Giuseppe Buttazzo
email: buttazzo@dm.unipi.it

Augusto Gerolin
email: gerolin@mail.dm.unipi.it

Bozhidar Berardo Ruffini
email: berardo.ruffini@sns.it

B Velichkov
email: b.velichkov@sns.it

Optimal Potentials For Schrodinger Operators

Keywords: Schrödinger operators, optimal potentials, spectral optimization, capacity. 2010 Mathematics Subject Classification: 49J45, 35J10, 49R05, 35P15, 35J05

come

Introduction

In this paper we consider optimization problems of the form min F (V ) :

V ∈ V , (1.1)
for functionals F , depending on the Schrödinger operator -∆ + V (x) with potential V belonging to a prescribed admissible class V of Lebesgue measurable functions on a 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 [START_REF] Ashbaugh | Maximal and minimal eigenvalues and their associated nonlinear equations[END_REF], Egnell [START_REF] Egnell | Extremal properties of the first eigenvalue of a class of elliptic eigenvalue problems[END_REF], Essen [START_REF] Essen | On estimating eigenvalues of a second order linear differential operator[END_REF], Harrell [START_REF] Harrell | Hamiltonian operators with maximal eigenvalues[END_REF],

Talenti [START_REF] Talenti | Estimates for eigenvalues of Sturm-Liouville problems[END_REF] and, more recently, by Carlen-Frank-Lieb [START_REF] Carlen | Stability estimates for the lowest eigenvalue of a Schrödinger operator[END_REF]. We refer to the monography [START_REF] Henrot | Extremum Problems for Eigenvalues of Elliptic Operators[END_REF], 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 ∈ L 2 (Ω) we consider the solution u V of the elliptic PDE -∆u + V u = f in Ω, u ∈ H 1 0 (Ω). The integral cost functionals we may consider are of the form

F (V ) = Ω j x, u V (x), ∇u V (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 , with a ∈ L 1 (Ω) and c smaller than the first Dirichlet eigenvalue of the Laplace operator -∆ in Ω. In particular, the energy E f (V ) defined by

E f (V ) = inf Ω 1 2 |∇u| 2 + 1 2 V (x)u 2 -f (x)u dx : u ∈ H 1 0 (Ω) , (1.2)
belongs to this class since, integrating by parts its Euler-Lagrange equation, we have

E f (V ) = - 1 2 Ω f (x)u V dx, 1
which corresponds to the integral functional above with j(x, s, z) = -1 2 f (x)s.

Spectral functionals. For every admissible potential V ≥ 0 we consider the spectrum Λ(V ) of the Schrödinger operator -∆ + V (x) on H 1 0 (Ω). If Ω is 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 ) = λ 1 (V ), λ 2 (V ), . . . ,
where λ k (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 functions Φ : R N → (-∞, +∞]. For instance, taking Φ(Λ) = λ k we obtain

F (V ) = λ k (V ).
The class of admissible potentials V we consider satisfies an integrability condition, namely

V = V : Ω → [0, +∞] : V Lebesgue measurable, Ω Ψ(V ) dx ≤ 1 . (1.3)
for a suitable function Ψ : [0, +∞] → [0, +∞]. It is worth remarking that the requirement 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 instance of optimization problem for signchanging potentials can be found in the recent work [START_REF] Carlen | Stability estimates for the lowest eigenvalue of a Schrödinger operator[END_REF], 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| ≤ constant . Indeed, if we set V E = 0 in E and V E = +∞ outside of E, then the Schrödinger operator -∆ + V E corresponds to the Dirichlet-Laplacian on the set E. This observation suggests that, by one side, 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 potential V E , corresponding to an optimal set E, 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 ) = λ 1 (V ), then the study of the optimization problem min F (V ) : V : Ω → [0, +∞], Ω V p dx = 1 1 For an introduction to the theory of shape optimization problems we refer to the survey paper [START_REF] Buttazzo | Spectral optimization problems[END_REF] and to the books [START_REF] Bucur | Variational Methods in Shape Optimization Problems[END_REF], [START_REF] Henrot | Extremum Problems for Eigenvalues of Elliptic Operators[END_REF] and [START_REF] Henrot | Variation et Optimisation de Formes[END_REF] naturally reduces to the one of ground states of the equation -∆ψ + |ψ| s ψ = λψ, ψ ∈ H 1 (Ω) ∩ L 2+s (Ω).

(1.4)

The case p > 0 corresponds to the superlinear case s > 0, while the case of negative exponent p < 0 corresponds to the sublinear case s < 0. Indeed, the potential V (x) = |ψ(x)| s satisfies an integrability condition inherited from the ground state ψ. In the superlinear case s > 0, we have V ∈ L p (Ω) with p = (s + 2)/s, while in the sublinear case s ∈ (-1, 0) we get Ω V -p dx < +∞ with p = -(s + 2)/s. The paper is organized as follows. In Section 2 we recall the concepts of capacitary measures and γ-convergence together with their main properties. Then we prove some preliminary results which will be exploited in the subsequent sections.

In Section 3 we prove two general results concerning the existence of optimal potentials in a bounded domain Ω ⊂ R d . In Theorem 3.1 we deal with constraints V 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 3.1 our assumptions allow to take F (V ) = -E f (V ) and thus the optimization problem becomes the maximization of E f under the constraint Ω V p dx ≤ 1. We prove that for p ≥ 1, there exists an optimal potential for the problem max E f (V ) :

Ω V p dx ≤ 1 .
(

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 [START_REF] Egnell | Extremal properties of the first eigenvalue of a class of elliptic eigenvalue problems[END_REF], summarized in [START_REF] Henrot | Extremum Problems for Eigenvalues of Elliptic Operators[END_REF]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 χ ω + -χ ω -,
where χ U indicates the characteristic function of the set U , f ∈ L 2 (Ω), M = u V L ∞ (Ω) , and

ω ± = {u = ±M }.
In Section 4 we deal with minimization problems of the form min F (V ) :

Ω Ψ(V ) dx ≤ 1 , (1.6)
and we prove existence for the problem (1.1) for a large class of functionals F and of constraints Ψ, 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 -p the 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 [START_REF] Bucur | On the characterization of the compact embedding of Sobolev spaces[END_REF]). 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 = E f or F = λ 1 , there exist solutions of problem (1.6) in R d , with Ψ(s) = s -p . Moreover, we characterize the optimal potential V as an explicit function of the solution u of 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 case F = E f , 1/V is compactly supported, provided f is compactly supported. In the case F = λ 1 the same holds and the optimal potential V 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 for F = λ 2 for problem (1.6) with Ψ(s) = s -p .

Capacitary measures and γ-convergence

For a subset E ⊂ R d its capacity is defined by

cap(E) = inf R d |∇u| 2 dx + R d u 2 dx : u ∈ H 1 (R d ), u ≥ 1 in a neighborhood of E .
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 almost everywhere (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 > 0 there exists an open subset 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 sequence of open sets (A n ) n such that cap(A n ) → 0 and the restriction u n of u to the set A c n is continuous (respectively lower semicontinuous). It is well known (see for instance [START_REF] Evans | Measure Theory and Fine Properties of Functions[END_REF]) that every function u ∈ H 1 (R d ) has a quasi-continuous representative u, which is uniquely defined up to a set of capacity zero, and given by

u(x) = lim ε→0 1 |B ε (x)| Bε(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 1 (R d ) with the (q.e.) equivalence class of quasi-continuous representatives u.

We denote by M + (R d ) the set of positive Borel measures on R d (not necessarily finite or 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 is a capacitary measure, the integral R d |u| 2 dµ is well-defined for each u ∈ H 1 (R d ), i.e. if u 1 and u 2 are two quasi-continuous representatives of u, then

R d | u 1 | 2 dµ = R d | u 2 | 2 dµ.
For a subset Ω ⊂ R d , we define the Sobolev space H 1 0 (Ω) as

H 1 0 (Ω) = u ∈ H 1 (R d ) : u = 0 q.e. on Ω c . (2.1)
Alternatively, by using the capacitary measure I Ω defined as

I Ω (E) = 0 if cap(E \ Ω) = 0 +∞ if cap(E \ Ω) > 0 for every Borel set E ⊂ R d , (2.2)
the Sobolev space H 1 0 (Ω) can be defined as

H 1 0 (Ω) = u ∈ H 1 (R d ) : R d |u| 2 dI Ω < +∞ .
More generally, for any capacitary measure µ ∈ M + cap (R d ), we define the space

H 1 µ = u ∈ H 1 (R d ) : R d |u| 2 dµ < +∞ ,
which is a Hilbert space when endowed with the norm u 1,µ , where

u 2 1,µ = R d |∇u| 2 dx + R d u 2 dx + R d u 2 dµ. If u / ∈ H 1 µ , then we set u 1,µ = +∞. For Ω ⊂ R d , we define M + cap (Ω) as the space of capacitary measures µ ∈ M + cap (R d ) such that µ(E) = +∞ for any set E ⊂ R d such that cap(E \ Ω) > 0. For µ ∈ M + cap (R d ), we denote with H 1 µ (Ω) the space H 1 µ∨I Ω = H 1 µ ∩ H 1 0 (Ω).
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 conditions are 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 [START_REF] Maso | An Introduction to Γ-convergence[END_REF]; 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) Definition 2.2. We say that the sequence of capacitary measures µ n ∈ M + cap (Ω), γ-converges to the capacitary measure µ ∈ M + cap (Ω) if the sequence of functionals • 1,µn Γ-converges to the functional • 1,µ in L 2 (Ω), i.e. if the following two conditions are satisfied:

• for every sequence u n → u in L 2 (Ω) we have

R d |∇u| 2 dx + R d u 2 dµ ≤ lim inf n→∞ R d |∇u n | 2 dx + R d u 2 n dµ n ;
• for every u ∈ L 2 (Ω), there exists

u n → u in L 2 (Ω) such that R d |∇u| 2 dx + R d u 2 dµ = lim n→∞ R d |∇u n | 2 dx + R d u 2 n dµ n . If µ ∈ M + cap (Ω) and f ∈ L 2 (Ω) we define the functional J µ (f, •) : L 2 (Ω) → R ∪ {+∞} by J µ (f, u) = 1 2 Ω |∇u| 2 dx + 1 2 Ω u 2 dµ - Ω f u dx. (2.4) If Ω ⊂ R d is a bounded open set, µ ∈ M + cap (Ω) and f ∈ L 2 (Ω), then the functional J µ (f, •) has a unique minimizer u ∈ H 1
µ that verifies the PDE formally written as

-∆u + µu = f, u ∈ H 1 µ (Ω), (2.5)
and whose precise meaning is given in the weak form

   Ω ∇u • ∇ϕ dx + Ω uϕ dµ = Ω f ϕ dx, ∀ϕ ∈ H 1 µ (Ω), u ∈ H 1 µ (Ω).
The resolvent operator of -∆ + µ, that is the map R µ that associates to every f ∈ L 2 (Ω) the solution u ∈ H 1 µ (Ω) ⊂ L 2 (Ω), is a compact linear operator in L 2 (Ω) and so, it has a discrete spectrum

0 < • • • ≤ Λ k ≤ • • • ≤ Λ 2 ≤ Λ 1 .
Their inverses 1/Λ k are denoted by λ k (µ) and are the eigenvalues of the operator -∆ + µ.

In the case f = 1 the solution will be denoted by w µ and when µ = I Ω we will use the notation w Ω instead of w I Ω . We also recall (see [START_REF] Bucur | Variational Methods in Shape Optimization Problems[END_REF]) that if Ω is bounded, then the strong L 2 -convergence of the minimizers w µn to w µ is equivalent to the γ-convergence of Definition 2.2.

Remark 2.3. An important well known characterization of the γ-convergence is the following: a sequence µ n γ-converges to µ, if and only if, the sequence of resolvent operators R µn associated to -∆ + µ n , converges (in the strong convergence of linear operators on L 2 ) 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.

Remark 2.4. The space M + cap (Ω) endowed with the γ-convergence is metrizable. If Ω is bounded, one may take d γ (µ, ν) = w µw ν L 2 . Moreover, in this case, in [START_REF] Maso | Wiener's criterion and Γ-convergence[END_REF] it is proved that the space M + cap (Ω) endowed with the metric d γ is compact.

Proposition 2.5. Let Ω ⊂ R d and let V n ∈ L 1 (Ω) be a sequence weakly converging in L 1 (Ω) to a 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 Vn (1) of -∆u n + V n (x)u n = 1 u ∈ H 1 0 (Ω) weakly converge in H 1 0 (Ω) to the solution u = R V (1) of -∆u + V (x)u = 1 u ∈ H 1 0 (Ω), or equivalently that the functionals J n (u) = Ω |∇u| 2 dx + Ω V n (x)u 2 dx Γ L 2 (Ω) -converge to the functional J(u) = Ω |∇u| 2 dx + Ω V (x)u 2 dx. The Γ-liminf inequality (Definition 2.1 (a)) is immediate since, if u n → u in L 2 (Ω), we have Ω |∇u| 2 dx ≤ lim inf n→∞ Ω |∇u n | 2 dx
by the lower semicontinuity of the H 1 (Ω) norm with respect to the L 2 (Ω)-convergence, and

Ω V (x)u 2 dx ≤ lim inf n→∞ Ω V n (x)u 2 n dx
by the strong-weak lower semicontinuity theorem for integral functionals (see for instance [START_REF] Buttazzo | Semicontinuity, relaxation and integral representation in the calculus of variations[END_REF]).

Let us now prove the Γ-limsup inequality (Definition 2.1 (b)) which consists, given u ∈

H 1 0 (Ω), in constructing a sequence u n → u in L 2 (Ω) such that lim sup n→∞ Ω |∇u n | 2 dx + Ω V n (x)u 2 n dx ≤ Ω |∇u| 2 dx + Ω V (x)u 2 dx. (2.6)
For every t > 0 let u t = (u ∧ t) ∨ (-t); then, by the weak convergence of V n , for t fixed we have

lim n→∞ Ω V n (x)|u t | 2 dx = Ω V (x)|u t | 2 dx, and
lim t→+∞ Ω V (x)|u t | 2 dx = Ω V (x)|u| 2 dx.
Then, by a diagonal argument, we can find a sequence t n → +∞ such that

lim n→∞ Ω V n (x)|u tn | 2 dx = Ω V (x)|u| 2 dx.
Taking now u n = u tn , and noticing that for every t > 0

Ω |∇u t | 2 dx ≤ Ω |∇u| 2 dx,
we obtain (2.6) 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 ∈ L 1 + (Ω) be two functions such that V ≥ W . Then, there is a sequence V n ∈ L 1 + (Ω), uniformly bounded in L 1 (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. 2 Without loss of generality we can suppose Ω (V -W ) dx = 1. Let µ n be a sequence of probability measures on Ω weakly* converging to (V -W ) dx and such that each µ n is a finite sum of Dirac masses. For each n ∈ N consider a sequence of positive functions

V n,m ∈ L 1 (Ω) such that 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 1/m | -1 χ B 1/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 +Vn,m converges in L 2 to w W , as m → ∞. Indeed, by the weak maximum principle, we have

w W +I Ωm,n ≤ w W +Vn,m ≤ w W ,
where Ω m,n = Ω \ ∪ j B 1/m (x j ) and I Ωm,n is as in (2.2).

the idea of this proof was suggested by Dorin Bucur

Since a point has zero capacity in R d (d ≥ 2) there exists a sequence φ m → 0 strongly in

H 1 (R d ) with φ m = 1 on B 1/m (0) and φ m = 0 outside B 1/ √ m (0). We have Ω |w W -w W +I Ωm,n | 2 dx ≤ 2 w W L ∞ Ω (w W -w W +I Ωm,n ) dx = 4 w W L ∞ E(W + I Ωm,n ) -E(W ) (2.7) ≤ 4 w W L ∞ Ω 1 2 |∇w m | 2 + 1 2 W w 2 m -w m dx - Ω 1 2 |∇w W | 2 + 1 2 W w 2 W -w W dx ,
where

w m is any function in ∈ H 1 0 (Ω m,n ). Taking w m (x) = w W (x) j (1 -φ m (x -x j )), since φ m → 0 strongly in H 1 (R d ), it is easy to see that w m → w W strongly in H 1 (Ω) and so, by (2.7), w W +I Ωm,n → w W in L 2 (Ω) as m → ∞.
Since the weak convergence of probability measures and the γ-convergence are both induced by metrics, a diagonal sequence argument brings to the conclusion. Remark 2.7. 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 compact embedding of H 1 0 (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 the following proposition, whose proof is contained in [12, Theorem 3.1] and we report it here for the sake of completeness.

Proposition 2.8. Let µ n ∈ M + cap (Ω) be a sequence of capacitary Radon measures weakly* converging 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 compact set. Let u be a nonnegative smooth function with compact support in Ω such that u ≤ 1 in Ω and u = 1 on K; we have

µ(K) ≤ Ω u 2 dµ ≤ lim inf n→∞ Ω u 2 dµ n = Ω u 2 dν ≤ ν ({u > 0}) .
Since u is arbitrary, we have the conclusion by the Borel regularity of ν.

3. Existence of optimal potentials in L p (Ω)

In this section we consider the optimization problem min F (V ) :

V : Ω → [0, +∞], Ω V p dx ≤ 1 , (3.1)
where p > 0 and F (V ) is a cost functional depending on the solution of some partial differential equation on Ω. Typically, F (V ) is the minimum of some functional J V : H 1 0 (Ω) → R depending on V . A natural assumption in this case is the lower semicontinuity of the functional F with respect to the γ-convergence, that is

F (µ) ≤ lim inf n→∞ F (µ n ), whenever µ n → γ µ. (3.2) Theorem 3.1. Let F : L 1 + (Ω)
→ R be a functional, lower semicontinuous with respect to the γ-convergence, and let V be a weakly L 1 (Ω) compact set. Then the problem min {F (V ) : V ∈ V} , (3.3) admits a solution.

Proof. Let (V n ) be a minimizing sequence in V. By the compactness assumption on V, we may assume that V n tends weakly L 1 (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.

Remark 3.2. 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 the γ-convergence.

Remark 3.3. In some special cases the solution of (3.1) can be written explicitly in terms of the solution of some partial differential equation on Ω. This is the case of the Dirichlet Energy, that we discuss in Subsection 3.1, and of the first eigenvalue of the Dirichlet Laplacian λ 1 (see [START_REF] Henrot | Minimization problems for eigenvalues of the Laplacian[END_REF]Chapter 8]).

The compactness assumption on the admissible class V for the weak L 1 (Ω) convergence in Theorem 3.1 is for instance satisfied if Ω has finite measure and V is a convex closed and bounded subset of L p (Ω), with p ≥ 1. In the case of measures an analogous result holds.

F (µ) : µ ∈ M + cap (Ω), µ(Ω) ≤ 1 , (3.4)
admits a solution.

Proof. Let (µ n ) be a minimizing sequence. Then, up to a subsequence µ n converges weakly* to some measure ν and γ-converges to some measure µ ∈ M + cap (Ω). By Proposition 2.8, we have that µ(Ω) ≤ ν(Ω) ≤ 1 and so, µ is a solution of (3.4).

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 (1.2). On the other hand, an explicit form for the optimal potential V (x) will be provided in Proposition 3.9 assuming that the right-hand side f is in L 2 (Ω).

Example 3.5. Let Ω = (-1, 1) and consider the functional

F (µ) = -min 1 2 Ω |u ′ | 2 dx + 1 2 Ω u 2 dµ -u(0) : u ∈ H 1 0 (Ω) .
Then, for any µ such that µ(Ω) ≤ 1, we have

F (µ) ≥ -min 1 2 Ω |u ′ | 2 dx + 1 2 sup Ω u 2 -u(0) : u ∈ H 1 0 (Ω), u ≥ 0 . (3.5)
By a symmetrization argument, the minimizer u of the right-hand side of (3.5) is radially decreasing; moreover, u is linear on the set u < M , where M = sup u, and so it is of the form

u(x) =        M 1-α x + M 1-α , x ∈ [-1, -α], M, x ∈ [-α, α], -M 1-α x + M 1-α , x ∈ [α, 1], (3.6)
for some α ∈ [0, 1]. A straightforward computation gives α = 0 and M = 1/3. Thus, u is also the minimizer of

F (δ 0 ) = -min 1 2 Ω |u ′ | 2 dx + 1 2 u(0) 2 -u(0) : u ∈ H 1 0 (Ω) ,
and so δ 0 is the solution of min {F (µ) : µ(Ω) ≤ 1} .

{-E f (V ) : V ∈ V} with V = V ≥ 0, Ω V p dx ≤ 1 , (3.7)
admits a solution, where E f (V ) is the energy functional defined in (1.2). We notice that, replacing -E f (V ) by E f (V ), makes problem (3.7) trivial, with the only solution V ≡ 0. Minimization problems for E f will be considered in Section 4 for admissible classes of the form

V = V ≥ 0, Ω V -p dx ≤ 1 .
Analogous results for F (V ) = -λ 1 (V ) were proved in [20, Theorem 8. 1) ,

V p = Ω |u p | 2p/(p-1) dx -1/p |u p | -1+(p+1)/(p-
where u p ∈ H 1 0 (Ω) ∩ L 2p/(p-1) (Ω)
is the minimizer of the functional

J p (u) := 1 2 Ω |∇u| 2 dx + 1 2 Ω |u| 2p/(p-1) dx (p-1)/p - Ω uf dx. (3.8
)

Moreover, we have E f (V p ) = J p (u p ).
Proof. We first show that we have max

V ∈V min u∈H 1 0 (Ω) Ω 1 2 |∇u| 2 + u 2 V -uf dx ≤ min u∈H 1 0 (Ω) max V ∈V Ω 1 2 |∇u| 2 + u 2 V -uf dx, (3.9)
where the maximums are taken over all positive functions V ∈ L p (Ω). For a fixed u ∈ H 1 0 (Ω), the maximum on the right-hand side (if finite) is achieved for a function V such that ΛpV p-1 = u 2 , where Λ is a Lagrange multiplier. By the condition Ω V p dx = 1 we obtain that the maximum is achieved for

V = Ω |u| 2p p-1 dx 1/p |u| 2 p-1 .
Substituting in (3.9), we obtain

max {E f (V ) : V ∈ V} ≤ min J p (u) : u ∈ H 1 0 (Ω) . (3.10)
Let u n be a minimizing sequence for J p . Since inf J p ≤ 0, we can assume J p (u n ) ≤ 0 for each n ∈ N. Thus, we have

1 2 Ω |∇u n | 2 dx + 1 2 Ω |u n | 2p/(p-1) dx (p-1)/p ≤ Ω u n f dx ≤ C f L 2 (Ω) ∇u n L 2 , (3.11)
where C is a constant depending on Ω. Thus we obtain

Ω |∇u n | 2 dx + Ω |u n | 2p/(p-1) dx (p-1)/p ≤ 4C 2 f 2 L 2 (Ω) , (3.12)
and so, up to subsequence u n converges weakly in H 1 0 (Ω) and L 2p/(p-1) (Ω) to some u p ∈ H 1 0 (Ω)∩ L 2p/(p-1) (Ω). By the semicontinuity of the L 2 -norm of the gradient and the L 2p p-1 -norm and the fact that Ω f u n dx → Ω f u p dx, as n → ∞, we have that u p is a minimizer of J p . By the strict 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 + Ω |u p | 2p/(p-1) dx -1/p |u p | 2/(p-1) u p = f. Setting V p = Ω |u p | 2p/(p-1) dx -1/p |u p | 2/(p-1) ,
we have that Ω V p p dx = 1 and u p is the solution of

-∆u p + V p u p = f. (3.13)
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 by the uniqueness of u p and the equality case in the Hölder inequality

Ω u 2 V dx ≤ Ω V p dx 1/p Ω |u| 2p/(p-1) dx (p-1)/p ≤ Ω |u| 2p/(p-1) dx (p-1)/p .
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. We also note that the analogous results were obtained in the case F = -λ 1 (see [START_REF] Henrot | Minimization problems for eigenvalues of the Laplacian[END_REF]Theorem 8.2.4]) and in the case F = -E f , where f is a positive function (see [START_REF] Buttazzo | Optimal measures for elliptic problems[END_REF]). Remark 3.7. Let u p be the minimizer of J p , defined in (3.8). By (3.12), we have the estimate

∇u p L 2 (Ω) + u p L 2p/(p-1) (Ω) ≤ 4C 2 f L 2 (Ω) , (3.14)
where C is the constant from (3.11). Moreover, we have u p ∈ H 2 loc (Ω) and for each open set Ω ′ ⊂⊂ Ω, there is a constant C not depending on p such that

u p H 2 (Ω ′ ) ≤ C(f, Ω ′ ). Indeed, u p satisfies the PDE -∆u + c|u| α u = f, (3.15)
with c > 0 and α = 2/(p -1), and standard elliptic regularity arguments (see [START_REF] Evans | Partial Differential Equations[END_REF]Section 6.3]) give that u ∈ H 2 loc (Ω). To show that u p H 2 (Ω ′ ) is bounded independently of p we apply the Nirenberg operator

∂ h k u = u(x+he k )-u(x)
h on both sides of (3.15), and multiplying by φ 2 ∂ h k u, where φ is an appropriate cut-off function which equals 1 on Ω ′ , we have

Ω φ 2 |∇∂ h k u| 2 dx + Ω ∇(∂ h k u) • ∇(φ 2 )∂ h k u dx + c(α + 1) Ω φ 2 |u| α |∂ h k u| 2 dx (3.16) = -f ∂ h k (φ 2 ∂ h k u) dx,
for all k = 1, . . . , d. Some straightforward manipulations now give

∇ 2 u 2 L 2 (Ω ′ ) ≤ d k=1 Ω φ 2 |∇∂ k u| 2 dx ≤ C(Ω ′ ) f L 2 ({φ 2 >0}) + ∇u L 2 (Ω) . (3.17) Lemma 3.8. Let Ω ⊂ R d be an open set and f ∈ L 2 (Ω). Consider the functional J 1 : L 2 (Ω) → R defined by J 1 (u) := 1 2 Ω |∇u| 2 dx + 1 2 u 2 ∞ - Ω uf dx, (3.18)
Then, J p Γ-converges in L 2 (Ω) to J 1 , as p → 1, where J p is defined in (3.8).

Proof. Let v n ∈ L 2 (Ω) be a sequence of positive functions converging in L 2 to v ∈ L 2 (Ω) and let α n → +∞. Then, we have that

v L ∞ (Ω) ≤ lim inf n→∞ v n L αn (Ω) . (3.19)
In fact, suppose first that v L ∞ = M < +∞ and let ω ε = {v > M -ε}, for some ε > 0. Then, we have

lim inf n→∞ v n L αn (Ω) ≥ lim n→∞ |ω ε | (1-αn)/αn ωε v n dx = |ω ε | -1 ωε v dx ≥ M -ε,
and so, letting ε → 0, we have lim inf n→∞ v n L αn (Ω) ≤ M . If v 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 2 (Ω). Then, by the semicontinuity of the L 2 norm of the gradient and (3.19) and the continuity of the term Ω uf dx, we have

J 1 (u) ≤ lim inf n→∞ J pn (u n ), (3.20)
for any decreasing sequence p n → 1. On the other hand, for any u ∈ L 2 , we have J pn (u) → J 1 (u) as n → ∞ and so, we have the conclusion.

Proposition 3.9. Let Ω ⊂ R d be a bounded open set and f ∈ L 2 (Ω). Then there is a unique solution of problem (3.7) with p = 1, given by

V 1 = 1 M χ ω + f -χ ω -f , where M = u 1 L ∞ (Ω) , ω + = {u 1 = M }, ω -= {u 1 = -M }, being u 1 ∈ H 1 0 (Ω) ∩ L ∞ (Ω)
the unique minimizer of the functional J 1 , defined in (3.18). In particular, ω + f dxω -f dx = M , f ≥ 0 on ω + and f ≤ 0 on ω -.

Proof. For any u ∈ H 1 0 (Ω) and any V ≥ 0 with Ω V dx ≤ 1 we have

Ω u 2 V dx ≤ u 2 ∞ Ω V dx ≤ u 2 ∞ ,
where for sake of simplicity, we write • ∞ instead of • L ∞ (Ω) . Arguing as in the proof of Proposition 3.6, we obtain the inequalities

1 2 Ω |∇u| 2 dx + 1 2 Ω u 2 V dx - Ω uf dx ≤ J 1 (u), max E f (V ) : Ω V ≤ 1 ≤ min J 1 (u) : u ∈ H 1 0 (Ω) .
As in (3.11), we have that a minimizing sequence of J 1 is bounded in H 1 0 (Ω) ∩ L ∞ (Ω) and thus by semicontinuity there is a minimizer u 1 ∈ H 1 0 (Ω) ∩ L ∞ (Ω) of J 1 , which is also unique, by the strict convexity of J 1 . Let u p denotes the minimizer of J p as in Proposition 3.6. Then, by Remark 3.7, we have that the family u p is bounded in H 1 0 (Ω) and in H 2 (Ω ′ ) for each Ω ′ ⊂⊂ Ω. Then, we have that each sequence u pn has a subsequence converging weakly in L 2 (Ω) to some u ∈ H 2 loc (Ω) ∩ H 1 0 (Ω). By Lemma 3.8, we have u = u 1 and so,

u 1 ∈ H 2 loc (Ω) ∩ H 1 0 (Ω). Thus u pn → u 1 in L 2 (Ω).
Let us define M = u 1 ∞ and ω = ω + ∪ ω -. We claim that u 1 satisfies, on Ω the PDE

-∆u + χ ω f = f. (3.21)
Indeed, setting Ω t = Ω ∩ {|u| < t} for t > 0, we compute the variation of J 1 with respect to any function ϕ ∈ H 1 0 (Ω M -ε ). Namely we consider functions of the form ϕ = ψw ε where w ε is the solution of -∆w ε = 1 on Ω M -ε , and w ε = 0 on ∂Ω M -ε . Thus we obtain that -∆u 1 = f on Ω M -ε and letting ε → 0 we conclude, thanks to the Monotone Convergence Theorem, that

-∆u 1 = f on Ω M = Ω \ ω.
Moreover, since u 1 ∈ H 2 loc (Ω), we have that ∆u 1 = 0 on ω and so, we obtain (3.21). Since u 1 is the minimizer of J 1 , we have that for each ε ∈ R, J 1 ((1 + ε)u 1 ) -J 1 (u 1 ) ≥ 0. Taking the derivative of this difference at ε = 0, we obtain

Ω |∇u 1 | 2 dx + M 2 = Ω f u 1 dx. ( 3
M = ω + f dx - ω - f dx. (3.23) Setting V 1 := 1 M χ ω + f -χ ω -f , we have that Ω V 1 dx = 1, -∆u 1 + V 1 u 1 = f in H -1 (Ω) and J 1 (u 1 ) = 1 2 Ω |∇u 1 | 2 dx + 1 2 Ω u 2 1 V 1 dx - Ω u 1 f dx.
We are left to prove that V 1 is admissible, i.e. V 1 ≥ 0. To do this, consider w ε the energy function 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 that 0 ≤ lim t→0 + J 1 (u 1 + tϕ) -J 1 (u 1 ) t = Ω ∇u 1 , ∇ϕ dx - Ω f ϕ dx.
This inequality holds for any ψ so that, integrating by parts, we obtain -∆u 1f ≥ 0 almost everywhere on {u 1 < M -ε}. In particular, since ∆u 1 = 0 almost everywhere on ω -= {u = -M }, we obtain that f ≤ 0 on ω -. Arguing in the same way, and considering test functions supported on {u 1 ≥ -M + ε}, we can prove that f ≥ 0 on ω + . This implies V 1 ≥ 0 as required.

Remark 3.10. Under some additional assumptions on Ω and f one can obtain some more precise regularity results for u 1 . In fact, in [14, Theorem A1] it was proved that if ∂Ω ∈ C 2 and if

f ∈ L ∞ (Ω) is positive, then u 1 ∈ C 1,1 (Ω).
Remark 3.11. In the case p < 1 problem (3.7) does not admit, in general, a solution, even for regular f and Ω. We give a counterexample in dimension one, which can be easily adapted to higher dimensions.

Let Ω = (0, 1), 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-1 k=1 +∞ δ x n,k ,
where δ x is the Dirac measure at the point x. Let w n be the minimizer of the functional J µn (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 1 2 1/n 0 |u ′ | 2 dx - 1/n 0 u dx : u ∈ H 1 0 (0, 1/n) = - C n 2 ,
where C > 0 is a constant. For any fixed n and j, let V n j be the sequence of positive functions such that

1 0 |V n j | p dx = 1, defined by V n j = C n n-1 k=1 j 1/p χ k n -1 j , k n + 1 j < n-1 k=1 I k n -1 j , k n + 1 j , (3.24)
where C n is a constant depending on n and I is as in (2.2). By the compactness of the γconvergence, we have that, up to a subsequence, V n j dx γ-converges to some capacitary measure µ as j → ∞. On the other hand it is easy to check that n-1 k=1 I k n -1 j , k n + 1 j (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 to check that each nonnegative function u ∈ H 1 0 ((0, 1)), for which u 2 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 is a sequence

u j ∈ H 1 0 (Ω) = H 1 V n j
(Ω) such that u j → u weakly in H 1 0 (Ω) and u 2 j V n j dx ≤ C, for some constant C not depending on j ∈ N. Since u j are uniformly 1/2-Hölder continuous, we can suppose that u j ≥ ε > 0 on some interval I containing k/n. But then for j large enough

I contains [k/n -1/j, k/n + 1/j] so that C ≥ 1 0 u 2 j V n j dx ≥ k/n+1/j k/n-1/j u 2 j V n j dx ≥ 2C n ε 2 j 1/p-1 ,
which is a contradiction for p < 1. Thus, we have that µ = µ n and so V n j γ-converges to µ n as j → ∞. In particular, E(µ n ) = lim j→∞ E 1 (V n j ) and since the left-hand side converges to zero as n → ∞, we can choose a diagonal sequence V n jn such that E(V n jn ) → 0 as n → ∞. Since there is no admissible functional V such that E 1 (V ) = 0, we have the conclusion.

Existence of optimal potentials for unbounded constraints

In this section we consider the optimization problem min {F (V ) : V ∈ V} , Then, for any functional F : M + cap (Ω) → R which is increasing and lower semicontinuous with respect to the γ-convergence, the problem (4.1) has a solution.

V = V : Ω → [0, +∞] : V Lebesgue measurable, Ω Ψ(V ) dx ≤ 1 .
Proof. Let V n ∈ V be a minimizing sequence for problem (4.1). Then, v n := Ψ(V n )

1/p is a bounded sequence in L p (Ω) and so, up to a subsequence, v n converges weakly in L p (Ω) to some function v. We will prove that V := Ψ -1 (v p ) is a solution of (4.1). Clearly V ∈ V and so it remains to prove that F (V ) ≤ lim inf n F (V n ). In view of the compactness of the γ-convergence on the class M 0 (Ω) of capacitary measures (see Section 2), we can suppose that, up to a subsequence, V n γ-converges to a capacitary measure µ ∈ M 0 (Ω). We claim that the following inequalities hold true:

F (V ) ≤ F (µ) ≤ lim inf n→∞ F (V n ). (4.2)
In fact, the second inequality in (4.2) is the lower semicontinuity of F with respect to the γconvergence, while the first needs a more careful examination. By the definition of γ-convergence, we have that for any u ∈ H 1 0 (Ω), there is a sequence u n ∈ H 1 0 (Ω) which converges to u in L 2 (Ω) and is such that

Ω |∇u| 2 dx + Ω u 2 dµ = lim n→∞ Ω |∇u n | 2 dx + Ω u 2 n V n dx = lim n→∞ Ω |∇u n | 2 dx + Ω u 2 n Ψ -1 (v p n ) dx ≥ Ω |∇u| 2 dx + Ω u 2 Ψ -1 (v p ) dx = Ω |∇u| 2 dx + Ω u 2 V dx, (4.3)
where the inequality in (4.3) is due to strong-weak lower semicontinuity of integral functionals (see for instance [START_REF] Buttazzo | Semicontinuity, relaxation and integral representation in the calculus of variations[END_REF]). Thus, for any u ∈ H 1 0 (Ω), we have

Ω u 2 dµ ≥ Ω u 2 V dx,
which gives V ≤ µ. Since F is assumed to be monotone increasing, we obtain the first inequality in (4.2) and so the conclusion.

Remark 4.2. The condition on the function Ψ in Theorem 4.1 is satisfied for instance by the following functions:

(1) Ψ(s) = s -p , for any p > 0;

(2) Ψ(s) = e -αs , for any α > 0.

4.1. Optimal potentials for the Dirichlet Energy and the first eigenvalue of the Dirichlet 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 occurs for instance when F = λ 1 or when F = E f , with f ∈ L 2 (Ω). We note that, by the variational formulation

λ 1 (V ) = min Ω |∇u| 2 dx + Ω u 2 V dx : u ∈ H 1 0 (Ω), Ω u 2 dx = 1 , (4.4)
we can rewrite problem (4.1) as min min

u 2 =1 Ω |∇u| 2 dx + Ω u 2 V dx : V ≥ 0, Ω Ψ(V ) dx ≤ 1 = min u 2 =1
min

Ω |∇u| 2 dx + Ω u 2 V dx : V ≥ 0, Ω Ψ(V ) dx ≤ 1 . (4.5)
One can compute that, if Ψ is differentiable with Ψ ′ invertible, then the second minimum in (4.5) is achieved for

V = (Ψ ′ ) -1 (Λ u u 2 ), (4.6)
where Λ u is a constant such that Ω Ψ (Ψ ′ ) -1 (Λ u u 2 ) dx = 1. Thus, the solution of the problem on the right hand side of (4.5) is given through the solution of min

Ω |∇u| 2 dx + Ω u 2 (Ψ ′ ) -1 (Λ u u 2 ) dx : u ∈ H 1 0 (Ω), Ω u 2 dx = 1 . (4.7)
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 of min

Ω 1 2 |∇u| 2 dx + Ω 1 2 u 2 (Ψ ′ ) -1 (Λ u u 2 ) dx - Ω f u dx : u ∈ H 1 0 (Ω) . (4.8)
Thus we obtain the following result.

Corollary 4.3. Under the assumptions of Theorem 4.1, for the functionals F = λ 1 and F = E f there exists a solution of (4.1) given by V = (Ψ ′ ) -1 (Λ u u 2 ), where u ∈ H 1 0 (Ω) is a minimizer of (4.7), in the case F = λ 1 , and of (4.8), in the case F = E f . Example 4.4. If Ψ(x) = x -p with p > 0, the optimal potentials for λ 1 and E f are given by

V = Ω |u| 2p/(p+1) dx 1/p u -2/(p+1) , (4.9)
where u is the minimizer of (4.7) and (4.8), respectively. We also note that, in this case

Ω u 2 (Ψ ′ ) -1 (Λ u u 2 ) dx = Ω |u| 2p/(p+1) dx (1+p)/p .
Example 4.5. If Ψ(x) = e -αx with α > 0, the optimal potentials for λ 1 and E f are given by

V = 1 α log Ω u 2 dx -log u 2 , (4.10)
where u is the minimizer of (4.7) and (4.8), respectively. We also note that, in this case

Ω u 2 (Ψ ′ ) -1 (Λ u u 2 ) dx = 1 α Ω u 2 dx Ω log u 2 dx - Ω u 2 log u 2 dx .

Optimization problems in unbounded domains

In this section we consider optimization problems for which the domain region is the entire Euclidean space R d . General existence results, in the case when the design region Ω is unbounded, are hard to achieve since most of the cost functionals are not semicontinuous with respect to the γ-convergence in these domains. For example, it is not hard to check that if µ is a capacitary measure, infinite outside the unit ball B 1 , then, for every x n → ∞, the sequence of translated measures µ n = µ(• + x n ) γ-converges to the capacitary measure

I ∅ (E) = 0, if cap(E) = 0, +∞, if cap(E) > 0.
Thus increasing and translation invariant functionals are never lower semicontinuous with respect to the γ-convergence. In some special cases, as the Dirichlet Energy or the first eigenvalue of the Dirichlet Laplacian, one can obtain existence results by a more direct methods, as those in Proposition 3.6. For a potential V ≥ 0 and a function f ∈ L q (R d ), we define the Dirichlet energy as

E f (V ) = inf R d 1 2 |∇u| 2 + 1 2 V (x)u 2 -f (x)u dx : u ∈ C ∞ c (R d ) . (5.1)
In some cases it is convenient to work with the space .

H 1 (R d ), obtained as the closure of C ∞ c (R d ) with respect to the L 2 norm of the gradient, instead of the classical Sobolev space H 1 (R d ). We recall that if d ≥ 3, the Gagliardo-Nirenberg-Sobolev inequality

u L 2d/(d-2) ≤ C d ∇u L 2 , ∀u ∈ . H 1 (R d ), (5.2)
holds, while in the cases d ≤ 2, we have respectively

u L ∞ ≤ r + 2 2 2/(r+2) u r/(r+2) L r u ′ 2/(r+2) L 2 , ∀r ≥ 1, ∀u ∈
.

H 1 (R);

(5.3)

u L r+2 ≤ r + 2 2 2/(r+2) u r/(r+2) L r ∇u 2/(r+2) L 2 , ∀r ≥ 1, ∀u ∈ . H 1 (R 2 ).
(5.4) 5.1. Optimal potentials in L p (R d ). In this section we consider optimization problems for the Dirichlet energy E f among potentials V ≥ 0 satisfying a constraint of the form V L p ≤ 1. We note that the results in this section hold in a generic unbounded domain Ω. Nevertheless, for sake of simplicity, we restrict our attention to the case Ω = R d .

Proposition 5.1. Let p > 1 and let q be in the interval with end-points a = 2p/(p + 1) and b = max{1, 2d/(d + 2)} (with a included for every d ≥ 1, and b included for every d = 2). Then, for every f ∈ L q (R d ), there is a unique solution of the problem

max E f (V ) : V ≥ 0, R d V p dx ≤ 1 .
(5.5)

Proof. Arguing as in Proposition 3.6, we have that for p > 1 the optimal potential V p is given by .

V p = R d |u p | 2p/(p-
H 1 (R d ) ∩ L 2p/(p-1) (R d ) .
Thus, it is enough to prove that there exists a solution of (5.7). For a minimizing sequence u n we have

1 2 R d |∇u n | 2 dx + 1 2 R d |u n | 2p/(p-1) dx (p-1)/p ≤ R d u n f dx ≤ C f L q u n L q ′ . (5.8)
Suppose that d ≥ 3. Interpolating q ′ between 2p/(p -1) and 2d/(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

1 2 R d |∇u n | 2 dx + 1 2 R d |u n | 2p/(p-1) dx (p-1)/p ≤ C.
Thus we can suppose that u n converges weakly in .

H 1 (R d ) and in L 2p/(p-1) (R d ) and so, the problem (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 u n L q ′ by means of ∇u n L 2 and u n L 2p/(p-1) .

Repeating the argument of Subsection 3.1, one obtains an existence result for (5.5) in the case p = 1, too. Proposition 5.2. Let f ∈ L q (R d ), where q ∈ [1, 2d d+2 ], if d ≥ 3, and q = 1, if d = 1, 2. Then there is a unique solution V 1 of problem (5.5) with p = 1, which is given by

V 1 = f M χ ω + -χ ω -, where M = u 1 L ∞ (R d ) , ω + = {u 1 = M }, ω -= {u 1 = -M }, and u 1 is the unique minimizer of min 1 2 R d |∇u| 2 dx + 1 2 u 2 L ∞ - R d uf dx : u ∈ . H 1 (R d ) ∩ L ∞ (R d ) .
(5.9)

In particular, ω + f dxω -f dx = M , f ≥ 0 on ω + and f ≤ 0 on ω -.

We note that, when p = 1, the support of the optimal potential V 1 is contained in the support of the function f . This is not the case if p > 1, as the following example shows.

Example 5.3. Let f = χ B(0,1) and p > 1. By our previous analysis we know that there exist a solution 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 ′′ (r) - d -1 r u ′ (r) + Cu α = 0, r ∈ (1, +∞),
where α = 2p/(p -1) > 2 and C is a positive constant. Thus, we have that

u p (r) = kr 2/(1-α) ,
where k is an explicit constant depending on C, d and α. In particular, we have that u p is not compactly supported on R d (see Figure 1). 5.2. Optimal potentials with unbounded constraint. In this subsection we consider the problems

min E f (V ) : V ≥ 0, R d V -p dx ≤ 1 , (5.10) min λ 1 (V ) : V ≥ 0, R d V -p dx ≤ 1 , (5.11)
for p > 0 and 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 the dimension d. Namely, we need q to satisfy the following conditions

q ∈ [ 2d d + 2 , 2p p -1 ], if d ≥ 3 and p > 1, q ∈ [ 2d d + 2 , +∞], if d ≥ 3 and p ≤ 1, q ∈ (1, 2p p -1
], if d = 2 and p > 1, (5.12)

q ∈ (1, +∞], if d = 2 and p ≤ 1, q ∈ [1, 2p p -1 ], if d = 1 and p > 1, q ∈ [1, +∞], if d = 1 and p ≤ 1.
We say that q = q(p, d) ∈ [1, +∞] is admissible if it satisfy (5.12). Note that q = 2 is admissible for any d ≥ 1 and any p > 0.

Proposition 5.4. Let p > 0 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 = R d |u p | 2p/(p+1) dx 1/p |u p | -2/(1+p) , (5.13)
where u p is a minimizer of

min 1 2 R d |∇u| 2 dx + 1 2 R d |u| 2p/(p+1) dx (p+1)/p - R d uf dx : (5.14) u ∈ . H 1 (R d ), |u| 2p/(p+1) ∈ L 1 (R d ) .
Moreover, if p ≥ 1, then the functional in (5.14) is convex, its minimizer is unique and so is the solution of (5.10).

Proof. By means of (5.2), (5.3) and (5.4), and thanks to the admissibility of q, we get the existence of a solution of (5.14) through an interpolation argument similar to the one used in the proof of Proposition 5.1. The existence of an optimal potential follows by the same argument as in Subsection 4.1.

In Example 5.3, we showed that the optimal potentials for (5.5), may be supported on the whole R d . The analogous question for the problem (5.10) is whether the optimal potentials given by (5.13) have a bounded set of finiteness {V p < +∞}. In order to answer this question, it is sufficient to study the support of the solutions u p of (5.14), which solve the equation

-∆u + C p |u| -2/(p+1) u = f, (5.15)
where C p > 0 is a constant depending on p.

Proposition 5.5. Let p > 0 and let f ∈ L q (R d ), for q > d/2, be a nonnegative function with a compact support. Then every solution u p of problem (5.14) has a compact support.

Proof. With no loss of generality we may assume that f is supported in the unit ball of R d . We first prove the result when f is radially decreasing. In this case u p is also radially decreasing and nonnegative. Let v be the function defined by v(|x|) = u p (x). Thus v satisfies the equation

   -v ′′ - d -1 r v ′ + C p v s = 0 r ∈ (1, +∞), v(1) = u p (1), (5.16)
where s = (p -1)/(p + 1) and C p > 0 is a constant depending on p. Since v ≥ 0 and v ′ ≤ 0, we have that v is convex. Moreover, since

+∞ 1 v 2 r d-1 dr < +∞, +∞ 1 |v ′ | 2 r d-1 dr < +∞,
we have that v, v ′ and v ′′ vanish at infinity. Multiplying (5.16) by v ′ we obtain

v ′ (r) 2 2 -C p v(r) s+1 s + 1 ′ = - d -1 r v ′ (r) 2 ≤ 0.
Thus the function v ′ (r) 2 /2 -C p v(r) s+1 /(s + 1) is decreasing and vanishing at infinity and thus nonnegative. Thus we have

-v ′ (r) ≥ Cv(r) (s+1)/2 , r ∈ (1, +∞), (5.17)
where C = 2C p /(s + 1) 1/2 . 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)/2 ≥ Ar + B,
where A = 2C/(1s) and B is determined by the initial datum v(1). This cannot occur, since the 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 on

each ball B R ⊂ R d , u p ≤ u, where u ∈ H 1 (B R ) is the solution of -∆u = f in B R , u = u p on ∂B R . Since f ∈ L q (R d ) with q > d/2,
= χ B 1 , in R d
, where C p is as in (5.16). Then, the function w t (x) = t 2/(1-s) w(x/t) is a solution of the equation -∆w t + C p w s t = t 2s/(1-s) χ Bt . Since u p is bounded, there exists some t ≥ 1 large enough such that w t ≥ u p on the ball B t . Moreover, w t minimizes (5.14) with f = t 2s/(1-s) χ Bt 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 step of the proof, w t has compact support.

The problems (5.11) and (5.10) are similar both in the questions of existence and the qualitative properties of the solutions. 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 1 (R d ) be a minimizing sequence of positive functions normalized in L 2 . Note that by the Pólya-Szegö inequality we may assume that each of these functions is radially decreasing in R d and so we will use the identification u n = u n (r). In order to prove that the minimum is achieved it is enough to show that the sequence u n converges in L 2 (R d ). Indeed, since u n is a radially decreasing minimizing sequence, there exists C > 0 such that for each r > 0 we have

u n (r) 2p/(p+1) ≤ 1 |B r | Br u 2p/(p+1) n dx ≤ C r d .
Thus, for each R > 0, we obtain

B c R u 2 n dx ≤ C 1 +∞ R r -d(p+1)/p r d-1 dr = C 2 R -1/p , (5.20)
where C 1 and C 2 do not depend on n and R. Since the sequence u n is bounded in H 1 (R d ), it converges locally in L 2 (R d ) and, by (5.20), this convergence is also strong in L 2 (R d ). Thus, we obtain the existence of a radially symmetric and decreasing solution u p of (5.19) and so, of an optimal 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: , where, in the last inequality, we used the fact that u p (r) → 0, as r → ∞, and s + 1 < 2.

Integrating both sides of the above inequality, we conclude that u p has a compact support. In Figure 2 we show the case d = 1 and f = χ (-1,1) . is achieved. Indeed, for any u ∈ H 1 (R d ) and any t > 0, we define u t (x) := t d/2 u(tx). Thus, we have that u L 2 = u t L 2 , for any t > 0. Moreover, up to a rescaling, we may assume that the function g : (0, +∞) → R, defined by , where C is a constant depending on p and d. In the case u = u p , the minimum of g is achieved for t = 1 and so, we have that u p is a solution also of min

g(t) = R d |∇u t | 2 dx + R d |u t | 2p/(p+1) dx (p+1)/p = t 2
: u ∈ H 1 (R d ), R d u 2 dx = 1 ,
and so, multiplying respectively by u + 2 and u - 2 , we obtain that λ 2 (V ) ≥ λ 1 (V + ), λ 2 (V ) ≥ λ 1 (V -), (6.4) where we have equalities, if and only if, u + 2 and u - 2 are the first eigenfunctions corresponding to λ 1 (V + ) and λ 1 (V -). Let now V + and V -be optimal potentials for λ 1 corresponding to the constraints

R d V -p + dx = R d V -p + dx, R d V -p -dx = R d V -p -dx.
By Proposition 5.6, the sets of finiteness of V + and V -are compact, hence we may assume (up to translations) that they are also disjoint. By the monotonicity of λ 1 , we have

max{λ 1 (V 1 ), λ 1 (V 2 )} ≤ max{λ 1 ( V + ), λ 1 ( V -)},
and so, we obtain λ 2 (min{V 1 , V 2 }) ≤ max{λ 1 ( V + ), λ 1 ( V -)} ≤ max{λ 1 (V + ), λ 1 (V -)} ≤ λ 2 (V ), as required. If u 2 does not change sign, then we consider V + = sup{V, ∞ {u 2 =0} } and V -= sup{V, ∞ {u 1 =0} }, where u 1 is the first eigenfunction of -∆ + V (x). Then the claim follows by the same argument as above.

Theorem 3 . 4 .

 34 Let Ω ⊂ R d be a bounded open set and let F : M + cap (Ω) → R be a functional lower semicontinuous with respect to the γ-convergence. Then the problem min

3. 1 .

 1 Minimization problems in L p concerning the Dirichlet Energy functional. Let Ω ⊂ R d be a bounded open set and let f ∈ L 2 (Ω). By Theorem 3.1, the problem min

Proposition 3 . 6 .

 36 Let Ω ⊂ R d be a bounded open set, 1 < p < ∞ and f ∈ L 2 (Ω). Then the problem (3.7) has a unique solution

. 22 )

 22 By(3.21), we have Ω |∇u 1 | 2 dx = Ω\ω f u 1 dx and so

(4. 1 )

 1 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 classes we study depend on a function Ψ : [0, +∞] → [0, +∞]

Theorem 4 . 1 .

 41 Let Ω ⊂ R d be a bounded open set and let Ψ : [0, +∞] → [0, +∞] be a function satisfying the conditions i) Ψ is strictly decreasing; ii) there exist p > 1 such that the function s → Ψ -1 (s p ) is convex.

Figure 1 .

 1 Figure 1. The solution u p of problem (5.7), with p > 1 and f = χ B(0,1) does not have a compact support.

Proposition 5 . 6 .

 56 For every p > 0 there is a solution of the problem(5.11) given byV p = R d |u p | 2p/(p+1) dx 1/p |u p | -2/(1+p) ,(5.18)where u p is a radially decreasing minimizer of minR d |∇u| 2 dx + R d |u| 2p/(p+1) dx (p+1)/p : u ∈ H 1 (R d ), R d u 2 dx = 1 .(5.[START_REF] Harrell | Hamiltonian operators with maximal eigenvalues[END_REF]

  + C p u s p = λu p , where s = (p-1)/(p+1) < 1 and C p > 0 is a constant depending on p. Arguing as in Proposition 5.5, we obtain that, for r large enough,

Figure 2 .

 2 Figure 2. The solution u p of problem (5.14), with p > 1 and f = χ (-1,1) .

R d |∇u| 2

 2 dx + t -d/p R d |u| 2p/(p+1) dx (p+1)/p, achieves its minimum in the interval (0, +∞) and, moreover, we have min

R d |∇u| 2

 2 dx d/(d+2p) R d |u| 2p/(p+1) dx 2(p+1)/(d+2p)

  1) dx

	where u p is the solution of the problem
	min	1 2 R d	|∇u| 2 dx +	1 2	R d	|u| 2p/(p-1) dx	(p-1)/p	-	R d	uf dx :	(5.7)
						u ∈
							-1/p	|u p | 2/(p-1) ,	(5.6)

  by[START_REF] Gilbarg | Elliptic partial differential equations of second order[END_REF] Theorem 9.11] and a standard bootstrap argument on the integrability of u, we have that u is continuous on B R/2 . As a consequence, u p is locally bounded in R d . In particular, it is bounded since u p ∧ M , where M = u p L ∞ (B 1 ) , is a better competitor than u p in(5.14). Let w be a radially decreasing minimizer of (5.14) with f = χ B 1 . Thus w is a solution of the PDE -∆w + C p w s

which is just another form of (5.21).

Remark 5.8. We conclude this section with a remark about the constraint Ψ(s) = e -αs . This type of constraint may be used to approximate shape optimization problems, in which the main unknown is a domain Ω, i.e. the potential V = I Ω is the capacitary measure of Ω. To get an example of this fact we recall the problem min E f (V ) + Λ Ω e -αV dx : V ∈ B(Ω) Borel , (5.22) where Λ is a Lagrange multiplier and B(Ω) is the class of nonnegative Borel measurable functions on Ω. As before, we note that the problem (5.22) is equivalent to min

We note that if u 2 ≥ Λk, then necessarily V = 0. On the other hand, if u 2 < Λk, then by the optimality of V , we have V > 0. Finally, we get

Substituting in (5.23), we obtain the problem min

or, equivalently, min

(5.24)

Note that the second term is actually positive and so, by a standard variational argument, we have that the problem (5.24) has a solution u α ∈ H 1 0 (Ω). Moreover, on the quasi-open set u 2 > Λα, we have -∆u = f . Let J α be the functional in (5.24), i.e.

Then J α Γ-converges in L 2 (Ω), as α → 0, to the functional

6. Further remarks on the optimal potentials for spectral functionals

We recall (see [START_REF] Bucur | On the characterization of the compact embedding of Sobolev spaces[END_REF]) that the injection

) is compact whenever the potential V satisfies R d V -p dx < +∞ for some 0 < p ≤ 1. In this case the spectrum of the Schrödinger operator -∆ + V (x) is discrete and we denote by λ k (V ) its eigenvalues. The existence of an optimal potential for spectral optimization problems of the form

was proved in [START_REF] Bucur | Spectral optimization problems for potentials and measures[END_REF], for any k ∈ N and for p ∈ (0, 1). This result cannot be deduced by the direct methods used in Subsection 5.2 and is based on a combination of a concentration-compactness argument and a fine estimate on the diameter of the set of finiteness {V k < +∞} of the optimal potential V k . We note that the existence of an optimal potential in the case k > 2 and p = 1 is still an open question.

In the case k = 2 the idea from Subsection 5.2 can still be applied to prove an existence result for (6.1) and to explicitly characterize the optimal potential. We first recall that, by Proposition 5.6, there exists optimal potential V p , for λ 1 , such that the set of finiteness {V p < +∞} is a ball. Thus, we have a situation analogous to the Faber-Krahn inequality, which states that the minimum min

is achieved for the ball of measure c. We recall that, starting from (6.2), one may deduce, by a simple argument (see for instance [START_REF] Henrot | Minimization problems for eigenvalues of the Laplacian[END_REF]), the Krahn-Szegö inequality, which states that the minimum min

is achieved for a disjoint union of equal balls. In the case of potentials one can find two optimal potentials for λ 1 with disjoint sets of finiteness and then apply the argument from the proof of the Krahn-Szegö inequality. In fact, we have the following result.

Theorem 6.1. There exists an optimal potential, solution of (6.1) with k = 2 and p ∈ (0, 1]. Moreover, any optimal potential is of the form min{V 1 , V 2 }, where V 1 and V 2 are optimal potentials for λ 1 which have disjoint sets of finiteness

Proof. Given V 1 and V 2 as above, we prove that for every V :

Indeed, let u 2 be the second eigenfunction of -∆ + V (x). We first suppose that u 2 changes sign on R d and consider the functions V + = sup{V, ∞ {u 2 ≤0} } and V -= sup{V, ∞ {u 2 ≥0} } where, for any measurable A ⊂ R d , we set

We note that 1 ≥

Moreover, on the sets {u 2 > 0} and {u 2 < 0}, the following equations are satisfied:


Olivier Cailloux
email: olivier.cailloux@uva.nl

Tommi Tervonen

Boris Verhaegen

François Picalausa

A Data Model for Algorithmic Multiple Criteria Decision Analysis

Keywords: Multiple Criteria Decision Analysis (MCDA), Data model, Multi-Attribute Value Theory, Utility Theory, ELECTRE methods, Stochastic Multicriteria Acceptability Analysis (SMAA), XML

Various software tools implementing Multiple Criteria Decision Analysis (MCDA) methods have appeared over the last decades. Although MCDA methods share common features, most of the implementing software have been developed independently from scratch. Majority of the tools have a proprietary storage format and exchanging data among software is cumbersome. Common data exchange standard would be useful for an analyst wanting to apply different methods on the same problem. The Decision Deck project has proposed to build components implementing MCDA methods in a reusable and interchangeable manner. A key element in this scheme is the XMCDA standard, a proposal that aims to standardize an XML encoding of common structures appearing in MCDA models, such as criteria and performance evaluations. Although XMCDA allows to present most data structures for MCDA models, it almost completely lacks data integrity checks. In this paper we present a new comprehensive data model for MCDA problems, implemented as an XML schema. The data model includes types that are sufficient to represent multi-attribute value/utility models, ELECTRE III/TRI models, and their stochastic (SMAA) extensions, and AHP. We also discuss use of the data model in algorithmic MCDA.

Introduction

The research in Multiple Criteria Decision Analysis (MCDA) has produced a large amount of methods to support decision making processes (Figueira et al 2005;[START_REF] Hwang | Multiple Attribute Decision Making: Methods and Applications; A State-of-the-Art Survey[END_REF][START_REF] Keeney | Decisions with multiple objectives: Preferences and value tradeoffs[END_REF]. Applicability of the methods has been shown in reported real-life applications [START_REF] Wallenius | Multiple criteria decision making, multiattribute utility theory: recent accomplishments and what lies ahead[END_REF], but most of them have been one-off decisions in disciplines where models can be constructed and calculated manually or with a general purpose software. The limited application of MCDA in new disciplines can be due to difficulty of integrating existing MCDA software with rest of the decision support technology [START_REF] Tervonen | JSMAA: open source software for SMAA computations[END_REF].

Various practical studies have proposed software frameworks for MCDA or Decision Support Systems (DSSs) implementing MCDA methods [START_REF] Fedorowicz | Representing modeling knowledge in an intelligent decision support system[END_REF][START_REF] Gauthier | SAGE: an object-oriented framework for the construction of farm decision support systems[END_REF][START_REF] Georgopoulou | Design and implementation of a group DSS for sustaining renewable energies exploitation[END_REF][START_REF] Jiménez | A generic multi-attribute analysis system[END_REF][START_REF] Jármai | Single-and multicriteria optimization as a tool of decision support system[END_REF][START_REF] Martin | Effective design and use of computer decision models[END_REF][START_REF] Minch | Computerized information systems supporting multicriteria decision making[END_REF][START_REF] Natividade-Jesus | A multicriteria decision support system for housing evaluation[END_REF][START_REF] Spengler | Development of a multiple criteria based decision support system for environmental assessment of recycling measures in the iron and steel making industry[END_REF][START_REF] Van Valkenhoef | ADDIS: a decision support system for evidence-based medicine[END_REF][START_REF] Zopounidis | PREFDIS: a multicriteria decision support system for sorting decision problems[END_REF]. However, these have been developed for specific problem types and/or methods, and aimed towards the end user. Furthermore, they lack a data interchange standard that could be used for sharing models over users and software. Theoretical interest has been shown for gathering several MCDA methods in a common software framework; already [START_REF] Teghem | An interactive decision support system (IDSS) for multicriteria decision aid[END_REF] analyzed several MCDA methods and studied the properties they satisfy. These were used for deriving a decision tree to choose the most appropriate method. Similarly, Hong and Vogel (1991) proposed a taxonomy of several MCDA methods and considered chaining compatible decision rules. However, neither of these works considered the sharing of input models over different MCDA methods and/or implementations. Considering the diversity of the needs an MCDA end user can face, it is hardly conceivable a single team of researchers could produce a software general enough to satisfy all of them. Therefore it should be made possible that different software tools would allow data exchange in a standard manner.

An appropriate data model for MCDA would allow evaluating results of different MCDA methods with similar inputs. This would be useful in an educational setting and it could also enable easier integration of MCDA methods in industrial applications. The approach of defining a common data interchange standard has proven succesful in the data mining community, who have defined the Predictive Model Markup Language (PMML) [START_REF] Guazzelli | PMML: An open standard for sharing models. The R Journal 1 Hong IB, Vogel DR[END_REF] as a means to encode predictive and data mining models in a vendor-independent way, thereby facilitating interoperability of software developed for the domain. Similarly, the OS standard has been proposed for representing mathematical programs together with a framework for their distributed computing [START_REF] Fourer | An XML-based schema for stochastic programs[END_REF](Fourer et al , 2010a,b),b). Initial steps for uniform data structures in MCDA have already been taken in the XMCDA standard (cf. www.decision-deck.org/xmcda).

PMML, OS and XMCDA allow encoding models in their corresponding domains, and therefore they all enable data sharing among various software tools. Unlike PMML and OS, XMCDA defines a single XML schema that can contain an arbitrary number of possibly unrelated elements. For example, a valid XMCDA instance could have a set of alternatives and a set of weights, or a performance table. Every implementation that allows XMCDA input has to use this universal schema. However, most MCDA methods have conceptually different input models. These differences in the inputs cannot be specified using XMCDA.

Moreover, instead of considering only complete models to be used as input to software tools, as in PMML and OS, the XMCDA standard aims to enable algorithmic MCDA: the design of computational components performing independent computational steps used in one or multiple MCDA methods. Algorithmic MCDA considers the decision support methods as workflows composed of sequential and parallel executable components. Such workflows can be managed e.g. with the diviz software (see [START_REF] Bigaret | Diviz: an MCDA workflow design, execution and sharing tool[END_REF] http://www.diviz.org). However, the components need to communicate through an interface, for which XMCDA is currently used. As different input types are not differenciated in the XMCDA schema, the middleware is unable to match outputs from one component to input to the next component in the workflow. Flow composition is thus non-trivial as the user has to know which component inputs and outputs can be connected. Such type-safe flow composition cannot in general be enables solely by designing a new data standard: the components themselves need input and output schemas defined with types of the data model. A mechanism need to be provided to enable this; only then can middleware detect incompatibilities in order to help the user in building correct workflow instances.

To overcome the shortcomings of XMCDA, this paper proposes a new core data model for MCDA data and results specification. The main idea of our proposal is that the data model contains only a small set of entities for modeling the main concepts (alternatives, attributes, criteria, relations) recurring in most MCDA methods, and that these concepts are grouped in sets to allow easy data interchange among components in algorithmic MCDA. We develop a model implemented as an XML schema and discuss its use in algorithmic computation of MCDA methods. Our data model does not provide representations for every data type used in all currently existing MCDA methods. Instead, it allows component developers to define the required inputs and outputs using a subset of the standard types, which enables both precise input and output specifications and interoperability of the components. The developed data model aims at enabling (i) easier exchange of data across different software packages, and (ii) decomposition of MCDA methods in smaller, individually executed components that communicate with interfaces defined through the data model. Our emphasis is on multi-attribute value/utility theory, ELECTRE-III/TRI methods, their stochastic extensions SMAA-2/TRI [START_REF] Lahdelma | SMAA-2: Stochastic multicriteria acceptability analysis for group decision making[END_REF][START_REF] Lahdelma | SMAA -stochastic multiobjective acceptability analysis[END_REF][START_REF] Tervonen | A stochastic method for robustness analysis in sorting problems[END_REF], and AHP. With this scope we cover two preference structures applied in majority of MCDA applications [START_REF] Wallenius | Multiple criteria decision making, multiattribute utility theory: recent accomplishments and what lies ahead[END_REF]: utility/value theory and outranking, all three problem statements: choice, ranking, and sorting [START_REF] Roy | Multicriteria Methodology for Decision Analysis[END_REF], and most methods currently used in practical applications [START_REF] Wallenius | Multiple criteria decision making, multiattribute utility theory: recent accomplishments and what lies ahead[END_REF].

Context

Most MCDA methods have the same base context. A set of decision alternatives A are evaluated on a set of attributes Z. The performance of alternative a ∈ A on attribute z ∈ Z is given by a performance function gz : A ⇒ Xz, where Xz is the evaluation scale bound to attribute z. The performances can be categorical, e.g. Xz = {"Bad", "Medium", "Good"}, or numerical, with Xz being a subset of R. Performances can be imprecise and modeled using, for example, an interval in R or a Gaussian distribution. MCDA methods also need a way to represent the preferences of the Decision Maker (DM). To achieve this, a set of criteria I, each corresponding to one attribute, is defined and preference information is attached to each i ∈ I. Most methods also include preference information that expresses relative importance of the criteria through weights. Usually MCDA methods are presented using only the concept of a criterion, but distinguishing criteria from attributes allows for clearer separation of performances and preference information. The former are mostly objective measurements whereas the latter model subjective views regarding outcome preferability, trade-offs or risk. Furthermore, clear separation of attributes and criteria allows to distinguish several preference model instances for a single attribute. This can be useful in a multiple DM context or if multiple ways to represent the DM preferences are needed.

In Utility Theory (UT) each criterion i, corresponding to an attribute z, is associated with a utility function u i : Xz → [0, 1]. For an evaluation x i ∈ Xz, u i (xz) is the DM's utility for performance xz according to criterion i. In the multiattribute additive utility model each criterion is also associated with a weight w i . This permits to compute the overall utility u(a) of an alternative a using the values given by the partial utility functions u i , i ∈ I, and the associated weights w i , i ∈ I. Once each alternative is associated with an overall utility, they can be rank-ordered. Note that not all UT models associate each criterion with a single weight, e.g. multilinear models include additional weights for sets of criteria. Value functions in value theory are different from utility functions in utility theory in that they do not quantify DM attitudes towards risk, but with respect to data modeling the two are equal. For this reason, in what follows, we use the term "utility" to refer uniformly to both utility and value (-functions, -theory).

In Outranking Theory (OT), including Electre [START_REF] Roy | The outranking approach and the foundations of ELECTRE methods[END_REF] and Promethee [START_REF] Brans | PROMETHEE: a new family of outranking methods in multicriteria analysis[END_REF] family of methods, criteria are associated with threshold parameters. Depending on the method, two or three types of threshold parameters are usually considered: preference, indifference, and possibly veto thresholds. Each criterion i ∈ I is associated with preference functions of pre-defined shapes, depending on the threshold values. Similarly to UT, these methods also include criteria weights to quantify their relative importances. Electre preference functions, and most Promethee preference functions, may be defined using solely the three threshold parameters included in our data model [START_REF] Cailloux | ELECTRE and PROMETHEE MCDA methods as reusable software components[END_REF]).

Data model

In this section, we present a data model appropriate for a set of common MCDA concepts. For each MCDA concept (such as an alternative or a performance table), we first present a UML diagram. This shows the attribute values that can be associated with the entity, and its relations to other entities. The XML schema specifications corresponding to all diagrams and the complete schema of the data model can be downloaded at http://github.com/tommite/pubs-code/tree/master/mcdadm-aor. Diagrams throughout this paper use UML 2.0 notation: the entities (Section 3) as class diagrams, the sample instantiations (Section 4) as object diagrams, and the method flow (Section 4) as a component diagram. All inheritance is implemented as "extension" in the XML schema, and therefore we omit the stereotype from the diagrams.

Fig. 1 Data model keyed entities: alternatives, attributes, categories and criteria.

Listing 1 A schema "example" with one attribute and one criterion. The header and namespace declarations have been omitted. <x s : s c h e m a> <x s : e l e m e n t name=" example "> <xs:complexType> <x s : s e q u e n c e> <x s : e l e m e n t name=" a t t r i b u t e " t y p e=" x 3 : a t t r i b u t e T y p e " /> <x s : e l e m e n t name=" c r i t e r i o n " t y p e=" x 3 : c r i t e r i o n T y p e " /> </ x s : s e q u e n c e> </ xs:complexType> </ x s : e l e m e n t> </ x s : s c h e m a> Listing 2 An instance of "example" where criterion c1 references attribute a1. The header and namespace declarations have been omitted, as well as the type of the criterion (criteria hierarchy are defined later in the text). <example> <a t t r i b u t e> <i d>a1</ i d> </ a t t r i b u t e> < c r i t e r i o n> <i d>c1</ i d> <a t t r i b u t e r e f=" a1 " /> </ c r i t e r i o n> </ example>

The most important entities of our data model are the four keyed ones: alternative, attribute, criterion and category. A keyed entity has an identifier that serves both as its name and as a unique identifier to refer to. We use inheritance with keyed entities to avoid duplication of other types described later on. Keyed entities are displayed in Figure 1.

Listing 1 provides an example XML schema that uses these entities: the example contains one attribute and a single criterion. This schema can be used for unambiguously defining such hypothetical models, and the developer of a MCDA software tool could define the accepted input models trough this schema. One such model is provided in Listing 2: there is a criterion c1, attribute a1, and c1 is explicitly referring to a1.

Our data model defines various measurement types, presented in Figure 2, to encode both performance and relation values. We use inheritance with measurements to enable uniform encoding of imprecise inputs in Stochastic Multicrite- ria Acceptability Analysis (SMAA) models [START_REF] Tervonen | A survey on stochastic multicriteria acceptability analysis methods[END_REF]. Many MCDA methods allow some model parameters to be imprecise or uncertain, and often these are represented as discrete or continuous probability distributions. Discrete uncertain outcomes can be encoded as imprecise nominal measurements, and continuous ones with intervals or Gaussian measurements. Note that the data model does not include more distributions or other ways to represent imprecise measurement, as even the most common distributions are quite numerous and to avoid bloating the model a non-trivial inclusion choice would have to be made. XML extension mechanisms allow users of our schema to define additional measurement types in external namespaces.

Any two keyed entities can be related with a measurement as a valued pair, e.g. alternative "car1" can have value 120.0 regarding attribute "speed", representing its max speed, or alternative "car1" can have a binary value true for alternative "car2", representing a holistic preference statement car1 ≻ car2. As XML inheritance does not allow generic types, the valued pair is repeated for each type of measurement, allowing to specify exactly what type of value the relation has to refer to. We refer to a valued pair with a measurement of type M using the notation <M>ValuedPair. ValuedPair refers to a <M>ValuedPair with measurement type ExactMeasurement. NominalValuedPair refers to a <M>ValuedPair with measurement type NominalMeasurement. Sets of valued pairs form valued relations, that represent a relation between two sets of objects (e.g. performances of a set of alternatives on a set of attributes). Keyed entities can also have a value connected to themselves only, which is represented as a valued entity (e.g. a criterion and a weight). Valued pairs, relations and entities are presented in Figure 3.

Associating two keyed entity types within a valued pair instead of more specific keyed entity types (such as alternative) loses type safety. A practical solution to this issue is to constrain references to strongly typed sets of keyed entities with XPath expressions and XML Schema keys. Listing 3 shows an example of such Fig. 3 Data model valued entity, pair and relation. Note that <M> represents a measurement type, and in the XML schema implementation the enclosing type is repeated for each sub-type of measurement (Exact, Interval, ...), as well as one for the abstract measurement type.

Listing 3 An XML schema with path constraints.

<e l e m e n t name=" exampleOne "> <complexType> <s e q u e n c e> <e l e m e n t name=" a l t e r n a t i v e S e t " t y p e=" x 3 : a l t e r n a t i v e S e t T y p e " /> <e l e m e n t name=" a t t r i b u t e S e t " t y p e=" x 3 : a t t r i b u t e S e t T y p e " /> <e l e m e n t name=" p e r f o r m a n c e T a b l e " t y p e=" x 3 : v a l u e d R e l a t i o n T y p e " /> </ s e q u e n c e> </ complexType> <key name=" a l t e r n a t i v e K e y ">

< s e l e c t o r xpath=" a l t e r n a t i v e S e t / a l t e r n a t i v e " /> < f i e l d xpath=" i d " /> </ key> <key name=" a t t r i b u t e K e y ">

< s e l e c t o r xpath=" a t t r i b u t e S e t / a t t r i b u t e " /> < f i e l d xpath=" i d " /> </ key> <k e y r e f name=" p e r f o r m a n c e T a b l e A l t e r n a t i v e K e y R e f " r e f e r=" x 3 e : a l t e r n a t i v e K e y ">

< s e l e c t o r xpath=" p e r f o r m a n c e T a b l e / v a l u e d P a i r / from " /> < f i e l d xpath=" @ r e f " /> </ k e y r e f> <k e y r e f name=" p e r f o r m a n c e T a b l e A t t r i b u t e K e y R e f " r e f e r=" x 3 e : a t t r i b u t e K e y "> < s e l e c t o r xpath=" p e r f o r m a n c e T a b l e / v a l u e d P a i r / t o " /> < f i e l d xpath=" @ r e f " /> </ k e y r e f> </ e l e m e n t> a schema defining a set of alternatives, a set of attributes, a set of valued pairs, and path constraints making sure the valued pairs refer to an alternative and an attribute. With no path constraints the schema would also validate if the valued pairs referred to e.g. two alternatives.

In order to increase readability and reduce verbosity of the encoding, the measurement is defined as optional in a binary relation: a missing measurement value Listing 4 An XML data file with a set of alternatives and a binary relation. The header and namespace declarations have been omitted. Assuming the binary relation is non reflexive and supposed to be defined on the whole set of alternatives, it may be deduced that the pairs (a1, a3) and (a2, a1) are not part of the relation.

<exampleTwo>

< a l t e r n a t i v e S e t> < a l t e r n a t i v e> <i d>a1</ i d> </ a l t e r n a t i v e> < a l t e r n a t i v e> <i d>a2</ i d> </ a l t e r n a t i v e> < a l t e r n a t i v e> <i d>a3</ i d> </ a l t e r n a t i v e> </ a l t e r n a t i v e S e t> < r e l a t i o n> <v a l u e d P a i r> <from r e f=" a1 " /> <t o r e f=" a2 " /> </ v a l u e d P a i r> <v a l u e d P a i r> <from r e f=" a2 " /> <t o r e f=" a3 " /> </ v a l u e d P a i r> <v a l u e d P a i r> <from r e f=" a3 " /> <t o r e f=" a1 " /> </ v a l u e d P a i r> <v a l u e d P a i r> <from r e f=" a3 " /> <t o r e f=" a2 " /> </ v a l u e d P a i r> </ r e l a t i o n> </exampleTwo> is considered to be true. Also, when the binary relation is supposed to be defined on some known set, for example, in the case of a binary relation to be defined on a set of alternatives given as input, missing valued pairs are considered to have a false value. With these rules the binary relations can be represented more naturally as only the pairs that are part of the relation have to be included. Listing 4 shows an example encoding of a binary relation. The binary relations may also be used to represent assignments in a sorting problem by using a relation on a set of alternatives and a set of categories. Path constraints may be used to ensure that an alternative is assigned to exactly one category if crisp assignments are required.

The data model criteria contain preference information related to measurements for a certain attribute. For example, if an attribute is top speed, we still need a criterion to represent the preference direction either implicitly with a utility criterion containing an increasing function, or explicitly with a directed criterion. Including meaningful names for the criteria allows to distinguish different parameterizations of the same preference model and to treat the criteria uniformly with other keyed entities. The data model criteria are presented in Figure 4. We do not distinguish between cardinal and nominal attributes. However, if alternatives evaluated with nominal measurements (represented by categories) are to be used with a criterion expecting cardinal measurements, a mapping of the nominal measurements to cardinal ones is necessary, as well as relating the new measurements with another attribute. The type used for this is presented in Figure 5. The mapping is defined thanks to valued entities associating values to categories.

We support three classical shapes for encoding partial utility functions u i . Affine linear functions of the form u i (x) = ax + b need only the slope and offset parameters. A piecewise linear function may be defined using at least a pair of two-dimensional points. Exponential functions u i (x) = 1e -ax require only a single parameter. Figure 6 shows these three types.

The data model contains set versions of most concepts: AlternativeSet, Cate-gorySet, AttributeSet, CriterionSet which may contain any type of criterion, Out-rankingCriterionSet which contains only outranking criteria, and similarly for the other criteria types, and ValuedEntitySet. Note that sets of valued pairs are valued relations. A summary of the data model is included in Appendix A.

Model application

In this section we present application of the model in two parts. First we illustrate with various use cases the encoding of attribute performance measurements and related preference structures, and afterwards how data model can be used in algorithmic MCDA to connect multiple components within a workflow representing a complete method.

Preference and measurement encoding

Let us consider a multi-criteria problem of choosing between two cars, car1 and car2. There are two relevant attributes for this decision, speed and color. Car1 has speed 140 and color "Blue", car2 has speed 120 and color "Red". The performance evaluations of car1 are presented in Figure 7 in terms of the data model. The choice between the two cars depends on the DM preferences that are elicited considering a certain preference model. Let us illustrate instantiation of the data model for utility theory (UT) and outranking theory (OT) models by considering the following use cases (we refer to "user" as the one encoding the model).

UC1: UT with nominal categories. The user wants to encode a utility function from the set of colors to the range [0, 1]. Assume that the DM considers blue color to be associated with the maximal utility (value 1) and red color with the minimal one (value 0). This is presented in Figure 8. The NominalUtilityCriterion is connected to the Attribute named Color to indicate that this criterion contains preference informations relating to that attribute. The NominalUtilityCriterion entity contains two valued entities. The first one links category Blue to the ExactMeasurement value one. The second one links category Red to value zero. UC2: UT, direct encoding. The user does not care of (or does not know) the exact color of each car, but she knows the utility of each car's color. She wants to encode directly the utility function from the set of alternatives to the range [0, 1] with no encoding of the color attribute. Note that this use case appears naturally when a proxy attribute is difficult to find (e.g. the beauty of a landscape, or the comfort of a car). The use case is presented in Figure 9. UC3: OT, with recoding. Assume the colors of each car are known, as displayed in Figure 10. The user wants to recode the colors (e.g. blue=3, red=10). To do this, she wants to encode the recoding function (i.e. the association between some set of possible colors and some set of numbers). Figure 11 presents the NominalToCardinal entity required for the recoding: two valued entities are used. The first one indicates that the category Blue maps to value 3 and the second one indicates that the category Red maps to value 10. Observe that this information together with the nominal performances defines a new attribute with cardinal performances. In algorithmic MCDA such conversions can be implemented as components taking as input the nominal performances and the recoding function represented by the NominalToCardinal entity, and outputting a new Attribute (here, named CardinalColor) with corresponding cardinal performances. This connects to the next use case. UC4: OT, with recoding and thresholds. The user wants to recode the colors (e.g. green=1, blue=3, red=10), and on top of that to define an outranking criterion quantifying the preference structure with thresholds. Figure 12 presents this case. Observe that an OutrankingCriterion is supposed to be linked with an Attribute that is associated with cardinal evaluations, otherwise its threshold values are not interpretable. Thus, for this use case, the user should encode the cars' measurements as cardinal values instead of nominal ones. This can be done for example by using the output of the component defined in the previous use case. UC5: UT, with cardinal performances. The user wants to code a utility function directly on the range of values of the car speeds using a linear utility function with utility 0 at speed ≤ 110 and 1 at speed ≥ 160. This is illustrated in Figure 13. UC6: UT or OT, with holistic evaluations. The user wants to provide holistic information car1 ≻ car2, that can subsequently be used to infer a set of preference models compatible with the information, for example as in the UTA family of methods [START_REF] Greco | Ordinal regression revisited: multiple criteria ranking using a set of additive value functions[END_REF]. For this case only the binary valued pair needs to be encoded as shown in Figure 14.

Encoding performance/preference information given as matrix of relative importances in methods such as AHP is trivial, as the valued relation can represent such information completely. We omit the presentation of these use cases for brevity.

Algorithmic MCDA

Algorithmic MCDA refers to composing the methods (workflows) from independent components communicating through strictly defined interfaces. Such components can be implemented as e.g. webservices. Workflow composition becomes considerably easier if there is only a small amount of different types available. For this purpose our XML implementation of the data model contains set versions of concepts presented in Section 3, such as AlternativeSet or AttributeSet. Although it is simple in XML to define such sets by using sequences, defining these types in the XML implementation of the data model permits to easily check for type compatibility when using the output of one component as an input to another component.

Let us now describe an example of composing an Electre III outranking relation computation using multiple components. The complete workflow is presented in Figure 15. Electre III, like most other outranking methods, consist of discrete steps of concordance computation, discordance computation, aggregation of the results into an outranking relation, and an exploitation of the outranking relation [START_REF] Roy | The outranking approach and the foundations of ELECTRE methods[END_REF]. The top component in Figure 15 computes the concordance valued relation. It requires a set of alternatives, a set of attributes, alternative performances on the given attributes, a set of criteria (one per attribute), criteria weights, and preference and indifference thresholds for each criterion. Therefore, the criteria used as input are outranking criteria. The component output is a valued relation, where the from and to attributes refer to alternatives given as input. This concordance valued relation may then be sent to a component called cut relation, which also takes as input a majority threshold parameter represented as an exact measurement, and cuts the valued relation into a binary one. Observe that this component is not specific to Electre and may hence be useful in other contexts as well.

The discordance component requires as input a set of alternatives, a set of attributes, performances for every alternative and attribute, and a set of criteria corresponding to the attributes. Each of the criteria may have an associated veto threshold. In Electre III, it is allowed to have criteria with no veto values defined on them, in which case the veto condition will never apply for that particular criterion. The discordance component computes a set of relations represented as valued relations, one for each criterion. To represent this, no ready made types exist in our data model, and a new type needs to be defined. However, it is easy to re-use existing types in the data model to define DiscordanceRelation elements to contain a sequence of pairs of criteria and valued relations. Finally, the binary concordance relation and valued discordance relations are sent to an outranking component that aggregates them and outputs an outranking relation in form of a binary relation.

Discussion

Although most MCDA methods share common elements such as alternatives, attributes and criteria, many also include method-specific concepts, and representing each of these in a standard data model would be pointless. The data model we propose in this paper contains the entities needed to represent data most commonly encountered in MCDA. This permits to have a reasonable size data model while enabling strong data typing. Because the inputs and outputs of different MCDA methods vary, each MCDA component should define its own input and output XML schemas. For example, the required input for electre concordance component is similar to the input required for electre discordance component, but not identical. The XML schemas may include elements having types defined in our data model, other standard types, or ad hoc types. For example, a component which outputs mathematical programs should reuse the adequate type to represent mathematical programs defined by the OS standard [START_REF] Fourer | An XML-based schema for stochastic programs[END_REF]. When no standard exists, for example due to the concept being specific to a new family of MCDA methods, a developer may define her own type. A separate XML namespace should be used to ensure that this new type has its own identity. Note that it is possible for the developer to share the newly defined type with other component developers working on similar methods.

As the data model enables strongly typed component interfaces, some types are not compatible even though subsets of their possible values have equal mathematical values. We suggest to use data converters in such cases. For example, an interval or an affine linear function can be represented in a particular case as an exact value. Should an ExactValue be used instead of an AffineLinear, as it does not inherit from AffineLinear, a data converter is required for transforming the ExactValue to an equivalent AffineLinear entity (i.e. one having the x coefficient zero). Similar technique is used in the rapidminer software.

Note that some of the types could be decomposed in looser coupled entities, e.g. outranking criteria could have the thresholds as separate types attached to the criterion entities. Such approach would allow for better decoupling and to define more data integrity constraints in component input schemas. However, the workflow design effort grows with the amount of decoupled types, and for this reason we have chosen to include the thresholds with the outranking criteria as optional attributes, and likewise functions within utility criteria.

The components of algorithmic MCDA could be implemented as web services and subsequently made available through a central registry (e.g. of the Decision Deck consortium). This would permit to create software that uses the components published as web services. The definition of standard types would make connecting components in a type safe manner trivial, for example to compose MCDA workflows in the diviz software, because it is easy to detect potential compatibility between the output of a component and the input of another one simply by looking at the types of the corresponding elements in the component XML schemas. Furthermore, our data model enables data interchange between different software and, for example, easy use of computational libraries of JSMAA [START_REF] Tervonen | JSMAA: open source software for SMAA computations[END_REF] and J-MCDA [START_REF] Cailloux | ELECTRE and PROMETHEE MCDA methods as reusable software components[END_REF]) in more domain-specific analysis software such as ADDIS [START_REF] Van Valkenhoef | ADDIS: a decision support system for evidence-based medicine[END_REF], that applies MCDA as the last component in a workflow of analyzing benefits and risks of medical treatments.

Fig. 2

 2 Fig. 2 Data model measurements.

Fig. 4

 4 Fig. 4 Data model criteria.

Fig. 5

 5 Fig. 5 Data model nominal to cardinal attribute conversion.

Fig. 6

 6 Fig. 6 Data model functions.

Fig. 7

 7 Fig.7Evaluation of car1 performances. The attributes of car1Speed and car1Color are actually references to the corresponding entities, that are omitted for brevity.

Fig. 8

 8 Fig. 8 Sample preference evaluation of UC1 (utility theory with nominal categories).

Fig. 9

 9 Fig. 9 Sample preference evaluation of UC2 (utility theory with direct encoding).

Fig. 10

 10 Fig. 10 Sample performance evaluation of UC3 (outranking theory with recoding). The cars are evaluated with a nominal attribute.

Fig. 11

 11 Fig. 11 Sample recoding function for UC3 (outranking theory with recoding).

Fig. 12

 12 Fig.12Sample preference evaluation of UC4 (outranking theory with recoding). The car evaluations have been recoded to a cardinal attribute (see Figures10 and 11), which permits to define thresholds.

Fig. 13

 13 Fig. 13 Sample preference evaluation of UC5 (utility theory with direct ratings).

Fig. 14

 14 Fig. 14 Sample preference evaluation of UC6 (holistic preference statement).

Fig. 15

 15 Fig. 15 Electre flow with four components: concordance computation, fuzzy-to-crisp relation cut, discordance computation, and outranking relation computation.

This representation omits the measurement hierarchy (displayed in Figure

2) and set types.

Appendix A: Data model 1