Nicola Soave

Hugo Tavares

AND Susanna Terracini

Alessandro Zilio

VARIATIONAL PROBLEMS WITH LONG-RANGE INTERACTION

Keywords: Free boundary condition, Lipschitz regularity, Long-range interaction, Optimal partition problems, Segregation problems, Variational methods

We consider a class of variational problems for densities that repel each other at distance. Typical examples are given by the Dirichlet functional and the Rayleigh functional

functions attaining some boundary conditions on ∂Ω, and subjected to the constraint dist({u i > 0}, {u j > 0}) 1 ∀i = j.

For these problems, we investigate the optimal regularity of the solutions, prove a free-boundary condition, and derive some preliminary results characterizing the free boundary ∂{ k i=1 u i > 0}.

Introduction

The object of this paper is the study of a class of minimal configurations for variational problems involving arbitrarily many densities related by long-range repulsive interactions. The mathematical setting we consider is described by the following two archetypical situations.

Problem (A)

Let Ω be a bounded domain of R N , N 2, and let

Ω 1 = x∈Ω B 1 (x) = {x ∈ R N : dist(x, Ω) < 1}.
Given k 2 nonnegative nontrivial functions f 1 , . . . , f k ∈ H 1 (Ω 1 ) ∩ C(Ω 1 ) satisfying 1 dist(supp f i , supp f j ) 1 ∀i = j, we consider the minimization problem

inf u∈H∞ J ∞ (u),
where the set H ∞ and the functional J ∞ are defined by (1.1)

H ∞ = u = (u 1 , . . . , u k ) ∈ H 1 (Ω 1 , R k ) dist(supp u i , supp u j ) 1 ∀i = j u i = f i a.e. in Ω 1 \ Ω , and
J ∞ (u) = k i=1 ˆΩ |∇u i | 2 .
The support of each component u i is taken in the weak sense: it corresponds to the complement in Ω 1 of the largest open set ω ⊆ R N where u i = 0 a.e. on ω (cf. [START_REF] Brezis | Functional analysis, Sobolev spaces and partial differential equations[END_REF]Proposition 4.17]). Notice also that the existence of f 1 , . . . , f k with the above properties imposes some conditions on Ω (for instance, the diameter of Ω cannot be too small), and we suppose that such conditions are satisfied. We are interested in existence and qualitative properties of minimizers.

Problem (B)

Let Ω be a bounded domain of R N , N 2, and let k 2. We consider the set of open partitions of Ω at distance 1, defined as

P k (Ω) = (ω 1 , . . . , ω k ) ω i
⊂ Ω is open and non-empty for every i, and dist(ω i , ω j ) 1 ∀i = j .

Then, for a cost function F ∈ C 1 ((R + ) k , R) satisfying • ∂ i F (x) > 0 for all x ∈ (R + ) k and i = 1, . . . , k, which in particular yields that F is component-wise increasing; • for any given i = 1, . . . , k, F (λ 1 (ω 1 ), . . . , λ 1 (ω k )),

where λ 1 (ω) is the first eigenvalue of the Laplace operator in ω with homogeneous Dirichlet boundary conditions. Problem (1.2) is a particular case of an optimal partition problem (cf. [START_REF] Bourdin | Optimal partitions for eigenvalues[END_REF][START_REF] Bucur | Existence results for some optimal partition problems[END_REF]). A typical case we have in mind is the cost function F (λ 1 (ω 1 ), . . . , λ 1 (ω k )) = k i=1 λ 1 (ω i ). We are interested in existence and qualitative properties of an optimal partition.

Our main results are, for problem (A):

• the existence of a minimizer;

• the optimal interior regularity of any minimizer;

• the derivation of several properties of the positivity sets {u i > 0};

• the derivation of a free boundary condition involving the normal derivatives of different components of any minimizers on the regular part of the free-boundary ∂{u i > 0}. For problem (B):

• the introduction of a weak formulation in terms of densities, and the existence of weak solutions; • the global optimal regularity of any weak solution, which leads in particular to the existence of a strong solution for the original problem; • the derivation of properties of the subsets ω i , and of a free boundary condition on the regular part of ∂ω i . In a forthcoming paper, we will study more in detail the regularity of the free-boundary.

We stress that, both in problems (A) and (B), the interaction among different densities takes place at distance: in problem (A) the positivity sets {u i > 0}, and in problem (B) the open subsets ω i , are indeed forced to stay at a fixed minimal distance from each other.

When the interaction among the densities takes place point-wisely, segregation problems analogue to (A) and (B) have been studied intensively, in connection with optimal partition problems for Laplacian eigenvalues [START_REF] Caffarelli | An optimal partition problem for eigenvalues[END_REF][START_REF] Conti | Nehari's problem and competing species systems[END_REF][START_REF] Conti | An optimal partition problem related to nonlinear eigenvalues[END_REF][START_REF] Conti | On a class of optimal partition problems related to the Fučík spectrum and to the monotonicity formulae[END_REF][START_REF] Ramos | Extremality conditions and regularity of solutions to optimal partition problems involving Laplacian eigenvalues[END_REF][START_REF] Tavares | Regularity of the nodal set of segregated critical configurations under a weak reflection law[END_REF][START_REF] Tavares | Sign-changing solutions of competition-diffusion elliptic systems and optimal partition problems[END_REF], with the regularity theory of harmonic maps into singular manifold [START_REF] Caffarelli | Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries[END_REF][START_REF] Conti | A variational problem for the spatial segregation of reaction-diffusion systems[END_REF][START_REF] Tavares | Regularity of the nodal set of segregated critical configurations under a weak reflection law[END_REF], and with segregation phenomena for systems of elliptic equations arising in quantum mechanics driven by strong competition [START_REF] Caffarelli | Singularly perturbed elliptic systems and multi-valued harmonic functions with free boundaries[END_REF][START_REF] Dancer | The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture[END_REF][START_REF] Noris | Uniform Hölder bounds for nonlinear Schrödinger systems with strong competition[END_REF][START_REF] Soave | Hölder bounds and regularity of emerging free boundaries for strongly competing Schrödinger equations with nontrivial grouping[END_REF][START_REF] Soave | Uniform bounds for strongly competing systems: the optimal Lipschitz case[END_REF][START_REF] Soave | On phase separation in systems of coupled elliptic equations: Asymptotic analysis and geometric aspects[END_REF][START_REF] Wei | Asymptotic behaviour of solutions of planar elliptic systems with strong competition[END_REF].

In contrast, the only results available so far regarding segregation problems driven by long-range competition are given in [START_REF] Caffarelli | On a long range segregation model[END_REF], where the authors analyze the spatial segregation for systems of type

(1.3) -∆u i,β = -βu i,β j =i (1 B1 |u j | p ) in Ω u i,β = f i ≥ 0 in Ω 1 \ Ω,
with 1 p +∞. In the above equation, 1 B1 denotes the characteristic function of B 1 , the ball2 of center 0 and radius 1, and stays for the convolution for p < +∞, so that

(1 B1 |u j | p )(x) = ˆB1(x) |u j (y)| p dy ∀x ∈ Ω, with 1 p < +∞;
in case p = +∞, we intend that the integral is replaced by the supremum over B 1 (x) of |u j |. In [START_REF] Caffarelli | On a long range segregation model[END_REF], the authors prove the equi-continuity of families of viscosity solutions {u β : β > 0} to (1.3), the local uniform convergence to a limit configuration u, and then study the free-boundary regularity of the positivity sets {u i > 0} in cases p = 1 and p = +∞, mostly in dimension N = 2. As we shall see, our problem (A) is strictly related with the asymptotic study of the solutions to (1.3) in case p = 2 (see the forthcoming Theorem 2.1); nevertheless, also in such a situation our approach is very different with respect to the one in [START_REF] Caffarelli | On a long range segregation model[END_REF], since we heavily rely on the variational nature of the problem. This gives differenti free boundary conditions which requires different techniques, and allows us to prove new results.

Regarding problem (1.3), we also refer to [START_REF] Bozorgnia | Uniqueness result for long range spatially segregation elliptic system[END_REF], where the author proves uniqueness results in the cases p = 1 and p = +∞.

1.1. Main results. We adopt the notation previously introduced. First of all, we have the following existence results for problems (A) and (B).

Theorem 1.1 (Problem (A))

. There exists a minimizer u = (u 1 , . . . , u k ) for inf H∞ J ∞ .

Theorem 1.2 (Problem (B))

. There exists a minimizer (ω 1 , . . . , ω k ) ∈ P k for (1.2).

Observe that, to each optimal partition (ω 1 , . . . , ω k ), we can associate a vector of signed first eigenfunctions. To fix ideas, from now on we always consider nonnegative eigenfunctions. The second part of our analysis concerns the properties satisfied by any minimizer of problems (A) and (B).

Theorem 1.3. Let u = (u 1 , . . . , u k ) be either any minimizer of J ∞ in H ∞ , or a vector of first eigenfunctions associated to an optimal partition (ω 1 , . . . , ω k ) of (1.2). Then u is a vector of nonnegative functions in Ω, and denoting by S i the positivity set {x ∈ Ω : u i > 0}, for every i = 1, . . . , k, we have:

(1) Subsolution in Ω: We have that

-∆u i 0 in distributional sense in Ω, if u is a solution to problem (A), -∆u i λ 1 (ω 1 )u i in distributional sense in Ω, if u is a solution to problem (B). (2) Solution in S i : We have that -∆u i = 0 in int(S i ), if u is a solution to problem (A), -∆u i = λ 1 (ω i ) in int(S i ), if u is a solution to problem (B).
(3) Exterior sphere condition for the positivity sets: S i satisfies the 1-uniform exterior sphere condition in Ω, in the following sense: for every x 0 ∈ ∂S i ∩ Ω there exists a ball B with radius 1 which is exterior to S i and tangent to S i at x 0 , i.e.

S i ∩ B = ∅ and x 0 ∈ S i ∩ B.
Moreover, in B ∩ B 1 (x 0 ) we have u j ≡ 0 for every j = 1, . . . , k (including j = i). (4) Lipschitz continuity: u i is Lipschitz continuous in Ω, and in particular S i is an open set, for every i. (5) Lebesgue measure of the free-boundary: the free-boundary ∂{u i > 0} has zero Lebesgue measure, and its Hausdorff dimension is strictly smaller than N . (6) Exact distance between the supports: for every x 0 ∈ ∂S i ∩ Ω there exists j = i such that

B 1 (x 0 ) ∩ ∂ supp u j = ∅.
Notice that, if y 0 ∈ ∂S j is such that |x 0 -y 0 | = 1, then B 1 (y 0 ) is an exterior sphere to S i at x 0 . Moreover, by the Hopf lemma, the interior Lipschitz regularity is optimal.

Regarding the regularity of a vector of eigenfunctions u of problem (B), if we ask that Ω satisfies the exterior sphere condition, then we have actually a stronger statement.

Theorem 1.4. Let u be a vector of first eigenfunctions associated to an optimal partition (ω 1 , . . . , ω k ) of (1.2). Assume that Ω satisfies the exterior sphere condition with radius r > 0. Then u is globally Lipschitz continuous in Ω.

Next, we establish a relation involving the normal derivatives of two "adjacent components" on the regular part of the free boundary.

In what follows, for each i, ν i (x) will denote the exterior normal at a point x ∈ ∂S i (at points where such a normal vector does exist).

Assumptions. Let x 0 ∈ ∂S i ∩ Ω, and let us assume that Γ R i := ∂S i ∩ B R (x 0 ) is a smooth hypersurface, for some R > 0. By the 1-uniform exterior sphere condition, we know that the principal curvatures of ∂S i in x 0 , denoted by χ i h (x 0 ), h = 1, . . . , N -1, are smaller than or equal to 1 (where we agree that outward is the positive direction). We further suppose that the strict inequality holds, that is there exists δ > 0 such that (1.4) χ i 1 (x 0 ), . . . , χ i N -1 (x 0 ) 1 -δ. We know that there exists j = i and y 0 ∈ ∂ supp u j such that |x 0 -y 0 | = 1.

Theorem 1.5. Let u = (u 1 , . . . , u k ) be either any minimizer of J ∞ in H ∞ , or a vector of first eigenfunctions associated to an optimal partition (ω 1 , . . . , ω k ) of (1.2). Under the previous assumptions and notations, we have that

y 0 = x 0 + ν i (x 0 ) is the unique point in k =i ∂ supp u k at distance 1 from x 0 . If y 0 ∈ ∂ supp u j ∩ Ω, then ∂ supp u j
is also smooth around y 0 , and

(1.5) (∂ ν u i (x 0 )) 2 (∂ ν u j (y 0 )) 2 =          N -1 h=1 χ i h (x0) =0 χ i h (x 0 ) χ j h (y 0 ) if χ i h (x 0 ) = 0 for some h, 1 if χ i h (x 0 )
= 0 for all h = 1, . . . , N -1. We stress that, since the sets S i and S j are at distance 1 from each other and (1.4) holds, χ i h (x 0 ) = 0 if and only if χ j h (y 0 ) = 0, and hence the term on the right hand side is always well defined.

The proof of Theorem 1.5 is based on the introduction of a family of domain variations for the minimizer u. As we shall see, the possibility of producing admissible domain variations, preserving the constraint on the distance of the supports in H ∞ , presents major difficulties. At the moment, we can only overcome such obstructions and produce more or less explicit variations supposing that ∂S i is locally regular. This is the main problem when trying to study the regularity of the free boundary. Regarding this point, we mention that the proofs of all our results (and also of those in [START_REF] Caffarelli | On a long range segregation model[END_REF], in a nonvariational case) are completely different with respect to the analogue counterpart in problems with point-wise interaction. Indeed, all the local techniques, such as blow-up analysis and monotonicity formulae, cannot be straightforwardly adapted when dealing with long-range interaction; the reason is that the interface between different positivity sets {u i > 0} and {u j > 0} with i = j is now a strip of width at least 1, and hence with a standard blow-up one cannot catch the interaction on the free-boundary at the limit.

We also mention that the validity of a uniform exterior sphere condition does not directly imply any extra regularity for ∂S i : if we could show that ∂S i is a set with positive reach (see [START_REF] Federer | Curvature measures[END_REF]), then we could argue as in [START_REF] Caffarelli | On a long range segregation model[END_REF]Corollary 6.3] and prove at least that the Hausdorff dimension of ∂S i is N -1 (see also [START_REF] Colombo | Differentiability properties for a class of non-convex functions[END_REF]Theorem 4.2] for a different proof of this fact), but on the other hand sets enjoying the uniform exterior sphere condition are not necessarily of positive reach, as shown in [START_REF] Nour | Proximal smoothness and the exterior sphere condition[END_REF]Section 2].

Remark 1.6. A very interesting feature of Theorem 1.5 stays in the fact that it reveals a deep difference between segregation models with point-wise interaction, and with long-range interaction. To explain this difference, let us consider a sequence {u β } of solutions to (1.3), with p = 1 and β → +∞. This is the setting studied in [START_REF] Caffarelli | On a long range segregation model[END_REF]. In [START_REF] Caffarelli | On a long range segregation model[END_REF]Theorem 9.1], the authors derive a free-boundary condition analogous to (1.5) for the limit configurations in case p = 1, but in their situation, the left hand side is replaced by the ratio between the normal derivatives, ∂ ν u i (x 0 )/∂ ν u j (y 0 ). This difference is in contrast with respect to segregation phenomena with point-wise interaction, where, as proved in [START_REF] Tavares | Regularity of the nodal set of segregated critical configurations under a weak reflection law[END_REF], limit configurations associated with

-∆u i = -βu i j =i u j or -∆u i = -βu i j =i u 2 j
belong to the same functional class [START_REF] Dancer | The limit equation for the Gross-Pitaevskii equations and S. Terracini's conjecture[END_REF][START_REF] Tavares | Regularity of the nodal set of segregated critical configurations under a weak reflection law[END_REF], and hence in particular satisfy the same free-boundary condition, that is |∂ ν u i (x 0 )| = |∂ ν u j (x 0 )| on the regular part of the free boundary. A similar difference has been observed in [START_REF] Terracini | Uniform Hölder regularity with small exponent in competition-fractional diffusion systems[END_REF][START_REF] Terracini | Uniform Hölder bounds for strongly competing systems involving the square root of the laplacian[END_REF][START_REF] Verzini | Strong competition versus fractional diffusion: the case of Lotka-Volterra interaction[END_REF] in the case of fractional operators, that is when the non-locality is in the differential operator.

Finally, in comparison with the free boundary condition derived in [START_REF] Caffarelli | On a long range segregation model[END_REF], it is worthwhile noticing that the analogue of (1.5) there involves the plain quotient of the normal derivatives, while here we find the squared one.

Remark 1.7. The previous result may fail if the right hand side in (1.4) is replaced by the constant 1. Indeed, if

∂S i ∩ B R (x 0 ) = ∂B 1 (0) ∩ B R (x 0 )
for some x 0 ∈ ∂B 1 (0) and R > 0, and the set S i is contained in the exterior of B 1 (0), then y 0 = 0 is a cusp for ∂S j .

1.2. Structure of the paper. We first treat problem (A). In Section 2 we prove Theorem 1.1 for this problem, relating this segregation problem with a variational competition-diffusion of type (1.3). Then some qualitative properties of any possible minimizer of problem (A) are shown in Section 3, where we prove Theorem 1.3 for this problem. Section 4 contains the proof of the free boundary condition contained in the statement of Theorem 1.5 for problem (A).

The analogous statements for problem (B) -existence and properties of minimizers, and free boundary condition -are proved in Section 5.

Finally, in Appendix A we state and prove an Hadamard's type formula which we need along this paper.

Existence of a minimizer for Problem (A)

In this section we prove Theorem 1.1. To this purpose, we introduce a competition parameter β > 0 which allows us to remove the segregation constraint. To be precise, let

H = {u ∈ H 1 (Ω 1 , R k ) : u i = f i a.e. in Ω 1 \ Ω} ⊃ H ∞ ,
and let β > 0. We consider the minimization of the functional

J β (u) = k i=1 ˆΩ |∇u i | 2 + 1 i<j k ¨Ω1×Ω1 β 1 B1 (x -y)u 2 i (x)u 2 j (y) dx dy
in the set H. With respect to the search of a minimizer for inf H∞ J ∞ , the advantage stays in the fact that we can get rid of the infinite dimensional constraint dist(supp u i , supp u j ) 1 for i = j, and we can easily show that a minimizer for J β in H does exists, and satisfies an Euler-Lagrange equation of type (1.3) with p = 2. This allows us to obtain Theorem 1.1 as a direct corollary of the following statement:

Theorem 2.1. For every β > 0, there exists a minimizer

u β = (u 1,β , . . . , u k,β ) for inf H J β , which is a solution of (2.1)      -∆u i = -βu i j =i (1 B1 u 2 j ) in Ω u i > 0 in Ω u i = f i in Ω 1 \ Ω.
The family

{u β : β > 0} is uniformly bounded in H 1 (Ω 1 , R k ) ∩ L ∞ (Ω 1 )
, and there exists u = (u 1 , . . . , u k ) ∈ H such that:

(1) u β → u strongly in H 1 (Ω) as β → +∞, up to a subsequence;

(2) dist(supp u i , supp u j ) 1 for every i = j, so that u ∈ H ∞ ;

(3) for every i = j,

lim β→+∞ ¨Ω1×Ω1 1 B1 (x -y)u 2 i,β (x)u 2 j,β (y) dx dy = 0 (4) u is a minimizer for inf H∞ J ∞ .
In particular, u is a solution to problem (A).

Remark 2.2. Without any additional complication, we can replace in the previous theorem the indicator function 1 B1 with a more general function

V ∈ L ∞ (R N ) satisfying V > 0 a.e. in B 1 , V = 0 a.e. on R N \ B 1 .
The proof of Theorem 2.1 is the object of the rest of the section. Before proceeding, we observe that, by the definition of support given in [3, Proposition 4.17], the set H ∞ can be defined in the following equivalent way:

H ∞ = u ∈ H : ¨Ω1×Ω1 1 B1 (x -y)u 2 i (x)u 2 j (y) dx dy = 0 ∀i = j
(see the proof of Lemma 3.1 below for more details).

Remark 2.3. Here it is worth to stress that we consider the functions u i as defined in Ω 1 , and hence the supports have to be considered in this set (and not only in Ω).

Proof of Theorem 2.1. The existence of a minimizer u β follows by the direct method of the calculus of variations, and the fact that minimizers solve (2.1) is straightforward. Observe that f i 0, hence the minimizers are positive in Ω, by the strong maximum principle.

For the uniform L ∞ estimate, since u i,β > 0 is subharmonic in Ω for every i = 1, . . . , k, by the maximum principle we have

u i,β L ∞ (Ω) f i L ∞ (∂Ω) . Let us set c β := inf H J β and c ∞ := inf H∞ J ∞ .
We observe that, since

J β (v) = J ∞ (v) for every v ∈ H ∞ , we have c β c ∞ .
Then, by the minimality of u β , for every β > 0 we have

J β (u β ) c ∞ . Since moreover u i,β ≡ f i in Ω 1 \ Ω, the uniform H 1 (Ω 1 , R k ) boundedness of {u β } follows. Hence, up to a subsequence, u β u weakly in H 1 (Ω 1 , R k ) and a.e. in Ω. Moreover lim β→+∞ ¨Ω1×Ω1 1 B1 (x -y)u 2 i (x)u 2 j (y) dx dy = 0 ∀i = j
and by the Fatou lemma we have 0

¨Ω1×Ω1 1 B1 (x -y)u 2 i (x)u 2 j (y) dx dy lim inf β→+∞ ¨Ω1×Ω1 1 B1 (x -y)u 2 i,β (x)u 2 j,β (y) dx dy = 0
for every i = j. This in particular proves point (2) in the thesis and implies that u ∈ H ∞ , defined in (1.1).

On the other hand, by the the minimality of u β and weak convergence,

c ∞ J ∞ (u) = k i=1 ˆΩ |∇u i | 2 lim inf β→∞ k i=1 ˆΩ |∇u i,β | 2 lim sup β→∞ k i=1 ˆΩ |∇u i,β | 2 lim sup β→∞ J β (u β ) = lim sup β→∞ c β c ∞ .
This means that all the previous inequalities are indeed equalities, and in particular:

• we have convergence ∇u i,β L 2 (Ω) → ∇u i L 2 (Ω)
, which together with the weak convergence ensures that u β → u strongly in H 1 (Ω, R k ) (recall that Ω is bounded); • point (3) of the thesis holds;

• we have c ∞ = J ∞ (u), which proves the minimality of u ∈ H ∞ .

Properties of minimizers for problem (A)

This section is devoted to the proof of Theorem 1.3 for the solutions of problem (A). Let then u be a minimizer for inf H∞ J ∞ . Theorem 1.1 (see also Theorem 2.1) does not give any information about the continuity of u i , and in particular we do not know if the sets S i = {x ∈ Ω : u i (x) > 0} are open. On the other hand it is reasonable to work at a first stage with the functions

Φ i : Ω → R, Φ i (x) := ˆB1(x) u 2 i (y) dy,
C i = Ω ∩   y∈{Φi=0} B 1 (y)   , D i := int (Ω \ C i ) , for i = 1, . . . , k, so that Ω = C i ∪ D i ∪ (∂D i ∩ Ω), and
∂D i ∩ Ω = ∂C i ∩ Ω.
Observe that, by the definition of Φ i , we have u i = 0 a.e. in C i . Moreover

D i = {x ∈ Ω : dist(x, {Φ i = 0}) > 1} ⊂ {Φ i > 0}.
The strategy of the proof of Theorem 1.3 can be summarized as follows:

• At first, we prove some simple properties of the set D i and of the restriction of u on D i .

• In particular, we show that S i is the union of connected components of D i , so that the regularity of u i in Ω is reduced to the regularity of u i on ∂D i . • Using the basic properties of D i , we show that u i is locally Lipschitz continuous across ∂D i , and hence in Ω. It follows in particular that S i is open, and directly inherits from D i properties ( 3) and ( 5) in Theorem 1.3. Moreover, points (1) and ( 2) holds. • As a last step, we prove point (6) by using the minimality of u.

Lemma 3.1. The function u i is harmonic in D i . In particular, if Di is any connected component of D i , then either u i ≡ 0 or u i > 0 in Di . Proof. The set D i is open. If we know that dist(D i , supp u j )
1, then we can consider any φ ∈ C ∞ c (D i ) and observe that, by the minimality of u for J ∞ on the set H ∞ , the function

f (ε) := J ∞ (u 1 , . . . , u i-1 , u i + εφ, u i+1 , . . . , u k )
has a minimum at ε = 0. This implies that u i is harmonic in D i , and all the other conclusions follow immediately. Therefore, in what follows we have to show that

(3.1) dist(D i , supp u j ) 1 ∀j = i.
By definition of H ∞ we have

u 2 i (x)u 2 j (y)1 B1 (x -y) = 0 for a.e. x, y ∈ Ω 1 , that is u 2 i (x)u 2 j (y) = 0 for a.e. x, y ∈ Ω 1 , |x -y| < 1.
As a consequence, u j (x)Φ i (x) = 0 for a.e. x ∈ Ω and every j = i. In particular, this implies that

(3.2) {Φ i > 0} ⊂ (Ω \ supp u j ) . Let x 0 ∈ D i .
Then by definition of D i , dist(x 0 , {Φ i = 0}) > 1, and hence B 1 (x 0 ) ⊂ {Φ i > 0}. But then, due to (3.2), and since x 0 has been arbitrarily chosen, we deduce that (3.1) holds.

Let A i be the union of the connected components of D i on which u i > 0, and let N i be the union of those on which u i ≡ 0, so that D i = A i ∪ N i . We know that u i is positive and harmonic in A i , while u i = 0 a.e. in N i ∪ C i . Since A i , N i and C i are open, this means that (if necessary replacing u i with a different representative in its same equivalence class) u i is continuous in A i , N i , and C i . To discuss the continuity of u i in Ω, we have to derive some properties of the boundary

∂D i ∩ Ω = (∂A i ∪ ∂N i ) ∩ Ω = ∂C i ∩ Ω.
In the next lemma we show that D i satisfies a uniform exterior sphere condition. Lemma 3.2. For each i, the set D i satisfies the 1-uniform exterior sphere condition in Ω, in the following sense: for every x 0 ∈ ∂D i ∩ Ω there exists a ball B of radius 1 such that

D i ∩ B = ∅ and x 0 ∈ D i ∩ B.
Moreover, in B we have u i ≡ 0.

Proof. This comes directly from the definitions: we have

∂D i ∩ Ω = ∂C i ∩ Ω = {x : dist(x, {Φ i = 0}) = 1} ∩ Ω.
Thus, given x ∈ ∂D i ∩ Ω, there exists y ∈ ∂B 1 (x) with Φ i (y) = 0. The ball B 1 (y) is the desired exterior tangent ball, since B 1 (y) ⊂ C i , and hence B 1 (y)

∩ D i = ∅.
The exterior sphere condition permits to deduce that ∂D i has zero Lebesgue measure.

Lemma 3.3. The boundary ∂D i is a porous set, and in particular it has 0 Lebesgue measure and dim H (∂D i ) < N .

For the definition of "porosity", we refer to [20, Section 3.2], while here and in what follows dim H denotes the Hausdorff dimension.

Proof. Since ∂D i ⊂ Ω is bounded, to prove its porosity it is sufficient to show that there exists δ > 0 such that: for every ball B r (x 0 ) with x 0 ∈ ∂D i , there exists y ∈ B r (x 0 ) with B δr (y) ⊂ B r (x 0 )\∂D i (see [START_REF] Petrosyan | Regularity of free boundaries in obstacle-type problems[END_REF]Exercise 3.4]).

The existence of such δ = 1/2 follows immediately by the exterior sphere condition: given x 0 ∈ ∂D i , there exists z ∈ Ω 1 such that B 1 (z) is exterior to D i . Let then y be the point on the segment x 0 z at distance r/2 from D i . The ball B r/2 (y) is contained both in Ω 1 \ ∂D i and in B r (x 0 ), and this proves that ∂D i is porous. The rest of the proof follows by [START_REF] Petrosyan | Regularity of free boundaries in obstacle-type problems[END_REF]Page 62].

It is not difficult now to deduce that u i is continuous at every point of ∂N i . Indeed, notice that ∂N i ⊂ ∂C i , and in both N i and C i we have u i ≡ 0. Since ∂N i ⊂ ∂D i has 0 Lebesgue measure, we deduce that u i = 0 a.e. in N i ∪ C i = Ω \ A i . That is, up to the choice of a different representative, u i ≡ 0 in Ω \ A i , and hence it is real analytic therein. At this stage, it remains to discuss the continuity of u i on ∂A i . This is the content of the forthcoming Corollary 3.6, where we show that actually u is locally Lipschitz continuous in Ω. We postpone the proof, proceeding here with the conclusion of Theorem 1.3. The continuity of u i implies in particular that {u i > 0} is open for every i, so that {u i > 0} = A i . Thus, Lemmas 3.1-3.3 establish the validity of points ( 2) and ( 5) in Theorem 1.3. The subharmonicity of u i , point (1), follows from [START_REF] Bozorgnia | Uniqueness result for long range spatially segregation elliptic system[END_REF].

Regarding point (3), the existence of an exterior sphere B of radius 1 for {u i > 0} at any boundary point x 0 comes directly from Lemma 3.2. We also know that u i ≡ 0 in B, and furthermore, by (3.1), B 1 (x 0 ) ∩ supp u j = ∅ for every j = i. This proves the validity of (3).

It remains only to show that also point (6) holds.

Proof of Theorem 1. 3-(6). This is a consequence of the minimality. Take x 0 ∈ ∂S i ∩ Ω and assume, in view of a contradiction, that dist(x 0 , supp u j ) > 1 for some x 0 ∈ ∂S i ∩ Ω, for every j = i. Then there exists ρ > 0 such that B ρ (x 0 ) ⊂ Ω and

(3.3) dist(B ρ (x 0 ), supp u j ) > 1 ∀j = i.
Let v be the harmonic extension of u i in B ρ (x 0 ):

∆v = 0 in B ρ (x 0 ) v = u i on ∂B ρ (x 0 ).
Since u i ≡ 0 on ∂B ρ (x 0 ), we infer that v > 0 in B ρ (x 0 ), and in particular v ≡ u i in B ρ (x 0 ). Let now ũ be defined by

ũi = u i in Ω \ B ρ (x 0 ) v in B ρ (x 0 ) , ũj = u j ∀j = i.
Due to (3.3), it belongs to H ∞ , so that by minimality J ∞ (u) J ∞ (ũ). On the other hand, by the definition of harmonic extension we have also J ∞ (ũ) < J ∞ (u) (the strict inequality comes from the fact that v ≡ u i in B ρ (x 0 )), a contradiction.

Remark 3.4. In [START_REF] Caffarelli | On a long range segregation model[END_REF], the authors proved harmonicity, local Lipschitz continuity, and exterior sphere condition for limits of any sequence of solutions to (2.1). Nevertheless, the result here is not contained in [START_REF] Caffarelli | On a long range segregation model[END_REF], since we establish harmonicity, Lipschitz continuity, and exterior sphere condition for any minimizer of inf H∞ J ∞ , independently on wether it can be approximated with a sequence of solutions to (2.1) or not. Also, it is worth to point out that the approach is completely different: while in [START_REF] Caffarelli | On a long range segregation model[END_REF] the authors proceed with careful uniform estimates for viscosity solution of (1.3), here we use the variational structure of the limit problem.

3.1. Lipschitz continuity of the minimizers. In this subsection we show that the solutions of problem (A) are Lipschitz continuous inside Ω, which is the highest regularity one can expect for the minimizers of J ∞ (by the Hopf lemma). This is a consequence of the following general statement.

Theorem 3.5. Let Λ be a domain of R N , and let A ⊂ Λ be an open subset, satisfying the r-uniform exterior sphere condition in Λ: for any x 0 ∈ ∂A ∩ Λ there exists a ball B with radius r which is exterior to A and tangent to ∂A at x 0 , i.e.

A ∩ B = ∅ and x 0 ∈ A ∩ B. Let f ∈ L ∞ (Λ), and let u ∈ H 1 (Λ) ∩ L ∞ (Λ) satisfy -∆u = f in A u = 0 a.e. in Λ \ A
Then u is locally Lipschitz continuous in Λ, and for every compact set K Λ there exists a constant C = C(r, N, K) > 0 such that

∇u L ∞ (K) C u L ∞ (Λ) + f L ∞ (Λ) .
For the sake of generality, we required no sign condition on the function u, even though we will apply the result only to nonnegative solutions. Corollary 3.6. Let u be any minimizer of J ∞ in H ∞ . Then u is locally Lipschitz continuous in Ω.

Proof. We apply Theorem 3.5 to the harmonic functions u i in A := A i , with Λ := Ω and r = 1.

The proof of Theorem 3.5 is based upon a simple barrier argument. For any R > 0, let us define

w R (x) := 1 2N (R 2 -|x| 2 ) + =⇒ -∆w R = 1 in B R w R = 0 in R N \ B R ,
and let

(3.4) w * R (x) := R |x| N -2 w R R 2 |x| 2 x = R N 2N |x| N |x| 2 -R 2 +
be its Kelvin transform with respect to the sphere of radius R. It is not difficult to check that

(3.5) -∆w * R (x) = - R |x| N +2 ∆w R R 2 |x| 2 x = R |x| N +2
.

With this preliminary observation, we can easily prove the following estimate:

Lemma 3.7. Let x 0 ∈ ∂A ∩ Λ, and let ρ > 0 be such that B ρ (x 0 ) Λ. Under the assumptions of Theorem 3.5, there exists a constant C > 0 depending on the dimension N , on r and on ρ,

such that |u(x)| C u L ∞ (Λ) + f L ∞ (Λ) |x -x 0 | ∀x ∈ B ρ (x 0 ).
Proof. Let y 0 ∈ R N be the center of the exterior sphere in x 0 :

A ∩ B r (y 0 ) = ∅ and x 0 ∈ A ∩ B r (y 0 )
Let z 0 be the medium point on the segment x 0 y 0 . Up to a rigid motion, we can suppose that z 0 = 0 and that x 0 = (0 , r/2), where 0 denotes the 0 vector in R N -1 . In this setting, we aim at proving that u w * r/2 in B ρ (x 0 ) ∩ A, with w * r/2 defined by (3.4). Since u = 0 a.e. in Ω \ A, we have (in the sense of traces) that u = 0 on ∂A ∩ B ρ (x 0 ). Moreover, since B r/2 (z 0 ) ⊂ B r (y 0 ), and ∂B r/2 (z 0 ) ∩ ∂B r (y 0 ) = {x 0 }, there exists a value δ = δ(r, ρ, N ) (independent on the point

x 0 ) such that dist(z 0 , A ∩ ∂B ρ (x 0 )) dist(z 0 , ∂B r (y 0 ) ∩ ∂B ρ (x 0 )) > r/2 + δ. Hence inf A∩∂Bρ(x0) w * r/2 m(r, ρ, N ) > 0,
and we can define

ϕ(x) := u L ∞ (Ω) m(r, ρ, N ) + 2ρ r N +2 f L ∞ (Λ) w * r/2 (x).
It is now not difficult to check that

-∆(ϕ -u) 0 in A ∩ B ρ (x 0 ) (ϕ -u) 0 on ∂(A ∩ B ρ (x 0 ))
Indeed, in A ∩ B ρ (x 0 ), by recalling (3.5),

-∆u = f ≤ f L ∞ (Λ) and -∆w * r/2 = r 2|x| N +2 ≥ r 2ρ N +2
The boundary ∂(A ∩ B ρ (x 0 )) splits into two parts. On the first part ∂A ∩ B ρ (x 0 ) we know that u = 0 in the sense of traces, and since ϕ 0 there, we have ϕ -u 0 on ∂A ∩ B ρ (x 0 ) in the sense of traces. On the remaining part A ∩ ∂B ρ(x0) , the function u can be evaluated point-wisely, since in the interior of A the function u is of class C 1,α ; therefore, it makes sense to write that u(x) u L ∞ (Ω) ϕ for any x ∈ A∩∂B ρ(x0) . All together, we obtain that u ϕ on ∂(A∩B ρ (x 0 )) in the sense of traces.

In conclusion, we have u ϕ in A ∩ B ρ (x 0 ) by the maximum principle. Observing that

r N 2 N +1 N |x| N |x| 2 - r 2 2 = r N 2 N +1 N |x| N |x| + r 2 |x| - r 2 r N 2 N +1 N (r/2) N ρ + r 2 (|x| -|x 0 |) 2 N -1 (2ρ + r) N |x -x 0 |.
for every x ∈ B ρ (x 0 ), we obtain the desired upper estimate for u. Arguing in the same way on -u, we obtain also the lower estimate, and the proof is complete.

As an immediate consequence:

Corollary 3.8. For every compact set K Λ there exists

C = C(K, r, N ) > 0 such that |u(x)| C u L ∞ (Λ) + f L ∞ (Λ) dist(x, ∂A) whenever x ∈ K with dist(x, ∂A) < dist(K, ∂Λ).
Proof. Let x ∈ A ∩ K such that dist(x, ∂A) < dist(K, ∂Λ). Then take x 0 ∈ ∂A such that |x -x 0 | = dist(x, ∂A). Then we can apply the previous theorem to B dist(K,∂Λ)/2 (x 0 ).

We are ready to proceed with the:

Proof of Theorem 3.5. Recall that -∆u = f in A, hence there the function u is of class C 1,α . Since moreover u ∈ H 1 (Λ) and ∇u = 0 a.e. in Λ \ A, it is sufficient to obtain a uniform estimate for ∇u in a neighborhood of ∂A (and actually only in A). Notice that in A it makes sense to consider point-wise values of the gradient of u.

We use the notation d x := dist(x, ∂A), for every x ∈ Ω. Take x 0 ∈ ∂A ∩ Λ and let δ > 0 be small enough such that, considering the compact set

K := x∈B δ (x0) B dx (x) then dist(x, ∂A) < dist(K, ∂Λ) ∀x ∈ K.
By Corollary 3.8, there exists C = C(K, N, r) > 0 such that

|u(x)| C u L ∞ (Λ) + f L ∞ (Λ) d x ∀x ∈ K.
In particular, for every

x ∈ A ∩ B δ (x 0 ), since B dx (x) ⊂ K, then (3.6) u L ∞ (B dx (x)) ≤ 2C u L ∞ (Λ) + f L ∞ (Λ) d x .
Now, let

Q x := y ∈ R N : |y i -x i | < d x m , i = 1, . . . , N ,
where m > 0 is chosen so large that the cube Q x is contained in the ball B dx (x) (m > 0 is a universal constant, depending only on the dimension N ). Since B dx (x) ⊂ A, then -∆u = f in B dx (x) and we can combine (3.6) with interior gradient estimates for the Poisson equation (see [15, Formula 3.15)]), deducing that

|∇u(x)| N m d x sup ∂Qx |u| + d x 2m sup Qx |f | C u L ∞ (Λ) + f L ∞ (Λ) ∀x ∈ A ∩ B δ (x 0 ).

Free-boundary condition for problem (A)

In this section we prove Theorem 1.5. We briefly recall the setting. Let x 0 ∈ ∂S i ∩ Ω, and let us assume that Γ R i := ∂S i ∩ B R (x 0 ) is a smooth hypersurface, for some R > 0. We suppose that, for a positive δ, condition (1.4) holds on Γ R i :

χ i 1 (x), . . . , χ i N -1 (x) 1 -δ ∀x ∈ Γ R i , where χ i 1 , . . . , χ i N -1 denote the principal curvatures of ∂S i . Without loss of generality, we can suppose that Γ R i is a graph: Γ R i = {(x , ψ(x )) : x ∈ B N -1 R (x 0 )}, and S i ∩ B R (x 0 ) = {(x , z) ∈ B R (x 0 ) : z ψ(x )} for a function ψ : B N -1 R (x 0 ) → R, where B N -1 R (x 0 ) denotes the ball of radius R in R N -1 centered at x 0 = (x 1 0 , . . . , x N -1 0
). We know from Theorem 1.3-( 6) that there exists j = i and y 0 ∈ ∂ supp u j such that |x 0 -y 0 | = 1.

The proof of Theorem 1.5 is divided into several steps. We start with the uniqueness and characterization of y 0 . Lemma 4.1. If x ∈ ∂S i ∩ Ω and ∂S i is smooth in a neighbourhood of x, then y = x + ν i (x) is the unique point in l =i ∂ supp u l at distance 1 from x.

Proof. By Theorem 1.3 (points (3) and ( 6)), we know that there exists a point y ∈ l =i ∂ supp u l such that |x -y| = 1, and |x -z| 1 for all z ∈

l =i ∂ supp u l .
This means that y -

x ∈ Q := {v : dist(x + v, S i ) = |v|}. By [14, Theorem 4.8-(2)],
Q is a subset of the normal cone to S i in x, and since ∂S i is smooth in x, we deduce that y -x = ν i (x).

The previous lemma implies that there exists a unique j and a unique y 0 ∈ ∂ supp u j at distance 1 from x 0 . In order to simplify the notation, let i = 1 and j = 1, and so

x 0 ∈ ∂S 1 ∩Ω, y 0 ∈ ∂ supp u 2 .
Assume from now on that y 0 ∈ Ω, so that

y 0 ∈ ∂S 2 ∩ Ω. We denote Γ R 1 := ∂S 1 ∩ B R (x 0 ) and Γ R 2 := {x + ν 1 (x) : x ∈ Γ R 1 }.
Notice that by Lemma 4.1 and by continuity, we have that

y 0 ∈ Γ R 2 ⊂ ∂S 2 ∩ Ω,
where the last inclusion holds for sufficiently small R > 0. Lemma 4.2. The set Γ R 2 is a smooth hypersurface.

Proof. The set Γ R 2 can be parametrized by Φ :

B N -1 R (x 0 ) → R N , Φ(x ) = (x , ψ(x )) + ν 1 (x , ψ(x )) = x - ∇ψ(x ) 1 + |∇ψ(x )| 2 , ψ(x ) + 1 1 + |∇ψ(x )| 2 ,
and hence we need to prove that DΦ(x ) has maximum rank. We have

(4.1) D x - ∇ψ(x ) 1 + |∇ψ(x )| 2 = Id N -1 -D ∇ψ(x ) 1 + |∇ψ(x )| 2 ,
where Id N -1 denotes the identity in R N -1 . Observe that D ∇ψ(x )/ 1 + |∇ψ(x )| 2 is the curvature tensor of Γ R 1 at (x , ψ(x )) (see for instance [15, p.356]). Assumption (1.4) implies that all its eigenvalues are strictly smaller than one. Then the determinant of (4.1) does not vanish, and the result follows.

Observe that, with the previous notations, (4.2)

ν 1 (x) = -ν 2 (x + ν 1 (x)) ∀x ∈ Γ R 1 and ν 2 (x) = -ν 1 (x + ν 2 (x)) ∀x ∈ Γ R 2 . Let η ∈ C ∞ c (B R (x 0
)) be a nonnegative test function. We define two deformations, one acting on S 1 , and the other on S 2 . The first one, which deforms S 1 , is a function denoted by Lemma 4.3. The set Γ R 1,ε is a smooth hypersurface. Moreover, if we denote its exterior normal at a point x

F 1,ε : R N → R N , ε ∈ [0, ε), such that, F 1,ε (x) = x if x ∈ B R (x 0 ) x + εη(x)ν 1 (x) if x ∈ Γ R 1 , extended to the whole R N in such a way that (ε, x) ∈ [0, ε) × R N → F 1,ε (x) is of class C 1 , and F 1,0 (•) = Id. We denote S 1,ε := F 1,ε (S 1 ) := S 1 ∪ {x + sη(x)ν 1 (x) : x ∈ Γ R 1 , 0 s < ε} and Γ R 1,ε := F 1,ε (Γ R 1 ) = {x + εη(x)ν 1 (x) : x ∈ Γ R 1 }.
+ εη(x)ν 1 (x) (for x ∈ Γ R 1 ) by ν ε (x), then ε → ν ε (x) is differentiable at ε = 0 and (4.3) d dε ν ε (x) ε=0 is orthogonal to ν 1 (x), for every x ∈ Γ R 1 .
Proof. By the smoothness of Γ R 1 and of the perturbation η, it follows that ν ε is differentiable in ε for ε small. By deriving the identity

|ν ε (x)| 2 = 1 in ε for each x ∈ Γ R 1 , we have d dε ν ε (x) • ν ε (x) = 0. Since ν 0 (x) = ν 1 (x), the statement (4.3) follows. Now we consider an open neighbourhood B R y0 of y 0 such that B R y0 ∩∂S 2 = Γ R 2 and dist(B R y0 , ∂Ω) > 0.
In order to deform S 2 , we take F 2,ε :

R N → R N , ε ∈ [0, ε), such that F 2,ε (y) = y if y ∈ B R y0 , x + εη(x)ν 1 (x) + ν ε (x), if x = y + ν 2 (y), y ∈ Γ R 2 extended to the whole R N in such a way that (ε, x) ∈ [0, ε) × R N → F 2,ε (x) is of class C 1 , and F 2,0 (•) = Id. Define S 2,ε := F 2,ε (S 2 ) := S 2 \{x + sη(x)ν 1 (x) + ν ε (x) : x ∈ Γ R 1 , 0 s < ε} and Γ R 2,ε := F 2,ε (Γ R 2 ) = {x + εη(x)ν 1 (x) + ν s (x) : x ∈ Γ R 1 }. Notice that, since η 0, we have Γ R 2,ε ⊂ S 2 for every ε > 0.
Remark 4.4. We observe that the map

x ∈ Γ R 1 → x ε := x + εη(x)ν 1 (x) ∈ Γ R 1
,ε is a diffeomorphism for ε > 0 small enough. For this reason, we can see the normal ν ε as defined on Γ R 1,ε , and use the notation

ν ε (x ε ) := ν ε (x) ⇐⇒ x ε = x + εη(x)ν(x).
The crucial point in our argument is the following: Lemma 4.5. We have dist(S 1,ε , S 2,ε ) 1. Moreover, dist(S i,ε , S j ) 1 for every i ∈ {1, 2}, j = 1, 2.

For the proof we will need the following elementary fact. Lemma 4.6. Let (x 1 , y 1 ), (x 2 , y 2 ), two points on the lower semi-circle ∂B - 1 := {x 2 +y 2 = 1, y < 0} in R 2 . Let γ be the graph of a C 2 function f : [x 1 , x 2 ] → R, and let us suppose that:

• the curvature of γ is strictly smaller than 1;

• f (x 1 ) = y 1 , i.e. (x 1 , y 1 ) is the initial point of γ;

• there exists ρ > 0 such that f (t) -√ 1 -t 2 for t ∈ (x 1 , x 1 + ρ). Then f (x 2 ) < y 2 , i.e. γ cannot contain any other point on ∂B 1 .

Proof. In terms of f , the curvature of γ is defined by

k(t) := f (t) (1 + (f (t)) 2 ) 3/2 .
Thus, by assumption:

f (t) < 1 + (f (t)) 2 3/2 in [x 1 , x 2 ], f (x 1 ) x 1 1 -x 2 1 and f (x 1 ) = y 1 . Recalling that v(t) = - √ 1 -t 2 solves v = (1 + (v ) 2 ) 3/2
, the thesis follows by a comparison argument for solutions to ODEs.

Proof of Lemma 4.5. The second statement of the lemma comes from the fact that dist(S i , S j ) ≥ 1 and dist(Γ R i , S j ) > 1 for i = 1, 2 and j > 2. As for the first statement, observe that it is enough to show that dist(∂S

1,ε ∩ Ω, ∂S 2,ε ∩ Ω) 1. By construction, ∂S i,ε \ Γ R i,ε = ∂S i \ Γ R i for i = 1, 2, and since dist(∂ supp u 1 , ∂ supp u 2 ) = 1, then dist(∂S 1,ε \ Γ R 1,ε , ∂S 2,ε \ Γ R 2,ε ) 1. Since every point in Γ R
i admits a unique point on ∂S j at distance exactly one, we have that dist(Γ R i , ∂S j \ Γ R j ) > 1 for every i = j, i, j ∈ {1, 2}. Thus, by the continuity of the deformations

F 1,ε , F 2,ε , dist(Γ R i,ε , ∂S j,ε \ Γ R j,ε ) > 1 ∀i = j, i, j ∈ {1, 2}. It remains to show that dist(Γ R 1,ε , Γ R 2,ε ) = 1.
This follows from the following property (we use the notation introduced in Remark 4.4):

(C) there exists ε > 0 small enough such that any point in y ∈ S c 1,ε such that dist(y, Γ R 1,ε ) = 1 has unique projection at minimal distance onto S 1,ε , this projection lies in Γ R 1,ε , and

moreover y = x ε + ν ε (x ε ) for some x ε ∈ Γ R 1,ε . Indeed, (C) implies, by definition of Γ R 2,ε , that {y ∈ S c 1,ε : dist(y, Γ R 1,ε ) = 1} = {y ∈ S 2 : dist(y, Γ R 1,ε ) = 1} = {y ∈ S 2 : y = x ε + ν ε (x ε ), x ε ∈ Γ R 1,ε } = Γ R 2
,ε , and completes the proof.

Let us now prove property (C). That any point at minimal distance from S 1,ε stays on Γ R 1,ε is a consequence of Lemma 4.1 for ε = 0; the case ε > 0 small follows by continuity of F 1,ε , and recalling that η has compact support. Take y ∈ S 2 ∩ {dist(z, Γ R 1,ε ) = 1}. To prove the uniqueness of the projection, suppose by contradiction that there exist two points x 1 and x 2 in Γ R 1,ε such that |x 1 -y| = |x 2 -y| = 1. Since our argument is local in nature, it is not restrictive to suppose that we chose R < 1/2 from the beginning, and hence in particular |x 1 -x 2 | < 1.

Let Π be the plane containing x 1 , x 2 and y, and let γ be the arc of the curve Γ R 1,ε ∩ Π connecting x 1 and x 2 . The basic idea which we develop in what follows is that the existence of both x 1 and x 2 is forbidden by the fact that, thanks to (1.4), the curvature at every point of γ is smaller than 1.

Since Γ R 1,ε is a graph of a function of x N , (4.4) also γ can be seen as the graph of a function of x N for ε small enough.

Also, since the principal curvatures of ∂S 1 are all smaller than 1 -δ on Γ R 1 , for ε small enough the principal curvatures of ∂S 1,ε are all smaller than 1 -δ/2) on Γ R 1,ε . Combining this with the fact that x 1 is a projection of y onto S 1,ε , it follows the existence of r > 0 small (possibly depending on ε) such that

(4.5) B r (x 1 ) ∩ Γ R 1,ε ∩ B 1 (y) = {x 1 }. Moreover, (4.6)
the (planar) curvature of γ is also smaller than 1 -δ/2.

Collecting together (4.4), (4.5), (4.6), we are in position to apply 3 Lemma 4.6 to the curve Γ on the plane Π, deducing that Γ cannot meet B 1 (y) in any other point than x 1 , in contradiction with the existence of x 2 .

It remains to show that y

= x ε + ν ε (x ε ) for some x ε ∈ Γ R 1,ε .
Having proved the uniqueness of the projection, this follows directly from [14, Theorem 4.8-( 2)] and the smoothness of Γ R 1,ε .

Lemma 4.5 is crucial since it allows us to produce a family of admissible variations of the minimizer u in the following way. For i ∈ {1, 2}, let u i,ε ∈ H 1 (S i,ε ) be such that

     ∆u i,ε = 0 in S i,ε u i,ε = u i on ∂S i,ε \Γ R i,ε = ∂S i \Γ R i u i,ε = 0 on Γ R i,ε
extended by zero to Ω\S i,ε . Observe that S i,ε = {x ∈ Ω : u i,ε (x) > 0}, and that for ε 0 small the vector (u 1,ε , u 2,ε , u 3 , . . . , u k ) belongs to the set H ∞ -defined in (1.1)-by Lemma 4.5.

Proposition 4.7. We have

d dε ˆΩ |∇u 1,ε | 2 ε=0 + = - ˆΓR 1 η(x)(∂ ν1 u 1 ) 2 , (4.7) d dε ˆΩ |∇u 2,ε | 2 ε=0 + = ˆΓR 2 η(x + ν 2 (x))(∂ ν2 u 2 ) 2 . (4.8)
Proof. The identity (4.7) is a direct consequence of Lemma A.2 in the appendix, with S := S 1 and ω = B R (x 0 ), since

Y 1 := d dε F 1,ε (x) ε=0 = η(x)ν 1 (x).
As for (4.8), we apply the same lemma with S = S 2 and ω = B R y0 . We have Proof of Theorem 1.5. Without loss of generality we work in the case i = 1 and j = 2, and use the notations previously introduced. Take, for ε 0 small, the vector (u 1,ε , u 2,ε , u 3 , . . . , u k ), which by Lemma 4.5 belongs to the set H ∞ . Since u 1,0 = u 1 and u 2,0 = u 2 , then by the minimality of u we have that

Y 2 (y) := d dε F 2,ε (y) ε=0 = η(y + ν 2 (y))ν 1 (y + ν 2 (y)) + d dε ν ε (y + ν 2 (y))
d dε J ∞ (u 1,ε , u 2,ε , u 3 , . . . , u k ) ε=0 + 0.
By Proposition 4.7, this is equivalent to ˆΓR

1 η(x)(∂ ν1 u 1 ) 2 ˆΓR 2 η(x + ν 2 (x))(∂ ν2 u 2 ) 2 .
This identity holds true for every nonnegative η ∈ C ∞ c (B R (x 0 )). In particular, by taking η = η δ such that η δ (x) = 1 for x ∈ B R-2δ (x 0 ) and η δ (x) = 0 in B R (x 0 )\B R-δ (x 0 ), and by making δ → 0, we can easily conclude that ˆΓR

1 (∂ ν1 u 1 ) 2 ˆΓR 2 (∂ ν2 u 2 ) 2 .
Arguing exactly in the same way, but deforming first Γ 2,R , and afterwards Γ 1,R , we can prove that also the opposite inequality holds, and hence ˆΓR

1 (∂ ν1 u 1 ) 2 = ˆΓR 2 (∂ ν2 u 2 ) 2 .
Therefore (4.9)

ffl Γ R 1 (∂ ν1 u 1 ) 2 ffl Γ R 2 (∂ ν2 u 2 ) 2 = |Γ R 2 | |Γ R 1 |
, and we can thus end the proof by applying [7, Lemma 9.3], which states that the right-hand-side of (4.9) tends to the right-hand-side of (1.5) as R → 0. We point out that, with respect to [START_REF] Caffarelli | On a long range segregation model[END_REF], the modulus is present in our formula (1.5). This is only a consequence of the different convention that we adopted regarding the sign of the curvatures.

Existence and properties of solutions to problem (B)

We focus now on problem (B). It is convenient to restate the problem as follows. Letting, for all

u ∈ H 1 0 (Ω; R k ), J(u) = F ˆΩ |∇u 1 | 2 , . . . , ˆΩ |∇u k | 2 , we define (5.1) c := inf u∈H∞ J(u)
where

H ∞ =    u = (u 1 , . . . , u k ) ∈ H 1 (Ω, R k ) dist(supp u i , supp u j ) 1 ∀i = j ˆΩ u 2 i = 1 ∀i    . (H1) t ∈ [0, T ) → F (t, •) ∈ W 1,∞ (R N , R N ) is differentiable at 0; (H2) F (0, •) = Id;
(H3) F (t, x) = x for every t ∈ [0, T ), x ∈ ∂S\ω. For notation convenience, we let F t (x) = F (t, x), while DF t (x) := D x F (t, x). We can assume that T > 0 is sufficiently small so that D x F (t, x) is an invertible matrix for (t, x) ∈ [0, T [×R N . Moreover, we define for all x ∈ R N , t ∈ [0, T ), ξ ∈ R N ; the map t ∈ [0, T ) → A t ∈ L ∞ (R N ) is differentiable at t = 0, and lim t→0 A t = A 0 = Id uniformly in R N ; and by recalling that Y := F 0 , we have by Jacobi's formula

Y = F 0 := d dt F t (•) t=0 ∈ W 1,∞ (R N , R N ), so that, by (H1), F t (x) = x + tY (x) + o(t) in W 1,∞ (R N , R N ),
d dt A t (x) t=0 = div Y Id -(DY + DY T ) uniformly in R N .
Step 2: Differentiability of the map t ∈ [0, T ) → v t ∈ H 1 (S) at t = 0. We introduce the incremental quotients

w t,0 := v t -v 0 t -0 = v t -u t ∈ H 1 0 (S), t ∈]0, T [.
Each w t,0 is a solution to (A.1) -div(A t ∇w t,0 ) = div At-Id t ∇u in S w t,0 = 0 on ∂S.

We introduce the function w 0 ∈ H 1 0 (S) solution to (A.2) -∆w 0 = div(A 0 ∇u) in S w 0 = 0 on ∂S.

and show that indeed w t,0 → w 0 as t → 0, strongly in H 1 0 (S), so that t → v t is differentiable at t = 0, with v 0 = w 0 . To do this, we subtract (A.2) from (A.1) and obtain the identity -div(A t ∇(w t,0 -w 0 )) = div((A t -A 0 )∇w 0 ) + div A t -A 0 t -A 0 ∇v 0 Testing this equation by w t,0 -w 0 ∈ H 1 0 (S), we can conclude that ˆS |∇(w t,0 -w 0 )| 2

1 λ A t -A 0 ∞ w 0 H 1 + A t -A 0 t -A 0 ∞ v 0 H 1
and the claim follows recalling the properties of the functions A t .

Step 3: Differentiability of the map t ∈ [0, T ) → I t ∈ R at t = 0. As a result of the previous step, the derivative of I t at t = 0 is equal to lim t→0 ˆS A t -A 0 t ∇v t + A 0 ∇w t,0 , ∇v t + A 0 ∇v 0 , ∇w t,0 = ˆS A 0 ∇u, ∇u + 2 ˆS ∇w 0 , ∇u By testing the equation of u by w 0 ∈ H 1 0 (S), we see that the last term in the previous expression is zero, and by exploiting the symmetry of the scalar product we obtain We now show that, if F t leaves invariant a neighborhood of ∂S\ω, then the derivatives in Lemma A.1 can be expressed only in terms of the value of the first order behavior of F around ∂S ∩ ω.

Lemma A.2. Assume (H1),(H2), and instead of (H3) assume the stronger condition (H3') F (t, x) = x for every t ∈ [0, T ), x ∈ S\ω , for some ω ω;

and assume also that ∂S ∩ ω is a smooth hypersurface Then we have

d dt I t t=0 = - ˆω∩∂S (Y • ν)(∂ ν u) 2
In particular, the first derivative of the energy at 0, I 0 , depends on F t only through the value of Y = F 0 over ω ∩ ∂S.

Proof. Observe that the assumptions imply that Y ∈ W 1,∞ (R N ) satisfies Y = 0 in S \ω. Moreover, since u is harmonic in S, u ∈ H

F

  (x 1 , . . . , xi-1 , x i , xi+1 . . . , xk ) = +∞ for all (x 1 , . . . , xi-1 , xi+1 . . . , xk ) ∈ (R + ) k-1 ,we consider the minimization problem (1.2) inf (ω1,...,ω k )∈P k (Ω)

  which are clearly continuous due to the Lebesgue dominated convergence theorem.Let us consider the open sets

Figure 1 .

 1 Figure 1. The picture on the left represents the deformation acting on S 1 . The picture on the right represents the deformation acting on S 2 .

ε=0 for every y ∈ Γ R 2 .

 2 Recalling (4.2) and taking into account (4.3), we haved dε ν ε (y + ν 2 (y)) ε=0 , ν 2 (y) = d dε ν ε (y + ν 2 (y)) ε=0 , -ν 1 (y + ν 2 (y)) = 0.Therefore, using (4.2) once again, Y 2 (y), ν 2 (y) = η(y + ν 2 (y)), and (4.8) follows by Lemma A.2.

  , ∇u = ˆS (div Y Id -2DY ) ∇u, ∇u

  2 (O), for every O ω ∩ S. Thus we can test the equation of u with Y • ∇u ∈ H 1 (S), obtaining0 = ˆS ∇u • ∇(Y • ∇u) -ˆω∩∂S (Y • ∇u)(ν • ∇u) = ˆω∩S ∇u, DY ∇u + ∇u, D 2 uY -ˆω∩∂S (Y • ∇u)(ν • ∇u) ∇u)(ν • ∇u)(the boundary term is well defined since ω ∩ ∂S is a smooth hypersurface). A further integration by parts and the observation that, since u = 0 on ω ∩ ∂S, we have |∇u| = |∂ ν u| and ∇u = (ν • ∇u)ν on ω ∩ ∂S, yields the identities ˆω∩S(div Y Id-2DY )∇u, ∇u = ˆω∩∂S (Y • ν)|∇u| 2 -2(Y • ∇u)(ν • ∇u) = -ˆω∩∂S (Y •ν)(∂ ν u) 2 .

  as t → 0. For every t ∈ [0, T ) we let S t = F t (S) and Γ t = F t (∂S ∩ ω). Let u t ∈ H 1 (S t ) be such that Under the previous assumptions, the function I t is differentiable at t = 0, with

	    	∆u t = 0 in S t u = ϕ on ∂S\ω u = 0 on Γ t	that is I t :=	ˆSt	|∇u t | 2 = min	   ˆSt	|∇v| 2 :	v ∈ H 1 (S t ), v = ϕ on ∂S\ω, v = 0 on Γ t	  
	Lemma A.1. d dt	I t

t=0

=

ˆS

(div Y Id -2DY )∇u, ∇u

Proof.

Step 1: Fixing the domain through a change of variables. For any t ∈ [0, T [, let v t ∈ H 1 (S) be defined as

v t := u t • F t . Observe that for every v ∈ H 1 (S t ) one has ˆSt |∇v(y)| 2 dy = ˆFt(S) |∇v(y)| 2 dy = ˆS |[(DF t (x)) -1 ] T ∇(v(F t (x))| 2 det(DF (x)) dx.

Thus v t is the minimizer of

I t = min ˆS det(DF t )|[(DF t ) -1 ] T ∇w| 2 : w ∈ H 1 (S), w = ϕ on ∂S

(recall that ϕ = 0 on ∂S ∩ ω) and a solution to the problem

-div(A t ∇v t ) = 0 in S v t = ϕ on ∂S with A t (x) = det(DF t (x))(DF t (x)) -1 [(DF t (x)) -1 ] T .

Observe that A t (x) is symmetric and there exist 0 < λ < Λ such that λ|ξ| 2 A t (x)ξ, ξ Λ|ξ| 2

We denote by Br(x) the ball of center x and radius r in R N . In case x = 0, we simply write Br.

Acknowledgments. The authors are partially supported by the ERC Advanced Grant 2013 n. 339958 "Complex Patterns for Strongly Interacting Dynamical Systems -COMPAT". N. Soave is partially supported by the PRIN-2015KB9WPT_010 Grant: "Variational methods, with applications to problems in mathematical physics and geometry". H. Tavares is partially supported by FCT -Portugal through the project PEst-OE/EEI/LA0009/2013. A. Zilio is partially supported by the ERC Advanced Grant 2013 n. 321186 "ReaDi -Reaction-Diffusion Equations, Propagation and Modelling".

Clearly, since to each set ω i of an element in P k we can associate an eigenvalue u i ∈ H 1 0 (ω i ), we have c inf (ω1,...,ω k )∈P k (Ω)

F (λ 1 (ω 1 ), . . . , λ 1 (ω k )).

We show below that these levels coincide.

5.1. Existence of a minimizer and its first properties. We first address the problem of existence of optimal partitions, and derive some preliminary properties of the sets composing the minimal solutions. This part is close the results in Section 2 and for this reason we shall only give a brief sketch of the methodology. We consider the auxiliary problem: for any u ∈ H 1 0 (Ω, R k ) we let

We have, similarly to Theorem 2.1:

Theorem 5.1. For every β > 0, there exists a nonnegative minimizer u β = (u 1,β , . . . , u k,β ) of J β in the set

There exist µ 1,β , . . . , µ k,β > 0 such that u β is a nonnegative solution of

Moreover, the family {u β : β > 0} is uniformly bounded in H 1 0 ∩ L ∞ (Ω, R k ), and there exists u = (u 1 , . . . , u k ) ∈ H such that:

(1) u β → u strongly in H 1 (Ω, R k ) as β → +∞, up to a subsequence;

(2) dist(supp u i , supp u j ) 1, for every i = j, so that u ∈ H ∞ ;

(3) for every i = j,

y) dx dy = 0;

(4) u is a minimizer for c, defined in (5.1).

Proof. All the listed properties can be shown by very similar arguments of Theorem 2.1, we shall only consider here those that are new. In particular, we focus on the uniform bounds on {u β }.

The existence of a nonnegative minimizer u β for J β on H is given by the direct method of the calculus of variations (J is lower-semicontinuous because

for any i = 1, . . . , k and β > 0.

Therefore, by the assumptions on F , there exists a > 0 such that

It follows, by the method of the Lagrange multipliers, that any minimizer u β is a weak solution to (5.2). Testing such equations by u β itself and using the uniform bound on J β (u β ), we obtain that the exists µ > 0 such that 0 < µ i,β < µ for any i = 1, . . . , k and β > 0.

The proof of the uniform L ∞ bounds is then a rather standard consequence of the Brezis-Kato iteration technique, since -∆u i,β ≤ µu i,β . The remaining properties can be shown reasoning exactly as in the proof of Theorem 2.1.

The previous result shows the existence of minimizers for problem c, in connection with an elliptic system with long-range competition. Since both H ∞ and J are invariant under the transformation (u 1 , . . . , u k ) → (|u 1 |, . . . , |u k |), we can work from now on, without loss of generality, with nonnegative functions. In what follows, we will show that all the minimizers for c are continuous (actually, we will show that they are Lipschitz continuous in Ω), and this will imply that (1.2) and (5.1) coincide, and there is a one-to-one correspondence between (open) optimal partitions (ω 1 , . . . , ω k ) of (1.2) and minimizers u of (5.1): for every u minimizer of c, the sets ω i = {u i > 0} constitute an optimal partition at distance 1 of Ω.

Proof of Theorems 1.3 and 1.4 for problem (B)

. By following exactly the same lines of the proof of Theorem 1.3, ( 1)-( 2)-( 3), ( 5)-( 6) for problem (A), we can show the exact same properties for any minimizer u of the level c.

Regarding the regularity of the eigenfunctions, using the notations of Section 3, we observe that u = 0 on ∂Ω, and that Ω satisfies the r-uniform exterior sphere condition for some r > 0. Then the Lipschitz continuity in Ω is a direct application of Theorem 3.5 with f = λ 1 (ω i )u i , Λ = R N and A := A i (this shows Theorem 1.4).

Observe that the continuity of u implies that then ω i = {u i > 0}, i = 1, . . . , k are minimizers for problem (B). Thus c and (1.2) coincide, and given any optimal partition of (1.2), then the conclusions of Theorem 1.3 hold also for the associated eigenvalues u.

Appendix A. Shape Derivatives

In this appendix we establish a formula which relates the change of the energy of the harmonic extension of a function ϕ, defined on a boundary ∂S and vanishing on a portion ∂S ∩ ω of ∂S. The domain variation is localized on ∂S ∩ ω. Although similar results are by now well known, and excellent references are available (we refer for instance to [17, Chapter 5]), we could not find exactly the result we needed, and therefore we provide here a short discussion for the sake of completeness.

Let S ⊂ R N be a open set, and let ω ⊂ R N be a bounded smooth domain such that ∂S∩int (ω) = ∅. For a function ϕ : ∂S → R such that ϕ ∈ Lip(∂S) and ϕ(x) = 0 if x ∈ ∂S ∩ ω, we consider its harmonic extension in S, that is the function u ∈ H 1 (S) solution to ∆u = 0 in S u = ϕ on ∂S or, equivalently, ˆS

The question we want to address is how a smooth deformation of a regular part of ∂S where u = 0 impacts the energy of the corresponding harmonic extension. We start by analyzing the derivative with respect to a global homotopy F : [0, T ) × R N → R N , for some T > 0, satisfying:


est destinée au dépôt

Trail Smelter arbitration case, 1 that recent case law has clarified while highlighting some of its implications. In particular, the obligation not to cause damage has been understood as a positive obligation, and more specifically a duty of due diligence (an obligation of conduct and not of result). States must act with due diligence in order to ensure to the highest possible extent that activities that are carried out on their territory or within their jurisdiction do not cause harmful consequences to other States or to areas beyond their national jurisdiction.

This obligation has been construed by the International Tribunal for the Law of the Sea (ITLOS)'s Seabed Disputes Chamber in 2011 as "an obligation to deploy adequate means, to exercise best possible efforts, to do the utmost, to obtain this result". amounts to a general principle of international law, rather than a customary obligation. In doing so, he would like to help to "address the significant questions left unanswered by the current view of prevention" (p. 6). How can we not agree with him when he states that, despite growing case law, there are still many grey areas as to the exact content and scope of these obligations. In his view, these uncertainties are so significant that they "undermine the relevance of the purported customary obligation to prevent transboundary environmental harm" (p. 7).

In Mario Gervasi's opinion, the classification of prevention of environmental harm as a customary obligation is unsatisfying and shows some weaknesses (first part). Adopting a functionalist approach to the notion of principle, he considers that the prevention principle operates as a general principle, inspiring the formation of more specific rules and performing a guiding function within international law cases. He argues that this reconstruction is "not only plausible, but is also best suited to fit the reality" (p. 14) (second part). Mario Gervasi then tries to identify the customary rules informing and embodying it, wondering if the multiplication of treaty regimes has not hindered the emergence of substantive customary obligations (third part). To conclude, he considers that the principle of prevention is being progressively overshadowed by the concept of sustainable development so that "the principle of prevention proves to be incapable of effective cutting into the economic exploitation of natural resources" (p. 398). Mario Gervasi asks what role the principle of prevention might play in the future, in particular in an -hypothetical -de-growth scenario.

Prevention of Environmental Harm under General International Law. An Alternative

Reconstruction is an original book in that it goes against current scholarship and even international case law. It is well-written and well-documented. In particular, Mario Gervasi demonstrates an in-depth knowledge of international litigation, which is abundantly commented on and exploited. He clearly outlines the inconsistencies in international jurisprudence. He also points out that it is a certain marginalisation of international practice that has led to our current understanding of the prevention principle. Doing so, he questions in a stimulating way many fundamental issues in international law: the notions of customary rules or general principles, the role of principles, the sources of international law, the protection of common interests or the potential role and limits of international law in the green transition. Even if the reader will not necessarily be convinced by the whole demonstration, it is without any doubt a very interesting and recommended reading, providing abundant food for thought.

Sandrine Maljean-Dubois

Research Director, CNRS and Aix-Marseille University (DICE-CERIC)

  Sandrine Maljean-Dubois. Bibliografia. MARIO GERVASI, Prevention of Environmental Harm under General International Law. An Alternative Reconstruction (Cultura giuridica e scambi internazionali, 7), Napoli/Baden-Baden, Edizioni scientifiche italiane/Nomos, 2021, pp. XVIII-430.

MARIO GERVASI, Prevention of Environmental Harm under General International Law. An Alternative Reconstruction (Cultura giuridica e scambi internazionali, 7), Napoli/Baden-Baden, Edizioni scientifiche italiane/Nomos, 2021, pp. XVIII-430.

	Rivista di diritto internazionale, n°4/2022, pp. 1239-1240.
	The 'no-harm' principle is without hesitation the bedrock of international
	environmental law. The obligation not to harm the environment in the territory of other States
	or in areas beyond national jurisdictions is a long-established rule, first identified in the 1941

Rivista di diritto internazionale, n°4/2022, pp. 1239-1240.. Rivista di diritto internazionale, 2022, 4, pp. 1239-1240. halshs-03884836 Book's review Bibliografia.

  The International Court of Justice sees it as a general -customary -obligation of due diligence from which a number of procedural obligations arise. Most scholars take the same view, considering that it is a customary obligation or even a set of customary obligations.

	In his book Prevention of Environmental Harm under General International Law, Mario
	Gervasi proposes an alternative view, arguing that prevention of environmental harm

2

It is seen by the International Court of Justice as a broad and demanding obligation "which entails not only the adoption of appropriate rules and measures, but also a certain level of vigilance in their enforcement and the exercise of administrative control applicable to public and private operators, such as the monitoring of activities undertaken by such operators". 3 As such, this due diligence obligation requires States to regulate the conduct of private actors.

1 Trail Smelter case (United States, Canada) (Award)

[16 April 1938 and 11 March 1941]

, Report of International Arbitral Awards, Vol III (1920).

2 ITLOS, Responsibilities and Obligations of States with respect to Activities in the Area, 2011, p. 41, §110.

3 ICJ, Pulp Mills on the River Uruguay (Argentina v Uruguay) above p. 14, §197.


come

Promoting the attractiveness of peri urban residential areas through public transport services: the strategy of

French real estate professionals

To be evasive on transport facilities in ads:

using the word "proximity to" and no more detail VS No information of transport facilities (more than 1 in 2 ads do not mention transport facilities)

HOW?

Photo credit S. Hasiak

•Conclusion••

  Under the influence of the metropolisis of Lille (1.72 millions of inhabitants), located 20km from Lille • Attractive in terms of housing, high standard of living • Low population density, a predominantly suburban fabric (more than 85% of individual houses) • Multi-motorized population • Attractive train services to reach the centre of Lille (frequency, travel time) Railway stations in periurban areas: an interesting and useful asset to promote periurban housing and residential attractiveness of periurban communities • Real estate professionals: a player in the valorization of territorial opportunities to some degree ✓ Aware of the need to adapt their economic strategy with the ecological transition issues ✓ But in a specific way of acting • Challenge of improving knowledge of people behaviors and their preferences for residence since the crisis due to the Covid-19: questioning dependency to public transport? 600 property ads published on websites from April 2019 to March 2020 • 10 promotional leaflets of new properties • 10 face-to-face interviews of real estate agents and notaries Conception MATRiS Cerema Hauts-de-FranceContext: Valuation of Regional Railway System • Challenge of both structuring periurban living and managing periurban mobilities through sustainable policies • Territorial policies of valuation of railway stations (densification, urban re-conquest) • 3 types of stakeholders involved: public stakeholders, individuals and the private sphere • Few knowledge about strategies of the private sphere Mistrust Aim of ads : to catch the attention of buyers, not to scare them away Rail station: seen as a real asset avoiding road congestion good level of train services WHY?