bounded.In what follows, we prove an existence result in the case k = 2. We first recall that,by Proposition 5.6, there exists optimal potential V p , for λ , such that the set of finiteness { V p < + ∞} is a ball. Thus, we have a situation analogous to the Faber-Krahn inequality, whichstates that the minimum min (cid:110) λ (Ω) : Ω ⊂ R d , | Ω | = c (cid:111) , (6.2) fk is achieved for the ball of measure c . We recall that, starting from (6.2), one may deduce, bya simple argument (see for instance [12]), the Krahn-Szeg¨o inequality, which states that theminimum min (cid:110) λ (Ω) : Ω ⊂ R d , | Ω | = c (cid:111) , (6.3) ks is achieved for a disjoint union of equal balls. In the case of potentials one can find two optimalpotentials for λ with disjoint sets of finiteness and then apply the argument from the proof ofthe Krahn-Szeg¨o inequality. In fact, we have the following result. potks Proposition 6.1. There exists an optimal potential, solution of (6.1) with k = 2 . Moreover,any optimal potential is of the form min { V , V } , where V and V are optimal potentials for λ which have disjoint sets of finiteness { V < + ∞} ∩ { V < + ∞} = ∅ and are such that (cid:82) R d V − p dx = (cid:82) R d V − p dx = 1 / .Proof. Given V and V as above, we prove that for every V : R d → [0 , + ∞ ] with (cid:82) R d V − p dx = 1,we have λ (min { V , V } ) ≤ λ ( V ) . Indeed, let u be the second eigenfunction of − ∆ + V . We first suppose that u changes signon R d and consider the functions V + = sup { V, ∞ { u ≤ } } and V − = sup { V, ∞ { u ≥ } } where, forany measurable A ⊂ R d , we set ∞ A ( x ) = (cid:40) + ∞ , x ∈ A, , x / ∈ A. We note that 1 = (cid:90) R d V − p dx = (cid:90) R d V − p + dx + (cid:90) R d V − p − dx. Moreover, on the sets { u > } and { u < } , the following equations are satisfied: − ∆ u +2 + V + u +2 = λ ( V ) u +2 , − ∆ u − + V − u − = λ ( V ) u − , and so, multiplying respectively by u +2 and u − , we obtain that λ ( V ) ≥ λ ( V + ) , λ ( V ) ≥ λ ( V − ) , (6.4) t61 where we have equalities if, and only if, u +2 and u − are the first eigenfunctions correspondingto λ ( V + ) and λ ( V − ). Let now (cid:101) V + and (cid:101) V − be optimal potentials for λ corresponding to theconstraints (cid:90) R d (cid:101) V − p + dx = (cid:90) R d V − p + dx, (cid:90) R d (cid:101) V − p − dx = (cid:90) R d V − p − dx. By Proposition 5.6, the sets of finiteness of (cid:101) V + and (cid:101) V − are compact, hence we may assume (upto translations) that they are also disjoint. By the monotonicity of λ , we havemax { λ ( V ) , λ ( V ) } ≤ max { λ ( (cid:101) V + ) , λ ( (cid:101) V − ) } , and so we obtain λ (min { V , V } ) ≤ max { λ ( (cid:101) V + ) , λ ( (cid:101) V − ) } ≤ max { λ ( V + ) , λ ( V − ) } ≤ λ ( V ) , as required. If u does not change sign, then we consider V + = sup { V, ∞ { u =0 } } and V − =sup { V, ∞ { u =0 } } , where u is the first eigenfunction of − ∆ + V . Then the claim follows by thesame argument as above. (cid:3) PTIMAL POTENTIALS FOR SCHR ¨ODINGER OPERATORS 23 For more general cost functionals F ( V ), the question if the optimization problemmin (cid:26) F ( V ) : V ≥ , (cid:90) R d V p dx ≤ (cid:27) admits a solution is, as far as we know, open. References [1] L. Ambrosio, N. Fusco, D. Pallara: Functions of Bounded Variation and Free Discontinuity Problems. Oxford University Press, Oxford (2000).[2] D. Bucur, G. Buttazzo: Variational Methods in Shape Optimization Problems. Progress in NonlinearDifferential Equations , Birkh¨auser Verlag, Basel (2005).[3] D. Bucur, G. Buttazzo: On the characterization of the compact embedding of Sobolev spaces. Calc. Var., (3) (2012), 455–475.[4] G. Buttazzo: Semicontinuity, relaxation and integral representation in the calculus of variations. PitmanResearch Notes in Mathematics , Longman, Harlow (1989).[5] G. Buttazzo: Spectral optimization problems. Rev. Mat. Complut., (2) (2011), 277–322.[6] G. Buttazzo, G. Dal Maso: Shape optimization for Dirichlet problems: relaxed formulation and opti-mality conditions. Appl. Math. Optim., (1991), 17–49.[7] G. Buttazzo, G. Dal Maso: An existence result for a class of shape optimization problems. Arch. RationalMech. Anal., (1993), 183–195.[8] G. Dal Maso: An Introduction to Γ -convergence. Birkh¨auser Verlag, Basel (1993).[9] G. Buttazzo, N. Varchon, H. Soubairi: Optimal measures for elliptic problems. Ann. Mat. Pura Appl., (2) (2006), 207–221.[10] G. Dal Maso, U. Mosco: Wiener’s criterion and Γ -convergence. Appl. Math. Optim., (1987), 15–63.[11] L. Evans: Partial Differential Equations. American Mathematical Society, Providence (1998).[12] A. Henrot: Minimization problems for eigenvalues of the Laplacian. J. Evol. Equ., (3) (2003), 443–461.[13] A. Henrot: Extremum Problems for Eigenvalues of Elliptic Operators. Frontiers in Mathematics,Birkh¨auser Verlag, Basel (2006).[14] A. Henrot, M. Pierre: Variation et Optimisation de Formes. Une Analyse G´eom´etrique. Math´ematiques& Applications , Springer-Verlag, Berlin (2005).[15] H. Lieb, M. Loss: Analysis. ,. Graduate Studies in Mathematics, AMS, 2000[16] J. Serrin, M. Tang: Uniqueness of ground states for quasilinear elliptic equations. Indiana Univ. Math.J., (3) (2000), 897-923.[17] H. Egnell: Extremal properties of the first eigenvalue of a class of elliptic eigenvalue problems. Ann. Sc.Norm. Super. Pisa Cl. Sci., (1) (1987), 1-48.[18] L. Evans, R. Gariepy: Measure Theory and Fine Properties of Functions. Studies in Advanced mathe-matics, Crc Press, Boca Raton (1991).[19] D. Gilbarg, N.S. Trudinger:: Elliptic partial differential equations of second order. Reprint of the 1998edition, Classics in Mathematics, Springer-Verlag, Berlin (2001). Dipartimento di Matematica, Universit`a di Pisa, Largo B. Pontecorvo 5, 56126 Pisa, ITALY E-mail address : [email protected] Dipartimento di Matematica, Universit`a di Pisa, Largo B. Pontecorvo 5, 56126 Pisa, ITALY E-mail address : [email protected] Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, ITALY E-mail address : [email protected] Scuola Normale Superiore di Pisa, Piazza dei Cavalieri 7, 56126 Pisa, ITALY E-mail address ::