Attractive Riesz potentials acting on hard spheres
aa r X i v : . [ m a t h . A P ] A p r ATTRACTIVE RIESZ POTENTIALS ACTING ONHARD SPHERES
A. KUBIN AND M. PONSIGLIONE
Abstract.
In this paper we introduce a model for hard spheres interact-ing through attractive Riesz type potentials, and we study its thermodynamiclimit. We show that the tail energy enforces optimal packing and round macro-scopic shapes.
Keywords.
Riesz kernels, hard spheres, optimal packing, crystallization, frac-tional perimeters, Γ-convergence, isoperimetric inequality.
AMS subject classifications.
Contents
1. Introduction 11.1. Notation of the paper 42. Hard spheres, optimal packing ahd empirical measures 42.1. Density of optimal packing 42.2. The empirical measures 53. Riesz interactions for σ ∈ ( − d,
0) 63.1. The energy functionals 73.2. Compactness 73.3. Γ-convergence 103.4. Asymptotic behaviour of minimizers 124. Riesz interactions for σ ∈ [0 ,
1) 134.1. The energy functionals 144.2. The continuous model 144.3. Compactness and Γ-convergence 17Acknowledgements 19References 191.
Introduction
In this paper we introduce and analyze variational models for hard spheres inter-acting through Riesz type attractive potentials. The model consists in minimizingnonlocal energies of the type(1.1) X i = j K p ( | x i − x j | ) , over all configurations of N points { x , . . . , x N } ⊂ R d satisfying | x i − x j | ≥ i = j ; here K p : R + → ( −∞ ,
0] is a power-law attractive potential K p ( r ) ≈ − r p for large r , with p ∈ (0 , d + 1). Eventually, we consider the thermodynamic limit N → + ∞ .As a consequence of the hard sphere constraint and of the attractive behaviourof the potential, the ground states of the system turn out to be optimal packedconfigurations of spheres filling a macroscopic set. The thermodynamic limit isdescribed by a nonlocal energy that is a Riesz type continuous counterpart of (1.1)for p ∈ (0 , d ); remarkably, in the case p ∈ [ d, d + 1) fractional perimeters arise inthe limit energy. In both cases p ∈ (0 , d ) and p ∈ [ d, d + 1), the optimal asymptoticshape is given (after scaling) by the Euclidean ball, and this is a consequence ofthe Riesz rearrangement inequality and of the fractional isoperimetric inequality,respectively. These results are obtained by providing a Γ-convergence expansion ofthe energy. The method allows to consider and understand also slight variants ofthe basic variational problem (1.1), taking into account also volume forcing terms,possibly enforcing different optimal shapes.The combination of the attractive potential together with the hard sphere con-straint provides a basic example of long range attractive/short range repulsive inter-actions. In this respect, the proposed model fits in the class of aggregation [14, 6, 8]and crystallization [5] problems, but with a substantial change of perspective dueto the fundamental role played in our model by the tail of the interaction energy.This is the case for both integrable and non-integrable tails, referred to as unstablepotentials in the crystallization community [5]. This is why in our model crystal-lization is replaced by the related but different concept of optimal packing , whilethe microscopic structure does not affect the macroscopic shape, that turns out tobe the Euclidean ball.To explain these new phenomena, we first provide an overview of the classicalcrystallization problem, focussing on two basic models in two dimensions. They arebased on minimization of an interaction energy as in (1.1), for some potential K thattends to infinity as r →
0, has a well at a specific length enforcing crystallizationand fixing the lattice spacing (and structure), and rapidly decays to 0 as r → + ∞ .The basic potential is provided by the Heitmann-Radin model [17] which consistsin systems of hard spheres whose pair-interaction energy is + ∞ if two balls overlap,it is equal to − N of discs, minimizers exhibit crystalline order: the centers ofthe discs lie on a subset of an equilateral triangular lattice. Moreover, for large N the discs fit a large hexagon. The first phenomenon is referred to as crystallization ,the second as macroscopic Wulff-shape . Crystallization is due to local optimizationof the potential around its well: almost each particle tends to maximize the numberof nearest neighbor particles. In view of the hard disc constraint, such a numberis 6. The macroscopic Wulff shape is the result of the minimization of the numberof boundary particles that have the wrong number of nearest neighbors. In thisrespect, the macroscopic shape minimizes an anisotropic perimeter energy; under avolume constraint, this is nothing but the anisotropic isoperimetric problem, whoseminimizer is the Wulff shape [15]. Recently, these phenomena have been analyzedin details in the solid formalism of Γ-convergence [3, 12].A less rigid and most popular model in elasticity is given by the polynomialLennard-Jones type potential; the hard sphere constraint is replaced by a repulsiveterm which is infinite only at 0; the only negative value in the Heitmann-Radinpotential is replaced by a narrow well, while the zero-long range interaction of the Heitmann-Radin potential is replaced by a rapidly decaying tail energy. In [21] itis proved that, if the well of the potential is very narrow, and the tail is a smallenough lower order term, then the crystallization property is preserved in average,while the Wulff shape problem is still open. Recently, it has been proved [4] thata slightly wider well in the potential favours the square lattice rather than thetriangular one. In higher dimensions the picture is much less clear.We pass to describe our model; since the tail energy will be predominant, it isconvenient to change length-scale, introducing a parameter ε >
0, whose inverse ε represents the size of the body filled by the hard spheres. Then, in order to deal, inthe thermodynamic limit, with a finite macroscopic body, we scale the spheres with ε . After this scaling the potential K p becomes integrable if and only if p ∈ (0 , d ).We discuss first the integrable case: we write p = d + σ for some σ ∈ ( − d, f σε : R + → R defined by(1.2) f σε ( r ) := + ∞ for r ∈ [0 , ε ) , − r d + σ for r ∈ [2 ε, ∞ ) . In this case the Γ-limit as ε → K p = f σε , isnothing but its continuous counterpart, defined on absolutely continuous measures,whose density is bounded from above by the density of the optimal packing problemin R d (see Theorem 3.3). This Γ-convergence result can be completed with suitableconfining volume forcing terms, ensuring compactness properties for minimizers.We prove that minimizers consist, in the limit as ε →
0, in optimal packed config-urations of balls filling a macroscopic set E , which is a ball whenever the volumeterm is radial.The non-integrable case is much more involved. In this case both the tail and thecore of the energy blow up as ε →
0, the first being the leading term. In order toprovide a first order expansion of the energy in terms of Γ-convergence, we need toregularize the potential, neglecting the core energy. More precisely, we introduce amesoscopic length-scale r ε ≫ ε with r ε → ε → r ε . Thecorresponding regularized Riesz type p -power-law potentials, with p = d + σ and σ ∈ [0 , f σε ( r ) := + ∞ for r ∈ [0 , ε ) , r ∈ [2 ε, r ε ) , − r d + σ for r ∈ [ r ε , + ∞ ) . Then, only the tail of the interaction energy remains, and the microscopic de-tails of the potential are neglected in the limit as ε →
0. This is consistent withthe integrable case (1.2), where the core contribution vanishes as a consequenceof the only integrability of the potential. Dividing the energy by the divergingtail contribution, we obtain the zero order term in the Γ-convergence expansion ofthe energy. This zero order Γ-limit still enforces optimal packing on minimizingsequences, but does not determine the macroscopic limit shape. Then, we look atthe next term in the Γ-convergence expansion. This consists in removing from thetotal energy the infinite volume-term energy per particle, so that a finite quantityreamins, which turns out to detect the macroscopic shape. In fact, the first order
A. KUBIN AND M. PONSIGLIONE
Γ-limit, provided in Theorem 4.3, is nothing but the σ -fractional perimeter, intro-duced in [7] for σ ∈ (0 , σ = 0. Such an analysis hasfirst been provided in a continuous setting in [13]; our results represent its discretecounterpart. Since fractional perimeters are minimized, under a volume constraint,by Euclidean balls, we deduce that, as ε →
0, minimizers are given by optimalpacked configurations of ε -spheres filling a macroscopic ball.The range of the parameter σ ∈ ( − d,
1) is somehow natural, since for σ < − d the potential becomes repulsive (and constant for d = 0); the case σ = 1 formallycorresponds to the Euclidean perimeter, being the limit of s -fractional perimetersas s → σ > p = 1, establishes thatthe ball is the optimal shape minimizing the potential energy of a fluid [18]. More-over, the hard sphere constraint could be relaxed, providing in the limit richermodels, accounting for density penalization terms, as in the rotating stars problem [20], as well as several attracting-repulsive potentials [6]. Our analysis could also beextended to discrete systems describing interactions between different populations,[11, 10].Finally, our analysis suggests the possible role of the tail energy as a new mech-anism enforcing optimal packing, and hence, in some respect, crystallization.1.1. Notation of the paper.
In this paper we use the following notation: ω d denotes the Lebesgue measure of the unit ball B (0) of R d . M ( R d ) denotes thefamily of Lebesgue measurable sets E ⊂ R d , while the corresponding Lebesguemeasure will be denoted by | E | . We set M f ( R d ) := { E ∈ M ( R d ) : | E | < + ∞} . M b ( R d ) denotes the space of (non negative) finite Radon measures in R d . TheDirac delta measure centered in x is denoted by δ x , while the Lebesgue measure by L d . We denote with C ( ⋆, · · · , ⋆ ) a constant that depends on ⋆, · · · , ⋆ ; this constantcan change in the steps of a proof. Finally, R := R ∪ {−∞ , + ∞} .2. Hard spheres, optimal packing ahd empirical measures
Here we introduce the admissible configurations of the variational model pro-posed in this paper, and revisit some concepts on optimal packed configurations wewill need in our analysis.2.1.
Density of optimal packing.Definition 2.1.
We denote by Ad d be the class of sets X ⊂ R d such that | x i − x j | ≥ for all x i , x j ∈ X with x i = x j . The volume density of optimal ball packings in R d is the constant C d defined by (2.1) C d := sup X ∈ Ad d lim sup r → + ∞ X ∩ rQ ) ω d | rQ | , where Q := [0 , d . Moreover, we say that T d ⊂ R d is an optimal configuration forthe (centers for the unit ball) optimal packing problem if T d ∈ Ad d and (2.2) lim r → + ∞ d ∩ rQ ) ω d | rQ | = C d . In [16] it is proved the existence of an optimal configuration, and that in defining C d and T d , Q can be replaced by any open bounded set A ⊂ R d with A = ∅ .Now we want to provide a rate of convergence in (2.1). To this purpose, forevery r > d ( rQ ) be the class of sets X ⊂ rQ such that | x i − x j | ≥ x i , x j ∈ X with x i = x j , and set(2.3) C dr := sup X ∈ Ad d ( rQ ) X ) ω d r d . It is easy to see that for all r > T dr . Lemma 2.2.
There exists C ( d ) > such that C d ≤ C dr ≤ C d + C ( d ) r for all r > .Proof. For every r > rQ = ∪ d i =1 rQ i where Q i = Q + v i , v i ∈ { , } d .Let T dr be any maximizer of (2.3), and setˆ T dr := { x ∈ T dr : dist ( x, r∂Q ) ≥ } , ˜ T d r := ∪ d i =1 ˆ T dr + v i , v i ∈ { , } d . It is easy to see that there exists a constant c ( d ) such that T dr − T dr ≤ c ( d ) r d − .Moreover, max { T d r ∩ rQ i , i = 1 , · · · , d } ≥ T d r d . Then we have r d C d r = T d r d ≤ r d C dr = T dr ≤ T d r d + c ( d ) r d − ≤ r d C d r + c ( d ) r d − . Therefore, for every r > n ∈ N we have C d n r ≤ C d n − r ≤ C d n r + c ( d )2 n − r , which by iteration over n yields C d n r ≤ C dr , C dr ≤ C d n r + n X k =1 c ( d )2 k − r . Sending n → + ∞ we deduce the claim. (cid:3) The empirical measures.
We introduce the family of empirical measures EM := (cid:26) N X i =1 δ x i : x i = x j for i = j, N ∈ N (cid:27) ⊂ M b ( R d ) . We consider the space M b ( R d ) endowed with the tight topology. Definition 2.3 (Tight convergence) . We say that a sequence { µ ε } ε ∈ (0 , ⊂ M b ( R d ) tightly converges to µ ∈ M b ( R d ) if µ ε ∗ ⇀ µ and µ ε ( R d ) → µ ( R d ) , as ε → + . Definition 2.4.
Let ε > , we define the set EM ε ⊂ EM as EM ε := (cid:26) µ ∈ EM : µ = N X i =1 δ x i with | x i − x j | ≥ ε for all i = j (cid:27) . Lemma 2.5.
Let { µ ε } ε ∈ (0 , ⊂ EM with µ ε ∈ EM ε for all ε ∈ (0 , be such that ε d ω d C d µ ε ∗ ⇀ µ for some µ ∈ M b ( R d ) , as ε → + (where C d is defined in (2.1) ).Then, there exists ρ ∈ L ∞ ( R d , [0 , such that µ = ρ L d . A. KUBIN AND M. PONSIGLIONE
Proof.
It is sufficient to prove that µ ( A ) ≤ | A | for all open set A . By the lowersemi-continuity of the total variation with respect to weak-star convergence, wehave µ ( A ) ≤ lim inf ε → + ε d ω d C d µ ε ( A ) = lim inf ε → + | A | C d ω d { A ∩ supp ( µ ε ) }| Aε |≤ | A | lim ε → + C d d ∩ Aε ) | Aε | = | A | , where the last inequality follows by (2.1) and (2.2) with Q replaced by A . (cid:3) Lemma 2.6.
For every ρ ∈ L ( R d , [0 , there exists a sequence { µ ε } ε ∈ (0 , ⊂ EM with µ ε ∈ EM ε for all ε ∈ (0 , such that ε d ω d C d µ ε → ρ L d tightly in M b ( R d ) .Proof. By a standard density argument, it is enough to prove the claim for ρ = aχ A for some a ∈ (0 ,
1) and some open set A ⊂ R d . Let µ ε := P i ∈ I ε δ x i where I ε := εa − d T d ∩ A . Then, it is easy to check that ε d ω d C d µ ε → aχ A L d tightly in M b ( R d ). (cid:3) For all µ := P Ni =1 δ x i in EM ε we set(2.4) ˆ µ := 1 C d N X i =1 χ B ε ( x i ) . Lemma 2.7.
Let { µ ε } ε ∈ (0 , ⊂ EM with µ ε ∈ EM ε for all ε ∈ (0 , , and let ρ ∈ L ( R d , [0 , be such that ε d ω d C d µ ε → ρ L d tightly in M b ( R d ) . Then, ˆ µ ε → ρ L d tightly.Proof. We observe that(2.5) lim ε → + ˆ µ ε ( R d ) = lim ε → + ε d ω d C d µ ε ( R d ) = Z R d ρ ( x ) dx. Therefore, up to a subsequence ˆ µ ε ∗ ⇀ g for some g ∈ M b ( R d ). We have to provethat g = ρ L d . To this purpose, notice that for all ϕ ∈ C c ( R d ) we have | ˆ µ ε ( ϕ ) − ρ L d ( ϕ ) | ≤ (cid:12)(cid:12)(cid:12)(cid:12) ˆ µ ε ( ϕ ) − ε d ω d C d µ ε ( ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ε d ω d C d µ ε ( ϕ ) − ρ L d ( ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) X x ∈ supp µ ε C d Z B ε ( x ) ϕ ( y ) − ϕ ( x ) dy (cid:12)(cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12)(cid:12) ε d ω d C d µ ε ( ϕ ) − ρ L d ( ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ X x ∈ supp µ ε C d Z B ε ( x ) | ϕ ( y ) − ϕ ( x ) | dy + (cid:12)(cid:12)(cid:12)(cid:12) ε d ω d C d µ ε ( ϕ ) − ρ L d ( ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ ε ε d ω d C d µ ε ( R d ) k∇ ϕ k L ∞ + (cid:12)(cid:12)(cid:12)(cid:12) ε d ω d C d µ ε ( ϕ ) − ρ L d ( ϕ ) (cid:12)(cid:12)(cid:12)(cid:12) . Since ε d ω d C d µ ε ∗ ⇀ ρ L d , the claim follows. (cid:3) Riesz interactions for σ ∈ ( − d, σ ∈ ( − d, The energy functionals.
For every ε > σ ∈ ( − d, f σε : [0 , + ∞ ) → R be defined by(3.1) f σε ( r ) := + ∞ for r ∈ [0 , ε ) , − r d + σ for r ∈ [2 ε, ∞ ) . Let C d be the volume density of the optimal ball packing in R d defined in (2.1).Let X = { x , · · · , x N } be a finite subset of R d . The corresponding energy F σε ( X )is defined as F σε ( X ) := X i = j f σε ( | x i − x j | ) (cid:18) ω d ε d C d (cid:19) . Clearly, there is a one-to-one correspondence, that we denote by A , between thefamily of finite subsets of R d and the family of empirical measures. We introducethe energy F σε : M b ( R d ) → R as a function of the empirical measure as follows:(3.2) F σε ( µ ) := ( F σε ( A ( µ )) if µ ∈ EM ε ,+ ∞ elsewhere.The functional F σε may also be rewritten as F σε ( µ ) = Z R d Z R d f σε ( | x − y | ) (cid:18) ε d ω d C d (cid:19) dµ ⊗ µ if µ ∈ EM ε ,+ ∞ elsewhere.We observe that the range of the functionals F σε is ( −∞ , ∪ { + ∞} . There-fore, we do not expect compactness properties for sequences with bounded en-ergy. In fact, it is easy to construct, adding more and more masses, a sequence { µ ε } ε ∈ (0 , ⊂ EM ε with ε d µ ε ( R d ) → + ∞ and F σε ( µ ε ) → −∞ as ε →
0. More-over, tight convergence can also fail by loss of mass at infinity, also for sequenceswith ε d µ ε ( R d ) ≤ C . Indeed, let T d be an optimal configuration for the optimalpacking, as in Definition 2.1. Let { z ε } ε ⊂ R d with | z ε | → + ∞ as ε →
0. Setting µ ε = P x ∈ ε T d ∩ B ( z ε ) δ x , we have that ε d µ ε ( R d ) ≤ C for some C independent of ε ,but in general ε d µ ε does not admit converging subsequences in the tight topology.Now we perturb the energy functionals by adding suitable confining forcing termsthat yield the desired compactness properties.Let g ∈ C ( R d ). Recalling that C d is the volume density defined in (2.1), for all ε ∈ (0 ,
1) we introduce the functionals T σε : M b ( R d ) → R defined as(3.3) T σε ( µ ) := F σε ( µ ) + G σε ( µ ) , where G σε ( µ ) := Z R d g ( x ) ε d ω d C d dµ . Compactness.
In this section we study compactness properties for the func-tionals T σε introduced in (3.3). We assume that(3.4) g ( x ) ≥ C + C | x | − σ , for some C ∈ R , C > . A. KUBIN AND M. PONSIGLIONE
Theorem 3.1 (Compactness for T σǫ ) . There exists a constant C ∗ ( σ, d ) > suchthat, if g satisfies (3.4) with C > C ∗ ( σ, d ) , then the following compactness propertyhold: let M > and let { µ ε } ε ∈ (0 , ⊂ M b ( R d ) be such that T σε ( µ ε ) ≤ M, for all ε > . Then, up to a subsequence, ε d ω d C d µ ε → ρ L d tightly in M b ( R d ) , for some ρ ∈ L ( R d , [0 , .Proof. In view of (3.4), it is enough to prove the theorem for g ( x ) = C + C | x | − σ with C > C ∗ ( σ, d ) for some C ∗ ( σ, d ) >
0. We divide the proof in several steps.
Step 1.
For all µ ∈ EM ε set K ε ( µ ) := ε d ω d µ ( R d ) and let R ε ( µ ) > R ε ( µ ) d ω d = K ε ( µ ). In this step we prove that there exists ˜ C ( σ, d ) > µ := P Ni =1 δ x i ∈ EM ε ( R d ) we have N X i =1 ε d ω d C d | x i | − σ ≥ ˜ C ( σ, d ) (cid:0) K ε ( µ ) (cid:1) d − σd . Here and later on we will assume without loss of generality (and whenever it willbe convenient) that | x i | ≥ ε for all x i ∈ supp( µ ). By triangular inequality we have | y | ≤ | x i | + ε ≤ | x i | for all y ∈ B ε ( x i ). Then, ω d ε d | x i | − σ = Z B ε ( x i ) | x i | − σ dy ≥ − σ Z B ε ( x i ) | y | − σ dy. Let A ε be the union of all the balls B ε ( x i ). We have N X i =1 ε d ω d C d | x i | − σ ≥ N X i =1 − σ C d Z B ε ( x i ) | y | − σ dy = 12 − σ C d Z A ε | y | − σ dy = 12 − σ C d Z A ε ∩ B Rε ( µ ) | y | − σ dy + 12 − σ C d Z A ε \ B Rε ( µ ) | y | − σ dy ≥ − σ C d Z B Rε ( µ ) | y | − σ dy = ˜ C ( σ, d ) (cid:0) K ε ( µ ) (cid:1) − σd , where in the last inequality we have used that | A ε | = K ε ( µ ) = | B R ε ( µ ) | , and that | y | − σ ≥ | y | − σ for all y ∈ A ε \ B R ε ( µ ) , y ∈ B R ε ( µ ) . Step 2.
Here we prove that there exists ˆ C ( σ, d ) > µ ∈ EM ε ,1( C d ) X i = j ( ε d ω d ) | x i − x j | d + σ ≤ ˆ C ( σ, d ) (cid:0) K ε ( µ ) (cid:1) − σd . First, we observe that by triangular inequality | x i − x j | ≥ | x − y | for all ( x, y ) ∈ B ε ( x i ) × B ε ( x j ). Then, there exists ˆ C ( σ, d ) > C d ) X i = j ( ε d ω d ) | x i − x j | d + σ ≤ ˆ C ( σ, d ) X i = j Z B ε ( x i ) Z B ε ( x j ) | x − y | d + σ dxdy ≤ ˆ C ( σ, d ) Z B Rε ( µε ) (0) Z B Rε ( µε ) (0) | x − y | d + σ dxdy ≤ ˆ C ( σ, d ) Z B Rε ( µε ) (0) dx Z B Rε ( µε ) (0) | z | d + σ dz = ˆ C ( σ, d )( K ε ( µ )) − σd , where the second inequality is nothing but Riesz inequality, see [19]. Step 3.
Here we prove that there exists C ∗ ( σ, d ) > C > C ∗ ( σ, d ),then the following implication holds:if lim sup ε T σε ( µ ε ) < + ∞ , then lim sup ε ε d ω d C d µ ε ( R d ) < + ∞ . By Step 1 and
Step 2 there exists C ( σ, d ) > T σε ( µ ε ) ≥ C C d K ε ( µ ) + ( − ˆ C ( σ, d ) + C ˜ C ( σ, d ))( K ε ( µ )) − σd . It is then sufficient to choose C large enough, so that ( − ˆ C ( σ, d ) + C ˜ C ( σ, d )) > Step 4.
We now prove the tight converge, up to a subsequence, of sequences { µ ε } ε with bounded energy. In view of Lemma 2.5, this step concludes the proof of thetheorem. By Step 3 we have that ε d ω d C d µ ε ( R d ) ≤ ˜ M for all ε ∈ (0 ,
1) and some˜
M >
0. Arguing by contradiction, assume that there exists δ > ε n → + and R n → + ∞ as n → + ∞ , such that(3.6) ε dn ω d C d µ ε n ( R d \ B R n (0)) ≥ δ ∀ n. Now let us split µ ε n into two components: µ ε n := µ ε n ⌊ B Rn (0) and µ ε n := µ ε n ⌊ R d \ B Rn (0) ;then T σε n ( µ ε n ) = T σε n ( µ ε n ) + T σε n ( µ ε n ) − Z B Rn (0) Z R d \ B Rn (0) | x − y | σ + d (cid:18) ε dn ω d C d (cid:19) dµ ε n ⊗ µ ε n . (3.7)From Step 2 we have that there exists
C > n such that(3.8) T σε n ( µ ε n ) ≥ − ˆ C ( σ, d )( K ε n ( µ ε n )) − σd ≥ C. Again by
Step 2 , applied now to µ ε n , we have that there exists C > n such that Z R d \ B Rn (0) Z R d \ B Rn (0) | x − y | d + σ (cid:18) ε dn ω d C d (cid:19) dµ ε n ⊗ µ ε n ≤ C. Therefore,(3.9) T ε n ( µ ε n ) ≥ − C − | C | ˜ M + C Z R d \ B Rn (0) R − σn ε dn ω d C d dµ ε n ≥ − C + C δR − σn . Finally, by Riesz inequality (or equivalently, arguing as in (3.5)) we have that thereexists
C > n such that(3.10) Z B Rn (0) Z R d \ B Rn (0) − | x − y | d + σ (cid:18) ε dn ω d C d (cid:19) dµ ε n ⊗ µ ε n ( x, y ) ≥ − C Now plugging (3.8),(3.9) and (3.10) into (3.7), we deduce that M ≥ T σε n ( µ ε n ) ≥ − C + C δR − σn , for some C independent of n , which clearly provides a contradiction for n largeenough. (cid:3) -convergence. In this section we study the Γ-convergence of the energyfunctionals defined in (3.2) and (3.3).
Proposition 3.2.
Let { µ ε } ε ∈ (0 , ⊂ M b ( R d ) with µ ε ∈ EM ε for all ε ∈ (0 , andlet ρ ∈ L ( R d ; [0 , be such that ε d ω d C d µ ε → ρ L d tightly. Let moreover h ( x, y ) := | x − y | d + σ for all x, y ∈ R d with x = y . Then, (cid:18) ε d ω d C d (cid:19) µ ε ⊗ µ ε ( h ) → ρ L d ⊗ ρ L d ( h ) , as ε → + . Proof.
The proof is divided in several steps:
Step 1.
Here we prove thatˆ µ ε ⊗ ˆ µ ε ( h ) → ρ L d ⊗ ρ L d ( h ) , as ε → + , where ˆ µ ε are defined as in (2.4) (with µ replaced by µ ε ).For all R > D ( R ) := [ x ∈ R d ( { x } × B R ( x )) . We have (cid:12)(cid:12)(cid:12)(cid:12) Z R d Z R d | x − y | d + σ d ˆ µ ε ⊗ ˆ µ ε − Z R d Z R d | x − y | d + σ ρ ( x ) ρ ( y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) ≤ Z D ( R ) | x − y | d + σ d ˆ µ ε ⊗ ˆ µ ε (3.12) + Z D ( R ) | x − y | d + σ ρ ( x ) ρ ( y ) dxdy (3.13) + (cid:12)(cid:12)(cid:12)(cid:12) Z R d \ D ( R ) | x − y | d + σ d ˆ µ ε ⊗ ˆ µ ε − Z R d \ D ( R ) | x − y | d + σ ρ ( x ) ρ ( y ) dxdy (cid:12)(cid:12)(cid:12)(cid:12) . (3.14)Moreover, we have Z D ( R ) | x − y | d + σ d ˆ µ ε ⊗ ˆ µ ε = Z R d d ˆ µ ε Z B R ( x ) | x − y | d + σ d ˆ µ ε ≤ Z R d d ˆ µ ε C d Z B R ( x ) | x − y | d + σ dy = ˆ µ ε ( R d ) ω ( R ) , where ω ( R ) → R →
0. This proves that the quantity in (3.12) tends to 0as R →
0, uniformly in ε ; a fully analogous argument shows that the same holdstrue also for the quantity in (3.13). Finally, the quantity in (3.14) tends to 0 as ε → R ) since | x − y | d + σ is continuous and bounded in R d \ D ( R ), andˆ µ e ⊗ ˆ µ ε → ρ L d ⊗ ρ L d tightly in R d , and hence also in R d \ D ( R ). Step 2.
Here we prove that (cid:18) ε d ω d C d (cid:19) µ ε ⊗ µ ε ( h ) − ˆ µ ε ⊗ ˆ µ ε ( h ) → ε → + . Let x i , x j ∈ supp( µ ε ), with i = j ; for all x ∈ B ε ( x i ) , y ∈ B ε ( x j ), by triangularinequality we have | x − y | ≤ | x i − x j | , and hence(3.15) (cid:18) ε d ω d C d (cid:19) | x i − x j | d + σ ≤ d + σ Z B ε ( x i ) Z B ε ( x j ) C d ) | x − y | d + σ dxdy. Let D ( R ) be the set defined in (3.11). We obtain that (cid:12)(cid:12)(cid:12)(cid:12) Z R d | x − y | d + σ (cid:18) ε d ω d C d (cid:19) dµ ε ⊗ µ ε − Z R d | x − y | d + σ d ˆ µ ε ⊗ ˆ µ ε (cid:12)(cid:12)(cid:12)(cid:12) (3.16) ≤ (cid:12)(cid:12)(cid:12)(cid:12) Z D ( R ) | x − y | d + σ (cid:18) ε d ω d C d (cid:19) dµ ε ⊗ µ ε (cid:12)(cid:12)(cid:12)(cid:12) (3.17) + (cid:12)(cid:12)(cid:12)(cid:12) Z D ( R ) | x − y | d + σ d ˆ µ ε ⊗ ˆ µ ε (cid:12)(cid:12)(cid:12)(cid:12) (3.18) + (cid:12)(cid:12)(cid:12)(cid:12) Z R d \ D ( R ) | x − y | d + σ (cid:18) ε d ω d C d (cid:19) dµ ε ⊗ µ ε (3.19) − Z R d \ D ( R ) | x − y | d + σ d ˆ µ ε ⊗ ˆ µ ε (cid:12)(cid:12)(cid:12)(cid:12) . By (3.15) we deduce that the quantiy in (3.17) is, up to a prefactor, less than orequal to the quantity in (3.18), which, as proved in
Step 1 , tends to zero as R → ε . Finally, since | x − y | d + σ is continuous and bounded in R d \ D ( R ), by Lemma 2.7 we easily deduce that, for any fixed R >
0, the quantityin (3.19) tends to zero as ε →
0. This concludes the proof of
Step 2.
The proof of the claim is clearly a consequence of
Step 1 and
Step 2 . (cid:3) We now introduce the candidate Γ-limit F σ : M b ( R d ) → R ∪ { + ∞} defined by F σ ( µ ) := Z R d Z R d − | x − y | d + σ dµ ⊗ µ if µ ≤ L d ,+ ∞ elsewhere. Theorem 3.3.
Let σ ∈ ( − d, . The following Γ -convergence result holds true. (1) ( Γ -liminf inequality) For every ρ ∈ L ( R d , [0 , and for every sequence { µ ε } ε ∈ (0 , ⊂ M b ( R d ) with ε d ω d C d µ ε → ρ L d tightly in M b ( R d ) it holds F σ ( ρ L d ) ≤ lim inf ε → + F σε ( µ ε ) . (2) ( Γ -limsup inequality) For every ρ ∈ L ( R d , [0 , , there exists a sequence { µ ε } ε ∈ (0 , ⊂ M b ( R d ) such that ε d ω d C d µ ε → ρ L d tightly in M b ( R d ) and F σ ( ρ L d ) ≥ lim sup ε → + F σε ( µ ε ) . Proof.
The Γ-liminf inequality is a direct consequence of Proposition 3.2 while theΓ-limsup inequaility is a direct consequence of Lemma 2.6 and again of Proposition3.2. (cid:3)
Now we introduce the Γ-limit T σ : M b ( R d ) → R of the functionals T σε introducedin (3.3), defined by T σ ( µ ) := F σ ( µ ) + Z R d g ( x ) dµ ( x ) if µ ≤ L d ,+ ∞ elsewhere. Theorem 3.4.
Let σ ∈ ( − d, , let g ∈ C ( R d ) satisfying g ( x ) ≥ for | x | largeenough, and let T σε be defined in (3.3) . The following Γ -convergence result holdstrue. (1) ( Γ -liminf inequality) For every ρ ∈ L ( R d , [0 , and for every sequence { µ ε } ε ∈ (0 , ⊂ M b ( R d ) with ε d ω d C d µ ε → ρ L d tightly in M b ( R d ) it holds T σ ( ρ L d ) ≤ lim inf ε → + T σε ( µ ε ) . (2) ( Γ -limsup inequality) For every ρ ∈ L ( R d , [0 , there exists a sequence { µ ε } ε ∈ (0 , such that ε d ω d C d µ ε → ρ L d tightly in M b ( R d ) and T σ ( ρ L d ) ≥ lim sup ε → + T σε ( µ ε ) . Proof.
We start by proving (1). It is easy to prove (see [2, Proposition 1.62])that the term R R d g ( x ) dµ is lower semicontinuous with respect to tight convergence.Then, by Theorem 3.3 we obtain that T σ ( ρ L d ) = F σ ( ρ L d ) + Z R d g ( x ) ρ ( x ) dx ≤ lim inf ε → + F σε ( µ ε ) + lim inf ε → + Z R d g ( x ) ε d ω d C d dµ ε ( x ) ≤ lim inf ε → + (cid:16) F σε ( µ ε ) + Z R d g ( x ) ε d ω d C d dµ ε ( x ) (cid:17) = lim inf ε → + T σε ( µ ε ) . We now prove (2). First consider the case ρ ∈ C c ( R d ). Let R > ρ ) ⊂ B R and let { µ ε } be the recovery sequence provided by Theorem3.3; then, it is easy to see that { µ ε χ B R } provides a recovery sequence also for thefunctionals T σε . The general case follows by a standard diagonalization argument.Indeed, for any sequence { ϕ n } ⊂ C ( R d ; [0 , ϕ in L we have F σ ( ϕ n L d ) → F σ ( ϕ L d ) (see for instance the proof of Proposition 3.2). Then, forany sequence { ρ n } ⊂ C c ( B R ; [0 , ρχ B r in L we have T σ ( ρ n L d ) →T σ ( ρχ B R L d ). Moreover, since ρ is nonnegative and g ( x ) is positive for | x | largeenough, we have that Z R d g ( x ) ρ ( x ) χ B R ( x ) dx → Z R d g ( x ) ρ ( x )( x ) dx as R → + ∞ . We deduce that T σ ( ρχ B R L d ) → T σ ( ρ L d ) as R → + ∞ . Therefore,there exists a sequence { ρ m } m ∈ N ⊂ C c ( R d ) such that ρ m → ρ in L ( R d ) and T σ ( ρ m L d ) → T σ ( ρ L d ) as m → + ∞ . (cid:3) Asymptotic behaviour of minimizers.
Here we analyze the asymptoticbehaviour of minimizers of the functionals T σε defined in (3.3). Proposition 3.5 (First variation) . Let ρ L d be a minimizer of T σ . For almostevery x ∈ R d such that < ρ ( x ) < we have g ( x ) − Z R d | x − y | d + σ ρ ( y ) dy = 0 . Proof.
Let h ( x, y ) := | x − y | − d − σ . Let 0 < α < β < E α,β := { x ∈ R d : α < ρ ( x ) < β } . Let E ⊆ E α,β , and set u := χ E . Then, for ε small enough the function ρ + εu takesvalues in (0 , ρ we deduce that0 ≤ T σ ( ρ + εu ) − T σ ( ρ )= ε Z R d g ( x ) u ( x ) dx − ε Z R d h ( x, y ) ρ ( y ) u ( x ) dy dx + o ( ε ) , where o ( ε ) /ε → ε →
0. We deduce that(3.20) Z R d g ( x ) u ( x ) dx − Z R d h ( x, y ) ρ ( y ) u ( x ) dy dx = 0 . Since the above inequality holds for u = χ E where E is any measurable set containedin { x ∈ R d : 0 < ρ ( x ) < } , by the fundamental lemma in the calculus of variationsand an easy density argument we deduce the claim. (cid:3) Theorem 3.6 (Behaviour of minimizers) . Let T σε be defined in (3.3) with g sat-isfying (3.4) for some C > C ∗ ( σ, d ) , where C ∗ ( σ, d ) is the constant provided byTheorem 3.1. Let moreover µ ε be minimizers of T σε for all ε > .Then, up to a subsequence, ε d ω d C d µ ε → χ E L d tightly in M b ( R d ) , for some set E ∈ M f ( R d ) . Moreover, χ E L d is a minimizer of T σ . Finally, if g ( x ) := G ( | x | ) for some increasing function G : R + → R , then E is a ball.Proof. By Theorem 3.1, up to a subsequence, ε d ω d C d µ ε → ρ L d tightly in M b ( R d ),for some ρ ∈ L ( R d ; [0 , ρ L d is a minimizer of T σ ; we have to prove that ρ is acharacteristic function.Let now ˜ ρ := χ supp( ρ ) and let u := ˜ ρ − ρ . By (3.20) we have0 ≤ T σ ( ρ + u ) − T σ ( ρ )= Z R d g ( x ) u ( x ) dx − Z R d h ( x, y ) ρ ( y ) u ( x ) dy dx − Z R d h ( x, y ) u ( y ) u ( x ) dy dx = − Z R d h ( x, y ) u ( y ) u ( x ) dy dx ≤ . We conlcude that the above inequalities are in fact all equalities, which in turnsimplies u = 0, i.e., ˜ ρ = ρ and ρ is a characteristic function.Finally, if g is radial and increasing with respect to | x | , then denoted by E ∗ theball centered at 0 with | E ∗ | = | E | , we have(3.21) F σ ( E ∗ L d ) < F σ ( E L d ) , Z E ∗ g ( x ) dx ≤ Z E g ( x ) dx, where the first (strict) inequality is a consequence of the uniqueness of the ball inthe Riesz inequality for characteristic functions interacting through strict increasingpotentials (see for instance [13, Theorem A4]). From (3.21) we easily conclude that E must be a ball. (cid:3) Riesz interactions for σ ∈ [0 , σ ∈ [0 , The energy functionals.
Let σ ∈ [0 , ε > r ε > r ε → ε → ε σ +1 r ε → ε → σ ∈ (0 , ε | log( r ε ) | r ε → ε → σ = 0 . (4.2)The regularized potentials are defined by(4.3) f σε ( r ) := + ∞ for r ∈ [0 , ε ) , r ∈ [2 ε, r ε ) , − r d + σ for r ∈ [ r ε , + ∞ ) , As in (3.2), we introduce the energy functionals(4.4) F σε ( µ ) := ( F σε ( A ( µ )) if µ ∈ EM ε ,+ ∞ elsewhere.We will also introduce suitable renormalized energy functionals. To this purpose,for all σ ∈ [0 ,
1) and r ∈ (0 ,
1] we set(4.5) γ σr := − Z B (0) \ B r (0) | z | d + σ dz. Notice that γ σr := dω d − r − σ σ if σ = 0 ,dω d log r if σ = 0 . (4.6)For σ ∈ [0 ,
1) the renormalized energy functionals ˆ F σε : M b ( R d ) → R are definedby ˆ F σε ( µ ) := F σε ( A ( µ )) − γ σr ε ε d ω d C d µ ( R d ) if µ ∈ EM ε ,+ ∞ elsewhere.The functional ˆ F σε may be also rewritten asˆ F σε ( µ ) = Z R d Z R d f σε ( | x − y | ) (cid:18) ε d ω d C d (cid:19) dµ ⊗ µ − γ σr ε ε d ω d C d µ ( R d ) if µ ∈ EM ε ,+ ∞ elsewhere.4.2. The continuous model.
Here we give a short overview of the Γ-convergenceanalysis of the continuous model for non-integrable Riesz potentials developed in[13].First, we introduce the fractional perimeters; for all σ ∈ (0 , σ -fractionalperimeter of E ∈ M ( R d ) is defined by P σ ( E ) = Z E Z R d \ E | x − y | d + σ dxdy. For σ = 0, a notion of 0-fractional perimeter has been introduced in [13] as follows. First, for all
R > γ R := Z B R (0) \ B (0) | z | d dz. Then, the following definition is well posed (namely, the following limit exists, [13]) P ( E ) := lim R → + ∞ Z E Z B R ( x ) \ E | x − y | d dx dy − γ R | E | . Now, we introduce the continuous Riesz functionals. For all r ∈ (0 ,
1) let J σr : M f ( R d ) → R be the functionals defined by J σr ( E ) := Z E Z E \ B r ( x ) − | x − y | d + σ dxdy. The renormalized functionals ˆ J σr : M f ( R d ) → R are defined by(4.8) ˆ J σr ( E ) := J σr ( E ) − γ σr | E | , where γ σr is the constant defined in (4.5).Now we introduce the candidate Γ-limits. For σ ∈ (0 ,
1) we define the functionalˆ F σ : M b ( R d ) → R as(4.9) ˆ F σ ( µ ) := ( P σ ( E ) − γ σ | E | if µ = χ E L d ,+ ∞ elsewhere,where γ σ = R R d \ B (0) 1 | z | d + σ dz .Moreover, for σ = 0 we define F : M b ( R d ) → R as(4.10) ˆ F ( µ ) := ( P ( E ) if µ = χ E L d ,+ ∞ elsewhere.The following theorem has been proved in [13]. Theorem 4.1.
The following compacntess and Γ -convergence results hold. Compactness:
Let σ ∈ [0 , and let r n → + . Let U ⊂ R d be an openbounded set and let { E n } n ∈ N ⊂ M f ( R d ) be such that E n ⊂ U for all n ∈ N .Finally, let C > .If ˆ J σr n ( E n ) ≤ C for all n ∈ N , then, up to a subsequence, χ E n → χ E in L ( R d ) for some E ∈ M f ( R d ) . Γ -convergence: The following Γ -convergence result holds true. (i) ( Γ -liminf inequality) For every E ∈ M f ( R d ) and for every sequence { E n } n ∈ N with χ E n → χ E strongly in L ( R d ) it holds ˆ F σ ( E ) ≤ lim inf n → + ∞ ˆ J σr n ( E n ) . (ii) ( Γ -limsup inequality) For every E ∈ M f ( R d ) , there exists a sequence { E n } n ∈ N such that χ E n → χ E strongly in L ( R d ) and ˆ F σ ( E ) ≥ lim sup n → + ∞ ˆ J σr n ( E n ) . Next proposition provides error estimates comparing the discrete functionals F σε with its continuous counterpart J σr ε . Proposition 4.2.
Let σ ∈ [0 , , and let { µ ε } ε ∈ (0 , ⊂ EM be such that µ ε ∈ EM ε for all ε ∈ (0 , and ε d ω d C d µ ε ( R d ) ≤ M for some M > .Then, there exists { E ε } ε ∈ (0 , ⊂ M f ( R d ) such that the following properties hold: (i) ε d ω d C d µ ε − χ E ε ∗ ⇀ as ε → ; (ii) || E ε | − ε d ω d C d µ ε ( R d ) | ≤ C ( M, d ) √ ε √ r ε ; (iii) |F σε ( µ ε ) − J σr ε ( E ε ) | ≤ C ( σ, d, M ) | γ σr ε | √ ε √ r ε .In particular, as a consequence of (4.1) , we have (iii’) | ˆ F σε ( µ ε ) − ˆ J σr ε ( E ε ) | → as ε → .Vice-versa, if { E ε } ε ∈ (0 , ⊂ M f ( R d ) is such that | E ε | ≤ M for some M > , thenthere exists { µ ε } ε ∈ (0 , ⊂ EM with µ ε ∈ EM ε for all ε ∈ (0 , and such that (i),(ii), (iii) and (iii’) hold.Proof. For every ε >
0, set ρ ε := √ εr ε . Let Q := [0 , d and set Q ρ ε := { ρ ε ( Q + v ) , v ∈ Z d } . Let moreover P ρ ε := { q ∈ Q ρ ε : ε d ω d C d µ ε ( q ) ≥ ρ dε } . For all q ∈ Q ρ ε we denote by ˜ q the square concentric to q and such that ˜ q = q if q ∈ P ρ ε , while | ˜ q | = ε d ω d C d µ ε ( q ) if q ∈ Q ρ ε \ P ρ ε .By Lemma 2.2 and by easy scaling arguments we deduce that(4.11) P ρ ε ≤ M ρ − dε , ≤ ε d ω d C d µ ε ( q ) − | ˜ q | ≤ C ( d ) ερ d − ε for all q ∈ Q ρ ε . We define E ε := ∪ q ∈ Q ρε ˜ q . By (4.11) we have that(4.12) || E ε | − ε d ω d C d µ ε ( R d ) | ≤ M C ( d ) ερ ε = M C ( d ) √ ε √ r ε , which proves property (ii).Let us pass to the proof of (i). Given ϕ ∈ C c ( R d ), by (4.11) we have(4.13) (cid:12)(cid:12)(cid:12) h ε d ω d C d µ ε − χ E ε , ϕ i (cid:12)(cid:12)(cid:12) ≤ C ( d, M ) k∇ ϕ k L ∞ ρ ε + k ϕ k L ∞ C ( d, M ) ερ ε , which tends to 0 as ε → | E ε | ≤ M + 1 for ε small enough. Then, by rearrangement (see for instance Lemma A.6 of [13]) it iseasy to see that − J σr ε ( E ε ) ≤ C ( σ, d, M ) | γ σr ε | . Therefore, in order to prove (iii) it isenough to show that − J σr ε ( E ε ) ≤ −F σε ( µ ε ) (cid:0) C ( σ, d ) √ ε √ r ε (cid:1) + C ( σ, d ) | γ σr ε | √ ε √ r ε , (4.14) − F σε ( µ ε ) ≤ − J σr ε ( E ε ) (cid:0) C ( σ, d ) √ ε √ r ε (cid:1) + C ( σ, d ) | γ σr ε | √ ε √ r ε . (4.15) We will prove only (4.15), the proof of (4.14) being fully analogous. For all p, q ∈ Q ρ ε with p = q , set I ( p, q ) := { ( x, y ) ∈ supp( µ ) × supp( µ ) ∩ p × q } ,R ε ( p, q ) := dist( p, q ) , ˜ R ε ( p, q ) := max x ∈ p,y ∈ q dist( x, y ) , m ε ( q ) := ε d ω d C d µ ε ( q ) . By (4.11) we have that(4.16) 1 ≤ m ε ( q ) | ˜ q | ≤ C ( d ) ερ ε for all q ∈ Q ρ ε . Moreover, since ˜ R ε ( p, q ) ≤ R ε ( p, q )+ C ( d ) ρ ε , it follows that there exists C ( σ, d ) > ε small enough,(4.17) (cid:16) ˜ R ε ( p, q ) R ε ( p, q ) (cid:17) d + σ ≤ (cid:0) C ( σ, d ) ρ ε R ε ( p, q ) (cid:1) for all q, p ∈ Q ρ ε : R ε ( p, q ) = 0 . Moreover, let Q + := { ( p, q ) ∈ Q ρ ε × Q ρ ε : R ε ( p, q ) > r ε } ; Q − := { ( p, q ) ∈ Q ρ ε × Q ρ ε : ˜ R ε ( p, q ) < r ε } ; Q = := Q ρ ε × Q ρ ε \ ( Q + ∪ Q − ) . Recalling that ε d ω d C d µ ε ( R d ) ≤ M and (4.11), it easily follows that, for ε small enough(4.18) ( ε d ω d C d ) X ( p,q ) ∈ Q = X ( x,y ) ∈ I ( p,q ) | x − y | − d − σ ≤ C ( σ, d ) r − d − σε r d − ε ρ ε = C ( σ, d ) r − σε √ ε √ r ε ≤ C ( σ, d ) | γ σr ε | √ ε √ r ε . By (4.16), (4.17) and (4.18) we have that, for ε small enough, − F σε ( µ ε ) ≤ ( ε d ω d C d ) X ( p,q ) ∈ Q + X ( x,y ) ∈ I ( p,q ) | x − y | − d − σ + C ( σ, d ) | γ σr ε | √ ε √ r ε ≤ X ( p,q ) ∈ Q + m ε ( p ) m ε ( q ) R ε ( p, q ) − d − σ + C ( σ, d ) | γ σr ε | √ ε √ r ε ≤ X ( p,q ) ∈ Q + (cid:16) C ( d ) ερ ε (cid:17) (cid:16) C ( σ, d ) ρ ε R ε ( p, q ) (cid:17) | ˜ p || ˜ q | ˜ R ε ( p, q ) − d − σ + C ( σ, d ) | γ σr ε | √ ε √ r ε ≤ (cid:16) C ( σ, d ) √ ε √ r ε (cid:17) ( − J σr ε ( E ε )) + C ( σ, d, M ) | γ σr ε | √ ε √ r ε . Finally, property (iii’) is an easy consequence of properties (ii), (iii) and of (4.1).The proof of the final claim of the proposition is fully analogoug to the proof ofthe first part of the proposition. (cid:3)
Compactness and Γ -convergence. Here we prove Γ-convergence and com-pactness properties for the functionals ˆ F σε defined in (4.9) and (4.10). Converselyto what done for the integrable case σ ∈ ( − d, Theorem 4.3.
Let σ ∈ [0 , . The following compactness and Γ -convergence resultshold. Compactness:
Let U ⊂ R d be an open bounded set and let M > . Let { µ ε } ε ∈ (0 , ⊂ M b ( R d ) be such that (4.19) ˆ F σε ( µ ε ) ≤ M, supp( µ ε ) ⊂ U ∀ ε ∈ (0 , . Then, ε d ω d C d µ ε → χ E L d tightly, as ε → + , for some measurable set E ⊂ U . Γ -convergence: The following Γ -convergence result holds true. (1) ( Γ -liminf inequality) For every E ∈ M f ( R d ) and for every { µ ε } ε ∈ (0 , ⊂M b ( R d ) with ε d ωC d µ ε → χ E L d tightly in M b ( R d ) , we have ˆ F σ ( χ E L d ) ≤ lim inf ε → + ˆ F σε ( µ ε ) . (2) ( Γ -limsup inequality) For every E ∈ M f ( R d ) , there exists a sequence { µ ε } ε ∈ (0 , with µ ε ∈ EM ε for all ε ∈ (0 , such that ε d ω d C d µ ε → χ E L d tightly in M b ( R d ) and ˆ F σ ( χ E L d ) ≥ lim sup ε → + ˆ F σε ( µ ε ) . Proof.
In order to prove the compactness property, first notice that by (4.19) wededuce that µ ε ∈ EM ε for all ε ∈ (0 , ε ∈ (0 ,
1) such that | ˆ F σε ( µ ε ) − ˆ J σr ε ( E ε ) | < ∀ ε < ε , where { E ε } ε is exactly the sequence of sets provided by Proposition 4.2. We deducethat ˆ J σr ε ( E ε ) is bounded; by Theorem 4.1 there exists E ∈ M f ( R d ) such that, upto a subsequence, χ E ε → χ E in L for ε → + . Therefore, again by Proposition 4.2 ε d ω d C d µ ε → χ E tightly as ε → + .Let us pass to the proof of the Γ-liminf inequality. By Proposition 4.2 and byTheorem 4.1 we obtain thatˆ F σ ( χ E L d ) ≤ lim inf ε → + ˆ J σr ε ( µ ε ) ≤ lim inf ε → + ( ˆ J σr ε ( E ε ) − ˆ F σε ( µ ε )) + lim inf ε → + ˆ F σε ( µ ε ) ≤ lim inf ε → + ˆ F σε ( µ ε ) . Hence the Γ-liminf inequality holds.We now prove the Γ-limsup inequality. Let { E ε } ε be the recovery sequenceprovided by Theorem 4.1; we haveˆ J σr ε ( E ε ) → ˆ F σ ( χ E L d ) as ε → . Let now { µ ε } ε ∈ (0 , be the sequence provided by the second part of Proposition4.2. Then, we have | ˆ F σε ( µ ε ) − ˆ F σ ( χ E L d ) | ≤ | ˆ F σε ( µ ε ) − ˆ J σr ε ( E ε ) | + | ˆ J σr ε ( E ) − ˆ F σ ( χ E L d ) | , which, in view of Proposition 4.2(iii’), tends to 0 as ε → (cid:3) Acknowledgements
The authors are members of the Gruppo Nazionale per l’Analisi Matematica,la Probabilit`a e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di AltaMatematica (INdAM).Tha authors thank L. De Luca and M. Novaga for useful discussions at the earlystage of this project.
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Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Univer-sit`a di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy
E-mail address , A. Kubin: [email protected] (Marcello Ponsiglione)
Dipartimento di Matematica “Guido Castelnuovo”, Sapienza Uni-versit`a di Roma, P.le Aldo Moro 5, I-00185 Roma, Italy
E-mail address , M. Ponsiglione:, M. Ponsiglione: