Optimal potentials for Schrödinger operators

& Bozhidar Velichkov

Optimal potentials for Schrödinger operators Tome 1 (2014), p. 71-100.

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OPTIMAL POTENTIALS FOR SCHRÖDINGER OPERATORS

by Giuseppe Buttazzo, Augusto Gerolin, Berardo Ruffini

& Bozhidar Velichkov

Abstract. — We consider the Schrödinger operator−∆ +V(x)onH₀¹(Ω), whereΩis a given domain of R^d. Our goal is to study some optimization problems where an optimal potential V >0has to be determined in some suitable admissible classes and for some suitable optimiza- tion criteria, like the energy or the Dirichlet eigenvalues.

Résumé(Potentiels optimaux pour les opérateurs de Schrödinger). — Nous considérons l’opé- rateur de Schrödinger−∆ +V(x)surH₀¹(Ω), oùΩest un domaine fixé deR^d. Nous étudions certains problèmes d’optimisation pour lesquels un potentiel optimalV >0doit être déterminé dans une certaine classe admissible et pour certains critères d’optimisation tels que l’énergie ou les valeurs propres de Dirichlet.

Contents

1. Introduction. . . 71

2. Capacitary measures andγ-convergence. . . 75

3. Existence of optimal potentials inL^p(Ω). . . 80

4. Existence of optimal potentials for unbounded constraints. . . 87

5. Optimization problems in unbounded domains. . . 89

6. Further remarks on the optimal potentials for spectral functionals. . . 97

References. . . 99

1. Introduction

In this paper we consider optimization problems of the form

(1.1) min

F(V) : V ∈ V ,

for functionalsF, depending on the Schrödinger operator−∆+V(x)withpotentialV belonging to a prescribed admissible classV of Lebesgue measurable functions on a

Mathematical subject classification (2010). — 49J45, 35J10, 49R05, 35P15, 35J05.

Keywords. — Schrödinger operators, optimal potentials, spectral optimization, capacity.

This work is part of the project 2010A2TFX2 “Calcolo delle Variazioni”funded by the Italian Ministry of Research and University. The first, second and third authors are members of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM).

set Ω⊂R^d, which is typically chosen to be a bounded open set or the entire space Ω =R^d. Problems of this type have been studied, for example, by Ashbaugh-Harrell [2], Egnell [15], Essen [16], Harrell [20], Talenti [24] and, more recently, by Carlen- Frank-Lieb [12]. We refer to the monograph [22], and to the references therein, for a complete list of references and as a comprehensive guide to the known results about the problem.

In our framework we include very general cost functionals, as for example the following.

Integral functionals. — Given a function f ∈L²(Ω) we consider the solution uV to the elliptic PDE

−∆u+V u=f inΩ, u∈H₀¹(Ω).

The integral cost functionals we may consider are of the form F(V) =

Z

Ω

j x, uV(x),∇uV(x) dx,

where j is a suitable integrand that we assume convex in the gradient variable and bounded from below. One may take, for example,

j(x, s, z)>−a(x)−c|s|²,

witha∈L¹(Ω) andcsmaller than the first Dirichlet eigenvalue of the Laplace oper- ator−∆ inΩ. In particular, the energyE_f(V)defined by

(1.2) E_f(V) = inf Z

Ω

1

2|∇u|²+1

2V(x)u²−f(x)u

dx: u∈H₀¹(Ω)

, belongs to this class since, integrating by parts its Euler-Lagrange equation, we have

E_f(V) =−1 2

Z

Ω

f(x)uV dx, which corresponds to the integral functional above with

j(x, s, z) =−1 2f(x)s.

Spectral functionals. — For every admissible potentialV >0 we consider the spec- trumΛ(V)of the Schrödinger operator−∆ +V(x)onH₀¹(Ω). IfΩis bounded or has finite measure, or if the potential V satisfies some suitable integrability properties, then the operator −∆ +V(x) has compact resolvent and so its spectrum Λ(V) is discrete:

Λ(V) = λ1(V), λ2(V), . . . ,

where λk(V) are the eigenvalues counted with their multiplicity. The spectral cost functionals we may consider are of the form

F(V) = Φ Λ(V) ,

for suitable functionsΦ :R^N→(−∞,+∞]. For instance, takingΦ(Λ) =λkwe obtain F(V) =λk(V).

Theclass of admissible potentialsV we consider satisfies an integrability condition, namely

(1.3) V =n

V : Ω→[0,+∞] : V Lebesgue measurable, Z

Ω

Ψ(V)dx61o , for a suitable functionΨ : [0,+∞]→[0,+∞]. It is worth remarking that the require- ment V > 0 is not, in general, necessary for the well-posedness of problem (1.1), but allowing V to change sign radically changes the conduct of the problem. An in- stance of optimization problem for sign-changing potentials can be found in the recent work [12], where the authors study a quantitative stability for the first eigenvalue of the Schrödinger operator. The integrability constraint in (1.3) naturally appears in the following cases.

Approximation of optimal sets. — In the case of spectral and energy functionals F as above, the optimization problems related to the Schrödinger operators may be linked to the classical shape optimization theory⁽¹⁾for problems of the form

min

F(E) : E⊂Ω, |E|6constant .

Indeed, if we setVE= 0inEandVE= +∞outside ofE, then the Schrödinger oper- ator−∆ +VE corresponds to the Dirichlet-Laplacian on the setE. This observation suggests, by one side, that we can approach problem (1.1) by means of techniques developed in the study of more classical shape optimization problems and, on the other hand, that we can approximate the potentialVE, corresponding to an optimal setE, by means of potentials that solve (1.1) under suitable constraints. We will show in Section 5 that a good approximation is given by the family of constraints

Ψ(V) =e^−αV.

Ground states of semilinear equations. — If the cost functional F is of energy type, as F(V) =λ1(V), then the study of the optimization problem

minn

F(V) : V : Ω−→[0,+∞], Z

Ω

V^pdx= 1o naturally reduces to the one of ground states of the equation (1.4) −∆ψ+|ψ|^sψ=λψ, ψ∈H¹(Ω)∩L^2+s(Ω).

The case p > 0 corresponds to the superlinear case s > 0, while the case of nega- tive exponent p < 0 corresponds to the sublinear cases < 0. Indeed, the potential V(x) =|ψ(x)|^ssatisfies an integrability condition inherited from the ground stateψ.

In the superlinear case s > 0, we have V ∈L^p(Ω) with p= (s+ 2)/s, while in the sublinear cases∈(−1,0)we getR

ΩV^−pdx <+∞withp=−(s+ 2)/s.

The paper is organized as follows. In Section 2 we recall the concepts of capacitary measures andγ-convergence together with their main properties. Then we prove some preliminary results which will be exploited in the subsequent sections.

(1)For an introduction to the theory of shape optimization problems we refer to the papers [8], [9], [10] and to the books [4], [22] and [23].

In Section 3 we prove two general results concerning the existence of optimal poten- tials in a bounded domainΩ⊂R^d. In Theorem 3.1 we deal with constraintsV which are bounded subsets of L^p(Ω), while Theorem 3.4 deals with the case of admissible classes consisting of suitable subsets of capacitary measures.

In Section 3 our assumptions allow to takeF(V) =−E_f(V)and thus the optimiza- tion problem becomes the maximization ofEf under the constraintR

ΩV^pdx61. We prove that forp>1, there exists an optimal potential for the problem

maxn E_f(V) :

Z

Ω

V^pdx61o .

The existence result is sharp in the sense that for p < 1 the maximum cannot be achieved (see Remark 3.11). For the existence issue in the case of a bounded domain, we follow the ideas of Egnell [15], summarized in [22, Chapter 8]. The case p= 1 is particularly interesting and we show that in this case the optimal potentials are of the form

V = f

M χω+−χω−

,

whereχUindicates the characteristic function of the setU,f∈L²(Ω),M=kuVk_L^∞(Ω), andω± ={u=±M}.

In Section 4 we deal with minimization problems of the form

(1.5) minn

F(V) : Z

Ω

Ψ(V)dx61o ,

and we prove existence for the problem (1.1) for a large class of functionalsF and of constraintsΨ, including the particular cases

Ψ(s) =s^−p and Ψ(s) =e^−αs.

These type of constraints are, as far as we know, new in the literature. In the case Ψ(s) =s^−pthe equation reduces, as already pointed out, to the sublinear case of (1.4).

In some cases the Schrödinger operator −∆ +V(x) is compact even if Ω is not bounded (see for instance [5]). This allows to consider spectral optimization problems in unbounded domains as Ω = R^d. We deal with this case in Section 5, where we prove that for F = Ef or F = λ1, there exist solutions to problem (1.5) in R^d, with Ψ(s) = s^−p. Moreover, we characterize the optimal potential V as an explicit function of the solutionuto a quasi-linear PDE of the form (1.4). Thus the qualitative properties of u immediately translate into qualitative properties for V. Thanks to this, we prove that, in the caseF =Ef,1/V is compactly supported, providedf is compactly supported. In the caseF =λ1the same holds and the optimal potentialV is an (explicit) function of the optimizers of a family of Gagliardo-Nirenberg-Sobolev inequalities (see Remark 5.7).

In the final Section 6 we make some further remarks about the state of the art of spectral optimization for Schrödinger operators on unbounded domains, and we apply the results of Section 5 to get, in Theorem 6.1, the qualitative behavior of the optimal potential forF =λ2 for problem (1.5) withΨ(s) =s^−p.

2. Capacitary measures andγ-convergence For a subsetE⊂R^d itscapacityis defined by

cap(E) = infnZ

R^d

|∇u|²dx+ Z

R^d

u²dx: u∈H¹(R^d), u>1in a neighborhood ofEo . If a property P(x) holds for all x ∈ Ω, except for the elements of a set E ⊂ Ω of capacity zero, we say thatP(x)holdsquasi-everywhere(shortlyq.e.) inΩ, whereas the expressionalmost everywhere(shortlya.e.) refers, as usual, to the Lebesgue measure, which we often denote by | · |.

A subsetA of R^d is said to be quasi-open if for every ε >0 there exists an open subset Aε of R^d, with A ⊂ Aε, such that cap(AεrA) < ε. Similarly, a function u:R^d →Ris said to bequasi-continuous (respectivelyquasi-lower semicontinuous) if there exists a decreasing sequence of open sets (An)n such that cap(An) → 0 and the restrictionun ofuto the complement A^c_n ofAn is continuous (respectively lower semicontinuous). It is well known (see for instance [18]) that every function u∈H¹(R^d)has a quasi-continuous representativeu, which is uniquely defined up toe a set of capacity zero, and given by

u(x) = lime

ε→0

1

|Bε(x)|

Z

Bε(x)

u(y)dy ,

whereBε(x)denotes the ball of radiusεcentered atx. We identify the (a.e.) equiva- lence classu∈H¹(R^d)with the (q.e.) equivalence class of quasi-continuous represen- tativesu.e

We denote byM⁺(R^d)the set of positive Borel measures on R^d (not necessarily finite or Radon) and byM⁺_cap(R^d)⊂ M⁺(R^d)the set ofcapacitary measures, i.e. the measures µ∈ M⁺(R^d)such that µ(E) = 0for any set E⊂R^d of capacity zero. We note that when µis a capacitary measure, the integralR

R^d|u|²dµis well-defined for eachu∈H¹(R^d), i.e. ifue1andeu2are two quasi-continuous representatives ofu, then R

R^d|ue1|²dµ=R

R^d|eu2|²dµ.

For a subsetΩ⊂R^d, we define the Sobolev spaceH₀¹(Ω)as H₀¹(Ω) =

u∈H¹(R^d) : u= 0q.e. onΩ^c . Alternatively, by using the capacitary measureIΩdefined as (2.1) IΩ(E) =

(0 if cap(ErΩ) = 0

+∞ if cap(ErΩ)>0 for every Borel setE⊂R^d, the Sobolev spaceH₀¹(Ω)can be defined as

H0¹(Ω) =n

u∈H¹(R^d) : Z

R^d

|u|²dIΩ<+∞o .

More generally, for any capacitary measureµ∈ M⁺_cap(R^d), we define the space H_µ¹=n

u∈H¹(R^d) : Z

R^d

|u|²dµ <+∞o ,

which is a Hilbert space when endowed with the normkuk1,µ, where kuk²_1,µ=

Z

R^d

|∇u|²dx+ Z

R^d

u²dx+ Z

R^d

u²dµ.

Ifu /∈H_µ¹, then we setkuk1,µ= +∞.

ForΩ⊂R^d, we defineM⁺_cap(Ω)as the space of capacitary measuresµ∈ M⁺_cap(R^d) such that µ(E) = +∞ for any set E ⊂ R^d such that cap(E rΩ) > 0. For µ∈ M⁺_cap(R^d), we denote with H_µ¹(Ω)the spaceH_µ∨I¹ _Ω =H_µ¹∩H₀¹(Ω).

Definition2.1. — Given a metric space(X, d)and sequence of functionalsJn:X → R∪ {+∞}, we say that Jn Γ-converges to the functionalJ :X →R∪ {+∞}, if the following two conditions are satisfied:

(a) for every sequencexn converging tox∈X, we have J(x)6lim inf

n→∞ Jn(xn);

(b) for everyx∈X, there exists a sequencexn converging tox, such that J(x) = lim

n→∞Jn(xn).

For all details and properties of Γ-convergence we refer to [13]; here we simply recall that, wheneverJn Γ-converges toJ,

x∈XminJ(x)6lim inf

n→∞ min

x∈XJn(x).

Definition2.2. — We say that the sequence of capacitary measuresµn ∈ M⁺_cap(Ω), γ-converges to the capacitary measure µ ∈ M⁺_cap(Ω) if the sequence of functionals k·k_1,µnΓ-converges to the functionalk·k_1,µinL²(Ω), i.e. if the following two conditions are satisfied:

• for every sequenceun→uinL²(Ω)we have Z

R^d

|∇u|²dx+ Z

R^d

u²dµ6lim inf

n→∞

Z

R^d

|∇un|²dx+ Z

R^d

u²_ndµn

;

• for everyu∈L²(Ω), there existsun→uin L²(Ω)such that Z

R^d

|∇u|²dx+ Z

R^d

u²dµ= lim

n→∞

Z

R^d

|∇un|²dx+ Z

R^d

u²_ndµn

.

Ifµ∈ M⁺_cap(Ω)andf∈L²(Ω)we define the functionalJµ(f,·) :L²(Ω)→R∪{+∞}

by

(2.2) Jµ(f, u) = 1 2

Z

Ω

|∇u|²dx+1 2 Z

Ω

u²dµ− Z

Ω

f u dx.

IfΩ⊂R^dis a bounded open set,µ∈ M⁺_cap(Ω)andf ∈L²(Ω), then the functional Jµ(f,·)has a unique minimizeru∈H_µ¹ that verifies the PDE formally written as

−∆u+µu=f, u∈H_µ¹(Ω),

and whose precise meaning is given in the weak form





 Z

Ω

∇u· ∇ϕ dx+ Z

Ω

uϕ dµ= Z

Ω

f ϕ dx, ∀ϕ∈H_µ¹(Ω), u∈H_µ¹(Ω).

The resolvent operator of −∆ +µ, that is the map R_µ that associates to every f ∈ L²(Ω) the solution u∈H_µ¹(Ω) ⊂L²(Ω), is a compact linear operator in L²(Ω) and so, it has a discrete spectrum

0<· · ·6Λk6· · ·6Λ26Λ1.

Their inverses 1/Λk are denoted by λk(µ) and are the eigenvalues of the operator

−∆ +µ.

In the casef = 1the solution will be denoted bywµ and whenµ=IΩwe will use the notationwΩinstead ofwIΩ. We also recall (see [4]) that ifΩis bounded, then the strongL²-convergence of the minimizerswµntowµis equivalent to theγ-convergence of Definition 2.2.

Remark 2.3. — An important well-known characterization of the γ-convergence is the following: a sequenceµnγ-converges toµ, if and only if, the sequence of resolvent operatorsRµn associated to−∆ +µn, converges (in the strong convergence of linear operators onL²) to the resolventR_µ of the operator−∆ +µ. A consequence of this fact is that the spectrum of the operator −∆ +µn converges (pointwise) to the one of−∆ +µ.

Remark2.4. — The space M⁺_cap(Ω) endowed with theγ-convergence is metrizable.

IfΩis bounded, one may takedγ(µ, ν) =kwµ−wνk_L2. Moreover, in this case, in [14]

it is proved that the spaceM⁺_cap(Ω) endowed with the metricdγ is compact.

Proposition2.5. — Let Ω⊂R^d and letVn∈L¹(Ω) be a sequence weakly converging inL¹(Ω) to a function V. Then the capacitary measuresVndx γ-converge to V dx.

Proof. — We have to prove that the solutionsun=RVn(1)to (−∆un+Vn(x)un= 1

u∈H0¹(Ω)

weakly converge inH₀¹(Ω) to the solutionu=RV(1) to (−∆u+V(x)u= 1

u∈H0¹(Ω), or equivalently that the functionals

Jn(u) = Z

Ω

|∇u|²dx+ Z

Ω

Vn(x)u²dx Γ-converge inL²(Ω)to the functional

J(u) = Z

Ω

|∇u|²dx+ Z

Ω

V(x)u²dx.

TheΓ-liminf inequality (Definition 2.1 (a)) is immediate since, ifun →uin L²(Ω), we have

Z

Ω

|∇u|²dx6lim inf

n→∞

Z

Ω

|∇un|²dx

by the lower semicontinuity of theH¹(Ω)norm with respect to theL²(Ω)-convergence,

and Z

Ω

V(x)u²dx6lim inf

n→∞

Z

Ω

Vn(x)u²_ndx

by the strong-weak lower semicontinuity theorem for integral functionals (see for instance [7]).

Let us now prove theΓ-limsup inequality (Definition 2.1 (b)) which consists, given u∈H₀¹(Ω), in constructing a sequenceun→uinL²(Ω)such that

(2.3) lim sup

n→∞

Z

Ω

|∇un|²dx+ Z

Ω

Vn(x)u²_ndx6 Z

Ω

|∇u|²dx+ Z

Ω

V(x)u²dx.

For every t > 0 let u^t = (u∧t)∨(−t); then, by the weak convergence of Vn, fort fixed we have

n→∞lim Z

Ω

Vn(x)|u^t|²dx= Z

Ω

V(x)|u^t|²dx, and

t→+∞lim Z

Ω

V(x)|u^t|²dx= Z

Ω

V(x)|u|²dx.

Then, by a diagonal argument, we can find a sequencetn →+∞such that

n→∞lim Z

Ω

Vn(x)|u^tn|²dx= Z

Ω

V(x)|u|²dx.

Taking nowun=u^tn, and noticing that for everyt >0 Z

Ω

|∇u^t|²dx6 Z

Ω

|∇u|²dx,

we obtain (2.3) and so the proof is complete.

In the case of weak* convergence of measures the statement of Proposition 2.5 is no longer true, as the following proposition shows.

Proposition 2.6. — Let Ω ⊂ R^d (d > 2) be a bounded open set and let V, W be two functions in the class L¹+(Ω) of nonnegative integrable functions on Ωsuch that V >W. Then, there exists a sequenceVn∈L¹₊(Ω), uniformly bounded inL¹(Ω), such that the sequence of measuresVn(x)dxconverges weakly* toV(x)dxandγ-converges toW(x)dx.

Proof⁽²⁾. — Without loss of generality we can suppose R

Ω(V −W)dx = 1. Let µn

be a sequence of probability measures on Ωweakly* converging to(V −W)dx and such that eachµn is a finite sum of Dirac masses. For eachn∈Nconsider a sequence of positive functions Vn,m ∈L¹(Ω) such that R

ΩVn,mdx = 1and Vn,mdx converges

(2)The idea of this proof was suggested by Dorin Bucur.

weakly* to µn as m → ∞. Moreover, we choose Vn,m as a convex combination of functions of the form|B1/m|⁻¹χB_1/m(xj).

We now prove that for fixed n ∈ N, (Vn,m+W)dx γ-converges, as m → ∞, to W dx or, equivalently, that the sequence wW+Vn,m converges in L² to wW, as m→ ∞. Indeed, by the weak maximum principle, we have

wW+I_Ωm,n 6wW+V_n,m 6wW, whereΩm,n= Ωr∪jB1/m(xj)andIΩm,n is as in (2.1).

Since a point has zero capacity in R^d (d > 2) there exists a sequence φm → 0 strongly inH¹(R^d)withφm= 1onB1/m(0)andφm= 0outsideB_1/^√_m(0). We have

Z

Ω

|wW −wW+IΩm,n|²dx62kwWk_L^∞ Z

Ω

(wW −wW+IΩm,n)dx

= 4kwWk_L^∞ E(W +IΩ_m,n)−E(W) 64kwWk_L^∞Z

Ω

1

2|∇wm|²+1

2W w²_m−wmdx

− Z

Ω

1

2|∇wW|²+1

2W w²_W −wWdx , (2.4)

wherewmis any function in ∈H₀¹(Ωm,n). Taking wm(x) =wW(x)Y

j

1−φm(x−xj) ,

sinceφm→0strongly inH¹(R^d), it is easy to see thatwm→wW strongly inH¹(Ω) and so, by (2.4),wW+IΩm,n →wW inL²(Ω)asm→ ∞. Since the weak convergence of probability measures and theγ-convergence are both induced by metrics, a diagonal

sequence argument brings to the conclusion.

Remark2.7. — Whend= 1, a result analogous to Proposition 2.5 is that any sequence (µn)weakly* converging toµ is alsoγ-converging toµ. This is an easy consequence of the compact embedding ofH0¹(Ω)into the space of continuous functions onΩ.

We note that the hypothesis V > W in Proposition 2.6 is necessary. Indeed, we have the following proposition, whose proof is contained in [11, Theorem 3.1] and we report it here for the sake of completeness.

Proposition2.8. — Let µn ∈ M⁺_cap(Ω) be a sequence of capacitary and Radon mea- sures weakly* converging to the measureνandγ-converging to the capacitary measure µ∈ M⁺_cap(Ω). Thenµ6ν inΩ.

Proof. — We note that it is enough to show thatµ(K)6ν(K)wheneverK⊂⊂Ωis a compact set. Let ube a nonnegative smooth function with compact support in Ω such thatu61in Ωandu= 1onK; we have

µ(K)6 Z

Ω

u²dµ6lim inf

n→∞

Z

Ω

u²dµn= Z

Ω

u²dν6ν({u >0}).

Sinceuis arbitrary, we have the conclusion by the Borel regularity ofν.

3. Existence of optimal potentials inL^p(Ω) In this section we consider the optimization problem

(3.1) minn

F(V) : V : Ω→[0,+∞], Z

Ω

V^pdx61o ,

wherep >0andF(V)is a cost functional acting on Schrödinger potentials, or more generally on capacitary measures. Typically,F(V)is the minimum of some functional JV : H₀¹(Ω) → R depending on V. A natural assumption in this case is the lower semicontinuity of the functionalF with respect to the γ-convergence, that is

F(µ)6lim inf

n→∞ F(µn), wheneverµn−→_γµ.

Theorem 3.1. — Let F : L¹₊(Ω) → R be a functional, lower semicontinuous with respect to the γ-convergence, and let V be a weakly L¹(Ω) compact set. Then the problem

min{F(V) : V ∈ V}, admits a solution.

Proof. — Let(Vn)be a minimizing sequence in V. By the compactness assumption onV, we may assume thatVntends weaklyL¹(Ω)to someV ∈ V. By Proposition 2.5, we have thatVn γ-converges to V and so, by the semicontinuity ofF,

F(V)6lim inf

n→∞ F(Vn),

which gives the conclusion.

Remark3.2. — Theorem 3.1 applies for instance to the integral functionals and to the spectral functionals considered in the introduction; it is not difficult to show that they are lower semicontinuous with respect to theγ-convergence.

Remark3.3. — In some special cases the solution to (3.1) can be written explicitly in terms of the solution to some partial differential equation onΩ. This is the case of the Dirichlet Energy (see Propositions 3.6 and 3.9), and of the first eigenvalue of the Dirichlet Laplacianλ1(see [21, Chapter 8]).

The compactness assumption on the admissible class V for the weak L¹(Ω) con- vergence in Theorem 3.1 is for instance satisfied if Ωhas finite measure and V is a convex closed and bounded subset ofL^p(Ω), with p >1. WhenV is only bounded in L¹(Ω)Theorem 3.1 does not apply, since minimizing sequences may weakly* converge to a measure. It is then convenient to extend our analysis to the case of functionals defined on capacitary measures, in which a result analogous to Theorem 3.1 holds.

Theorem3.4. — Let Ω⊂R^d be a bounded open set and letF :M⁺_cap(Ω) →Rbe a functional lower semicontinuous with respect to theγ-convergence. Then the problem

(3.2) min

F(µ) : µ∈ M⁺_cap(Ω), µ(Ω)61 , admits a solution.

Proof. — Let(µn)be a minimizing sequence. Then, up to a subsequenceµnconverges weakly* to some measure ν and γ-converges to some measure µ ∈ M⁺_cap(Ω). By Proposition 2.8, we have thatµ(Ω)6ν(Ω)61and so,µis a solution to (3.2).

We notice that, since the class of Schrödinger potentials is dense, with respect to the γ-convergence, in the class M⁺_cap(Ω) of capacitary measures (see [14]), the minimum in (3.2) coincides with

infn

F(V) : V >0, Z

Ω

V dx61o wheneverF is a γ-continuous cost functional.

The following example shows that the optimal solution to problem (3.2) is not, in general, a function V(x), even when the optimization criterion is the energy E_f introduced in (1.2). On the other hand, an explicit form for the optimal potentialV(x) will be provided in Proposition 3.9 assuming that the right-hand sidef is inL²(Ω).

Example3.5. — LetΩ = (−1,1) and consider the functional F(µ) =−min

1 2

Z 1

−1

|u⁰|²dx+1 2

Z 1

−1

u²dµ−u(0) : u∈H₀¹(−1,1)

. Then, for anyµsuch thatµ(Ω)61, we have

(3.3) F(µ)>−min 1

2 Z 1

−1

|u⁰|²dx+1 2 sup

(−1,1)

u²

−u(0) : u∈H₀¹(−1,1), u>0

. By a symmetrization argument, the minimizer u of the right-hand side of (3.3) is radially decreasing; moreover,uis linear on the setu < M, whereM = supu, and so it is of the form

u(x) =













 M

1−αx+ M

1−α, x∈[−1,−α],

M, x∈[−α, α],

− M

1−αx+ M

1−α, x∈[α,1],

for some α ∈ [0,1]. A straightforward computation gives α = 0 and M = 1/3.

Thus,uis also the minimizer of F(δ0) =−min

1 2

Z 1

−1

|u⁰|²dx+1

2u(0)²−u(0) : u∈H₀¹(−1,1)

, and soδ0 is the solution to

min{F(µ) : µ(Ω)61}.

In the rest of this section we consider the particular caseF(V) =−E_f(V), in which we can identify the optimal potential through the solution to a nonlinear PDE. Let Ω⊂R^d be a bounded open set and letf ∈L²(Ω). By Theorem 3.1, the problem (3.4) min{−E_f(V) : V ∈ V} with V=n

V >0, Z

Ω

V^pdx61o ,

admits a solution, where E_f(V) is the energy functional defined in (1.2). We no- tice that, replacing −Ef(V) by Ef(V), makes problem (3.4) trivial, with the only solution V ≡ 0. Minimization problems for Ef will be considered in Section 4 for admissible classes of the form

V =n V >0,

Z

Ω

V^−pdx61o .

Analogous results forF(V) =−λ1(V)were proved in [21, Theorem 8.2.3].

Proposition3.6. — Let Ω⊂R^d be a bounded open set, 1< p <∞ andf ∈L²(Ω).

Then the problem (3.4)has a unique solution Vp=

Z

Ω

|up|^2p/(p−1)dx −1/p

|up|^2/(p−1), whereup∈H₀¹(Ω)∩L^2p/(p−1)(Ω) is the minimizer of the functional (3.5) Jp(u) := 1

2 Z

Ω

|∇u|²dx+1 2

Z

Ω

|u|^2p/(p−1)dx

(p−1)/p

− Z

Ω

uf dx.

Moreover, we have E_f(Vp) =Jp(up).

Proof. — We first note that we have (3.6) max

V∈V min

u∈H₀¹(Ω)

Z

Ω

1

2|∇u|²+1

2u²V −uf dx

6 min

u∈H₀¹(Ω)max

V∈V

Z

Ω

1

2|∇u|²+1

2u²V −uf dx, where the maximums are taken over all positive functionsV∈L^p(Ω)withR

ΩV^pdx61.

For a fixed u ∈ H₀¹(Ω), the maximum on the right-hand side (if finite) is achieved for a function V such thatΛpV^p−1 =u², where Λ is a Lagrange multiplier. By the conditionR

ΩV^pdx= 1 we obtain that the maximum is achieved for V =

Z

Ω

|u|^2p/(p−1)dx −1/p

|u|^2/(p−1). Substituting in (3.6), we obtain

max{E_f(V) : V ∈ V}6min

Jp(u) : u∈H₀¹(Ω) .

Letun be a minimizing sequence forJp. Sinceinf Jp60, we can assumeJp(un)60 for eachn∈N. Thus, we have

(3.7) 1 2 Z

Ω

|∇un|²dx+1 2

Z

Ω

|un|^2p/(p−1)dx

^(p−1)/p

6 Z

Ω

unf dx6Ckfk_L2(Ω)k∇unk_L2, whereC is a constant depending onΩ. Thus we obtain

(3.8)

Z

Ω

|∇un|²dx+ Z

Ω

|un|^2p/(p−1)dx

(p−1)/p

64C²kfk²_L2(Ω),

and so, up to subsequenceun converges weakly inH₀¹(Ω) and L^2p/(p−1)(Ω) to some up ∈ H₀¹(Ω)∩L^2p/(p−1)(Ω). By the semicontinuity of the L²-norm of the gradient and the L^2p/(p−1)-norm and the fact that R

Ωf undx → R

Ωf updx, as n → ∞, we have that up is a minimizer of Jp. By the strict convexity of Jp, we have that up is unique. Moreover, by (3.7) and (3.8),Jp(up)>−∞. Writing down the Euler-Lagrange equation forup, we obtain

−∆up+ Z

Ω

|up|^2p/(p−1)dx −1/p

|up|^2/(p−1)up=f.

Setting

Vp= Z

Ω

|up|^2p/(p−1)dx −1/p

|up|^2/(p−1), we have thatR

ΩV_p^pdx= 1andup is the solution to

−∆up+Vpup=f.

In particular, we have Jp(up) =Ef(Vp)and soVp solves (3.4). The uniqueness ofVp

follows by the uniqueness ofupand the equality case in the Hölder inequality Z

Ω

u²V dx6 Z

Ω

V^pdx

1/pZ

Ω

|u|^2p/(p−1)dx

(p−1)/p

6 Z

Ω

|u|^2p/(p−1)dx

(p−1)/p

.

When the functionalF is−E_f, then the existence result holds also in the casep= 1.

Before we give the proof of this fact in Proposition 3.9, we need some preliminary results. We also note that the analogous results were obtained in the case F =−λ1

(see [21, Theorem 8.2.4]) and in the case F = −E_f, where f is a positive function (see [11]).

Remark3.7. — Letupbe the minimizer ofJp, defined in (3.5). By (3.8), we have the estimate

k∇upk_L2(Ω)+kupk_L2p/(p−1)(Ω)62√

2Ckfk_L2(Ω),

where C is the constant from (3.7). Moreover, we have up ∈ H_loc² (Ω) and for each open set Ω⁰ ⊂⊂Ω, there is a constantC not depending onpsuch that

kupk_H²(Ω⁰)6C(f,Ω⁰).

Indeed,up satisfies the PDE

(3.9) −∆u+c|u|^αu=f,

with c > 0 and α = 2/(p−1), and standard elliptic regularity arguments (see [17, Section 6.3]) give thatu∈H_loc² (Ω). To show thatkupk_H²(Ω⁰)is bounded independently ofpwe apply the Nirenberg operator∂_k^hu=^u(x+he_h^k)−u(x)on both sides of (3.9), and

multiplying byφ²∂^h_ku, whereφis an appropriate cut-off function which equals1onΩ⁰, we have

Z

Ω

φ²|∇∂_k^hu|²dx+ Z

Ω

∇(∂_k^hu)∇(φ²)∂^h_ku dx+c(α+ 1) Z

Ω

φ²|u|^α|∂_k^hu|²dx

=− Z

f ∂_k^h(φ²∂_k^hu)dx, for allk= 1, . . . , d. Some straightforward manipulations now give

k∇²uk²_L2(Ω⁰)6

d

X

k=1

Z

Ω

φ²|∇∂ku|²dx6C(Ω⁰) kfk_L2({φ²>0})+k∇ukL²(Ω)

.

Lemma3.8. — Let Ω ⊂R^d be a bounded open set and let f ∈ L²(Ω). Consider the functionalJ1:L²(Ω)→Rdefined by

(3.10) J1(u) := 1

2 Z

Ω

|∇u|²dx+1

2kuk²_∞− Z

Ω

uf dx.

Then, Jp Γ-converges in L²(Ω)toJ1, asp→1, whereJp is defined in (3.5).

Proof. — Let vn ∈ L²(Ω) be a sequence of positive functions converging in L² to v∈L²(Ω)and letαn→+∞. Then, we have that

(3.11) kvkL^∞(Ω)6lim inf

n→∞ kvnk_Lαn(Ω).

In fact, suppose first that kvkL^∞ =M <+∞and let ωε ={v > M −ε}, for some ε >0. Then, we have

lim inf

n→∞ kvnk_Lαn(Ω)> lim

n→∞|ωε|^{(1−αn)/αn} Z

ωε

vndx=|ωε|⁻¹ Z

ωε

v dx>M−ε, and so, lettingε →0, we have lim infn→∞kvnk_Lαn(Ω) >M. If kvkL^∞ = +∞, then settingωk ={v > k}, for anyk>1, and arguing as above, we obtain (3.11).

Letun →uin L²(Ω). Then, by the semicontinuity of theL² norm of the gradient and (3.11) and the continuity of the termR

Ωuf dx, we have J1(u)6lim inf

n→∞ Jpn(un),

for any decreasing sequence pn → 1. On the other hand, for any u ∈ L², we have Jpn(u)→J1(u)as n→ ∞and so, we have the conclusion.

Proposition3.9. — LetΩ⊂R^d be a bounded open set andf ∈L²(Ω). Then there is a unique solution to problem (3.4)with p= 1, given by

V1= 1

M χω+f−χω−f ,

whereM=ku1k_L^∞_(Ω),ω+={u1=M},ω−={u1=−M}, beingu1∈H₀¹(Ω)∩L^∞(Ω) the unique minimizer of the functional J1, defined in (3.10). In particular, R

ω+f dx−R

ω−f dx=M,f >0on ω+ andf 60 onω−.

Proof. — For anyu∈H₀¹(Ω)and any V >0withR

ΩV dx61we have Z

Ω

u²V dx6kuk²_∞ Z

Ω

V dx6kuk²_∞,

where for sake of simplicity, we write k · k∞ instead of k · k_L^∞(Ω). Arguing as in the proof of Proposition 3.6, we obtain the inequalities

1 2

Z

Ω

|∇u|²dx+1 2

Z

Ω

u²V dx− Z

Ω

uf dx6J1(u), maxn

E_f(V) : Z

Ω

V 61o

6min

J1(u) : u∈H₀¹(Ω) . As in (3.7), we have that a minimizing sequence ofJ1 is bounded inH₀¹(Ω)∩L^∞(Ω) and thus by semicontinuity there is a minimizer u1 ∈ H₀¹(Ω)∩L^∞(Ω) of J1, which is also unique, by the strict convexity of J1. Let up denotes the minimizer of Jp as in Proposition 3.6. Then, by Remark 3.7, we have that the family up is bounded in H₀¹(Ω)and inH²(Ω⁰)for eachΩ⁰ ⊂⊂Ω. Then, we have that each sequenceupn has a subsequence converging weakly inL²(Ω)to someu∈H_loc² (Ω)∩H₀¹(Ω). By Lemma 3.8, we have u=u1and so,u1∈H_loc² (Ω)∩H0¹(Ω). Thusupn→u1 inL²(Ω).

Let us define M = ku1k_∞ and ω = ω+∪ω−. We claim that u1 satisfies, on Ω, the PDE

(3.12) −∆u+χωf =f.

Indeed, setting Ωt = Ω∩ {|u| < t} for t > 0, we compute the variation of J1 with respect to any function ϕ∈ H₀¹(ΩM−ε). Namely we consider functions of the form ϕ=ψwε where wε is the solution to−∆wε = 1 onΩM−ε, and wε= 0 on ∂ΩM−ε. Thus we obtain that−∆u1=f onΩM−ε and lettingε→0 we conclude, thanks to the Monotone Convergence Theorem, that

−∆u1=f onΩM = Ωrω.

Moreover, sinceu1∈H_loc² (Ω), we have that∆u1 = 0onω and so, we obtain (3.12).

Sinceu1is the minimizer ofJ1, we have that for eachε∈R,J1((1+ε)u1)−J1(u1)>0. Taking the derivative of this difference atε= 0, we obtain

Z

Ω

|∇u1|²dx+M²= Z

Ω

f u1dx.

By (3.12), we haveR

Ω|∇u1|²dx=R

Ωrωf u1dx and so M =

Z

ω+

f dx− Z

ω−

f dx.

Setting V1 := _M¹ χω+f−χω−f

, we have that R

ΩV1dx = 1, −∆u1+V1u1 =f in H⁻¹(Ω)and

J1(u1) =1 2

Z

Ω

|∇u1|²dx+1 2 Z

Ω

u²₁V1dx− Z

Ω

u1f dx.

We are left to prove thatV1is admissible, i.e.V1>0. To do this, considerwεthe energy function of the quasi-open set {u < M −ε} and letϕ =wεψ where ψ ∈ C_c^∞(R^d), ψ>0. Sinceϕ>0, we get that

06 lim

t→0⁺

J1(u1+tϕ)−J1(u1)

t =

Z

Ω

h∇u1,∇ϕidx− Z

Ω

f ϕ dx.

This inequality holds for anyψso that, integrating by parts, we obtain

−∆u1−f >0

almost everywhere on{u1< M−ε}. In particular, since∆u1= 0almost everywhere on ω− = {u = −M}, we obtain that f 6 0 on ω−. Arguing in the same way, and considering test functions supported on {u1 >−M +ε}, we can prove that f >0

onω+. This impliesV1>0 as required.

Remark3.10. — Under some additional assumptions onΩandf one can obtain some more precise regularity results foru1. In fact, in [15, Theorem A1] it was proved that if∂Ω∈C² and iff ∈L^∞(Ω)is positive, thenu1∈C^1,1(Ω).

Remark3.11. — In the casep <1problem (3.4) does not admit, in general, a solution, even for regularf andΩ. We give a counterexample in dimension one, which can be easily adapted to higher dimensions.

LetΩ = (0,1),f = 1, and letxn,k =k/nfor anyn∈Nandk= 1, . . . , n−1. We define the capacitary measuresµn=IΩrK_nwhereKn={k/n: k=k= 1, . . . , n−1}

and IΩrKn is defined in (2.1). Let wn be the minimizer of the functional Jµn(1,·), defined in (2.2). Thenwn vanishes atxn,k, fork= 1, . . . , n−1, and so we have

E(µn) =nmin (1

2 Z 1/n

0

|u⁰|²dx− Z 1/n

0

u dx: u∈H₀¹(0,1/n) )

=−C n², whereC >0is a constant.

For any fixed n and j, let V_jⁿ be the sequence of positive functions such that R1

0 |V_jⁿ|^pdx= 1, defined by (3.13) V_jⁿ=Cn

n−1

X

k=1

j^1/pχ[^kn−¹_j,_n^k+¹_j]<

n−1

X

k=1

I_Ωr[^kn−¹_j,^k_n+¹_j],

where Cn is a constant depending on n, and I is as in (2.1). By the compact- ness of the γ-convergence, we have that, up to a subsequence, V_jⁿdx γ-converges to some capacitary measureµ asj → ∞. On the other hand it is easy to check that Pn−1

k=1I_Ωr[n^k−¹_j,^k_n+¹_j] γ-converges toµn as j → ∞. By (3.13), we have thatµ6µn. In order to show thatµ =µn it is enough to check that each nonnegative function u∈H₀¹((0,1)), for whichR

u²dµ <+∞, vanishes atxn,k for k= 1, . . . , n−1. Sup- pose that u(k/n) > 0. By the definition of the γ-convergence, there is a sequence uj ∈H0¹(Ω) = H_V¹ⁿ

j (Ω) such thatuj →uweakly in H0¹(Ω) and R

u²_jV_jⁿdx 6C, for

some constantC not depending onj∈N. Sinceuj are uniformly 1/2-Hölder contin- uous, we can suppose thatuj >ε > 0on some interval Acontainingk/n. But then forj large enoughAcontains[k/n−1/j, k/n+ 1/j]so that

C>

Z 1 0

u²_jV_jⁿdx>

Z k/n+1/j k/n−1/j

u²_jV_jⁿdx>2Cnε²j^1/p−1,

which is a contradiction forp <1. Thus, we have thatµ=µn and soV_jⁿγ-converges to µn as j → ∞. In particular, E(µn) = limj→∞E1(V_jⁿ) and since the left-hand side converges to zero as n→ ∞, we can choose a diagonal sequence V_jⁿ_n such that E(V_jⁿ_n)→0asn→ ∞. Since there is no admissible functionalV such thatE₁(V) = 0, we have the conclusion.

4. Existence of optimal potentials for unbounded constraints In this section we consider the optimization problem

(4.1) min{F(V) : V ∈ V},

where V is an admissible class of nonnegative Borel functions on the bounded open setΩ⊂R^d andF is a cost functional on the family of capacitary measuresM⁺_cap(Ω).

The admissible classes we study depend on a functionΨ : [0,+∞]→[0,+∞]

V =n

V : Ω→[0,+∞] : V Lebesgue measurable, Z

Ω

Ψ(V)dx61o .

Theorem4.1. — Let Ω⊂R^d be a bounded open set and letΨ : [0,+∞]→[0,+∞]be an injective function satisfying the condition

(4.2) there existp >1such that the function s7→Ψ⁻¹(s^p)is convex.

Then, for any functional F : M⁺_cap(Ω) →Rwhich is increasing and lower semicon- tinuous with respect to theγ-convergence, the problem (4.1) has a solution, provided the admissible setV is nonempty.

Proof. — Let Vn ∈ V be a minimizing sequence for problem (4.1). Then, vn :=

Ψ(Vn)1/p

is a bounded sequence inL^p(Ω)and so, up to a subsequence,vnconverges weakly in L^p(Ω) to some function v. We will prove thatV := Ψ⁻¹(v^p) is a solution to (4.1). Clearly V ∈ V and so it remains to prove thatF(V)6 lim infnF(Vn). In view of the compactness of the γ-convergence on the class M⁺_cap(Ω) of capacitary measures (see Section 2), we can suppose that, up to a subsequence,Vn γ-converges to a capacitary measureµ∈ M⁺_cap(Ω). We claim that the following inequalities hold true:

(4.3) F(V)6F(µ)6lim inf

n→∞ F(Vn).

In fact, the second inequality in (4.3) is the lower semicontinuity ofF with respect to theγ-convergence, while the first needs a more careful examination. By the definition

of γ-convergence, we have that for any u∈ H₀¹(Ω), there is a sequence un ∈H₀¹(Ω) which converges touinL²(Ω)and is such that

Z

Ω

|∇u|²dx+ Z

Ω

u²dµ= lim

n→∞

Z

Ω

|∇un|²dx+ Z

Ω

u²_nVndx

= lim

n→∞

Z

Ω

|∇un|²dx+ Z

Ω

u²_nΨ⁻¹(v_n^p)dx

>

Z

Ω

|∇u|²dx+ Z

Ω

u²Ψ⁻¹(v^p)dx

= Z

Ω

|∇u|²dx+ Z

Ω

u²V dx, (4.4)

where the inequality in (4.4) is due to strong-weak lower semicontinuity of integral functionals (see for instance [7]), which follows by assumption (4.2). Thus, for any u∈H₀¹(Ω), we have

Z

Ω

u²dµ>

Z

Ω

u²V dx,

which givesV 6µ. SinceF is assumed to be monotone increasing, we obtain the first

inequality in (4.3) and so the conclusion.

Remark4.2. — The condition on the function Ψin Theorem 4.1 is satisfied for in- stance by the following functions:

(1) Ψ(s) =s^−p, for anyp >0;

(2) Ψ(s) =e^−αs, for anyα >0.

In some special cases, the solution to the optimization problem (4.1) can be com- puted explicitly through the solution to some PDE, as in Proposition 3.6. This occurs for instance when F =λ1 or when F =Ef, with f ∈ L²(Ω). We note that, by the variational formulation

λ1(V) = minnZ

Ω

|∇u|²dx+ Z

Ω

u²V dx: u∈H0¹(Ω), Z

Ω

u²dx= 1o , we can rewrite problem (4.1) as

(4.5) min

kukmin2=1

nZ

Ω

|∇u|²dx+ Z

Ω

u²V dxo

: V >0, Z

Ω

Ψ(V)dx61

= min

kuk2=1

minnZ

Ω

|∇u|²dx+ Z

Ω

u²V dx: V >0, Z

Ω

Ψ(V)dx61o . One can compute that, ifΨis differentiable withΨ⁰ invertible, then the second min- imum in (4.5) is achieved for

(4.6) V = (Ψ⁰)⁻¹(Λuu²),

whereΛu is a constant such thatR

ΩΨ (Ψ⁰)⁻¹(Λuu²)

dx= 1. Thus, the solution to the problem on the right hand side of (4.5) is given through the solution to

(4.7) min Z

Ω

|∇u|²dx+ Z

Ω

u²(Ψ⁰)⁻¹(Λuu²)dx: u∈H₀¹(Ω), Z

Ω

u²dx= 1

.