Compactness by Coarse-Graining in Long-Range Lattice Systems

<jats:title>Abstract</jats:title>                <jats:p>We consider energies on a periodic set <jats:inline-formula id="j_ans-2020-2100_ineq_9999">                      <jats:alternatives>                         <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">                            <m:mi mathvariant="script">ℒ</m:mi>                         </m:math>                         <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2100_inl_001.png" />                         <jats:tex-math>{\mathcal{L}}</jats:tex-math>                      </jats:alternatives>                   </jats:inline-formula> of the form <jats:inline-formula id="j_ans-2020-2100_ineq_9998">                      <jats:alternatives>                         <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">                            <m:mrow>                               <m:msub>                                  <m:mo largeop="true" symmetric="true">∑</m:mo>                                  <m:mrow>                                     <m:mrow>                                        <m:mi>i</m:mi>                                        <m:mo>,</m:mo>                                        <m:mi>j</m:mi>                                     </m:mrow>                                     <m:mo>∈</m:mo>                                     <m:mi mathvariant="script">ℒ</m:mi>                                  </m:mrow>                               </m:msub>                               <m:mrow>                                  <m:msubsup>                                     <m:mi>a</m:mi>                                     <m:mrow>                                        <m:mi>i</m:mi>                                        <m:mo>⁢</m:mo>                                        <m:mi>j</m:mi>                                     </m:mrow>                                     <m:mi>ε</m:mi>                                  </m:msubsup>                                  <m:mo>⁢</m:mo>                                  <m:mrow>                                     <m:mo fence="true" stretchy="false">|</m:mo>                                     <m:mrow>                                        <m:msub>                                           <m:mi>u</m:mi>                                           <m:mi>i</m:mi>                                        </m:msub>                                        <m:mo>-</m:mo>                                        <m:msub>                                           <m:mi>u</m:mi>                                           <m:mi>j</m:mi>                                        </m:msub>                                     </m:mrow>                                     <m:mo fence="true" stretchy="false">|</m:mo>                                  </m:mrow>                               </m:mrow>                            </m:mrow>                         </m:math>                         <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2100_inl_002.png" />                         <jats:tex-math>{\sum_{i,j\in\mathcal{L}}a^{\varepsilon}_{ij}\lvert u_{i}-u_{j}\rvert}</jats:tex-math>                      </jats:alternatives>                   </jats:inline-formula>, defined on spin functions <jats:inline-formula id="j_ans-2020-2100_ineq_9997">                      <jats:alternatives>                         <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">                            <m:mrow>                               <m:msub>                                  <m:mi>u</m:mi>                                  <m:mi>i</m:mi>                               </m:msub>                               <m:mo>∈</m:mo>                               <m:mrow>                                  <m:mo stretchy="false">{</m:mo>                                  <m:mn>0</m:mn>                                  <m:mo>,</m:mo>                                  <m:mn>1</m:mn>                                  <m:mo stretchy="false">}</m:mo>                               </m:mrow>                            </m:mrow>                         </m:math>                         <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2100_inl_003.png" />                         <jats:tex-math>{u_{i}\in\{0,1\}}</jats:tex-math>                      </jats:alternatives>                   </jats:inline-formula>, and we suppose that the typical range of the interactions is <jats:inline-formula id="j_ans-2020-2100_ineq_9996">                      <jats:alternatives>                         <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">                            <m:msub>                               <m:mi>R</m:mi>                               <m:mi>ε</m:mi>                            </m:msub>                         </m:math>                         <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2100_inl_004.png" />                         <jats:tex-math>{R_{\varepsilon}}</jats:tex-math>                      </jats:alternatives>                   </jats:inline-formula> with <jats:inline-formula id="j_ans-2020-2100_ineq_9995">                      <jats:alternatives>                         <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">                            <m:mrow>                               <m:msub>                                  <m:mi>R</m:mi>                                  <m:mi>ε</m:mi>                               </m:msub>                               <m:mo>→</m:mo>                               <m:mrow>                                  <m:mo>+</m:mo>                                  <m:mi mathvariant="normal">∞</m:mi>                               </m:mrow>                            </m:mrow>                         </m:math>                         <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2100_inl_005.png" />                         <jats:tex-math>{R_{\varepsilon}\to+\infty}</jats:tex-math>                      </jats:alternatives>                   </jats:inline-formula>, i.e., if <jats:inline-formula id="j_ans-2020-2100_ineq_9994">                      <jats:alternatives>                         <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">                            <m:mrow>                               <m:mrow>                                  <m:mo fence="true" stretchy="false">|</m:mo>                                  <m:mrow>                                     <m:mi>i</m:mi>                                     <m:mo>-</m:mo>                                     <m:mi>j</m:mi>                                  </m:mrow>                                  <m:mo fence="true" stretchy="false">|</m:mo>                               </m:mrow>                               <m:mo>≤</m:mo>                               <m:msub>                                  <m:mi>R</m:mi>                                  <m:mi>ε</m:mi>                               </m:msub>                            </m:mrow>                         </m:math>                         <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2100_inl_006.png" />                         <jats:tex-math>{\lvert i-j\rvert\leq R_{\varepsilon}}</jats:tex-math>                      </jats:alternatives>                   </jats:inline-formula>, then <jats:inline-formula id="j_ans-2020-2100_ineq_9993">                      <jats:alternatives>                         <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">                            <m:mrow>                               <m:msubsup>                                  <m:mi>a</m:mi>                                  <m:mrow>                                     <m:mi>i</m:mi>                                     <m:mo>⁢</m:mo>                                     <m:mi>j</m:mi>                                  </m:mrow>                                  <m:mi>ε</m:mi>                               </m:msubsup>                               <m:mo>≥</m:mo>                               <m:mi>c</m:mi>                               <m:mo>&gt;</m:mo>                               <m:mn>0</m:mn>                            </m:mrow>                         </m:math>                         <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2100_inl_007.png" />                         <jats:tex-math>{a^{\varepsilon}_{ij}\geq c&gt;0}</jats:tex-math>                      </jats:alternatives>                   </jats:inline-formula>. In a discrete-to-continuum analysis, we prove that the overall behavior as <jats:inline-formula id="j_ans-2020-2100_ineq_9992">                      <jats:alternatives>                         <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">                            <m:mrow>                               <m:mi>ε</m:mi>                               <m:mo>→</m:mo>                               <m:mn>0</m:mn>                            </m:mrow>                         </m:math>                         <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2100_inl_008.png" />                         <jats:tex-math>{\varepsilon\to 0}</jats:tex-math>                      </jats:alternatives>                   </jats:inline-formula> of such functionals is that of an interfacial energy. The proof is performed using a coarse-graining procedure which associates to scaled functions defined on <jats:inline-formula id="j_ans-2020-2100_ineq_9991">                      <jats:alternatives>                         <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">                            <m:mrow>                               <m:mi>ε</m:mi>                               <m:mo>⁢</m:mo>                               <m:mi mathvariant="script">ℒ</m:mi>                            </m:mrow>                         </m:math>                         <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2100_inl_009.png" />                         <jats:tex-math>{\varepsilon\mathcal{L}}</jats:tex-math>                      </jats:alternatives>                   </jats:inline-formula> with equibounded energy a family of sets with equibounded perimeter. This agrees with the case of equibounded <jats:inline-formula id="j_ans-2020-2100_ineq_9990">                      <jats:alternatives>                         <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">                            <m:msub>                               <m:mi>R</m:mi>                               <m:mi>ε</m:mi>                            </m:msub>                         </m:math>                         <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2100_inl_010.png" />                         <jats:tex-math>{R_{\varepsilon}}</jats:tex-math>                      </jats:alternatives>                   </jats:inline-formula> and can be seen as an extension of coerciveness result for short-range interactions, but is different from that of other long-range interaction energies, whose limit exits the class of surface energies. A computation of the limit energy is performed in the case <jats:inline-formula id="j_ans-2020-2100_ineq_9989">                      <jats:alternatives>                         <m:math xmlns:m="http://www.w3.org/1998/Math/MathML">                            <m:mrow>                               <m:mi mathvariant="script">ℒ</m:mi>                               <m:mo>=</m:mo>                               <m:msup>                                  <m:mi>ℤ</m:mi>                                  <m:mi>d</m:mi>                               </m:msup>                            </m:mrow>                         </m:math>                         <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_ans-2020-2100_inl_011.png" />                         <jats:tex-math>{\mathcal{L}=\mathbb{Z}^{d}}</jats:tex-math>                      </jats:alternatives>                   </jats:inline-formula>.</jats:p>

section_content: In this paper, we give a contribution to the general problem of the asymptotic analysis of systems of lattice interactions of the form

where L is a periodic lattice in ℝ d, ε > 0 is a parameter tending to 0 and a ε ij are non-negative coefficients. These functionals depend on (scalar) "spin functions" with u i ∈ {0, 1}, somehow related to ferromagnetic energies in the terminology of statistical mechanics (where usually u i ∈ {-1, 1}). 

We investigate coerciveness properties related to such energies in a discrete-to-continuum process, where the values u ε i are identified as the values u ε (εi) of a function defined on εL. In this way, a continuum limit of u ε can be defined as a limit of their piecewise-constant interpolations, e. g., defined as u ε (x) = u ε i if the point of minimum distance of εL from x is εi. Coerciveness is established by exhibiting scales s ε such that if

then the interpolations u ε are precompact in some topology and their limit points are in general non-trivial. This can be expressed by proving that the domain of the Γ-limit of the scaled energies 1

in that topology is not trivial. 

The simplest case that has been previously treated [1, 16] is nearest-neighbor interactions; i. e., when a ε ij are strictly positive only when i, j are nearest neighbors (n. n. for short) in the Delaunay triangulation of L (e. g., |i -j| = 1 if L = ℤ d ). In this case, choosing s ε = ε 1-d gives that the scaled energies ∑ i,j∈L i,j n. n. 

can be directly seen as a (possibly anisotropic) perimeter of the sets {x : u ε (x) = 1} defined through the piecewise-constant interpolation of u ε from the scaled lattice εL. Then the compactness properties of sets of equibounded perimeter ensure the coerciveness in L 1 loc (ℝ d ) and the limits are characteristic functions. Moreover, the Γ-limit of the energies can be described by an energy defined on sets of finite perimeter A, which, in the simplest homogeneous case, takes the form

(1. 1)

The same scaling works for finite-range interactions, i. e., when a ε ij is 0 if |i -j| > R for some R, even though the energies in that case must be interpreted as a non-local perimeter [14]. The finiteness of the range of the interactions can be weakened to a decay condition that can be quantified as sup{

even though the limit energies may have a non-local part if the "tails" of these series are not uniformly negligible [2]. We note that such analysis is valid beyond pair potentials and generalizes to classes of many-point interactions (see [12] ). 

If the decay assumptions (1. 2) do not hold, then the "natural" scaling for the energies may be different from the "surface scaling" ε d-1, and we might exit the class of interfacial energies. An extreme case is that of "dense graphs", which is better stated in a bounded domain, i. e., when considering energies

with Q a cube in ℝ d, and suppose that a ε ij ≥ c > 0 for (a positive percentage of) all interactions. In that case, the scaling is s ε = ε -2d, and the limit behavior is described by a more abstract limit functional called a "graphon" energy [3, 5, 20, 21], which can be viewed as a relaxation of a double integral on (0, 1) of the form ∫ (0,1)×(0,1)
section_name: Introduction

section_content: defined on BV((0, 1); {0, 1}) after a complex and rather abstract relabeling procedure and identification of functions defined on Q with functions defined on (0, 1) (see [9] ). For sparse graphs (i. e., graphs which are not dense according to the definition above) and interactions not satisfying decay conditions (1. 2), the correct scaling, the relative convergence and the form of the Γ-limit is a complex open problem. In [8], an example is given of one-dimensional energies with range R ε = 1/√ε such that a non-trivial Γ-limit exists for s ε = 1/√ε with respect to the L ∞ -weak * convergence, but it is defined on all functions of bounded variation with values in [0, 1] (and not only those with values in {0, 1}). In that example, a crucial issue is the topology of the graph of the connections where a ε ij ̸ = 0. In this paper, we consider an intermediate case, i. e., when the decay condition described above does not hold, and a ε ij ≥ c > 0 when |i -j| ≤ R ε with R ε ≫ 1 but the topology of the interactions within that range is that of a "dense" graph. We further make the assumption εR ε ≪ 1 so that the discrete-to-continuum process makes sense. We note that this latter condition is not restrictive upon a redefinition of ε in terms of R ε, e. g. taking R -1/2 ε in the place of ε. We keep the dependence of our system on R ε and ε separate since these parameters may be defined independently in applications. Under these conditions, we have

and with this scaling, functions of equibounded energy interpolated on the lattice εL converge to a characteristic function of a set of finite perimeter. The main argument for obtaining this result is by coarse-graining. Namely, we average the values of u ε for interaction on cubes with side length of order R ε so that we can think of those averages as labeled on εR ε ℤ d. We prove first that those labels for which averages are not essentially close to 0 and 1 are negligible; hence, we may regard such functions as spin functions defined on a cubic lattice. Then we show that the arguments used for nearest-neighbor interactions of [1] can be adapted for the interpolated functions of the averages. Once a limit set of finite perimeter is obtained, we can prove the convergence of the interpolations of the original functions to the same set. 

As an application of this scaling argument, we show that, for

where a is a positive function with ∫ a(ξ)|ξ| dξ finite, the Γ-limit of the energies

defined on the cubic lattice of ℝ d, is given by an energy as in (1. 1) with

In particular, if a is radially symmetric, then (1. 1) is simply a multiple of the perimeter of A. It is interesting to note that, in a sense, the case R ε → +∞ can be seen as a limit of the case of R ε finite, for which the Γlimit is of form (1. 1) with the integrand φ(ν) given by a discretization of the integral in (1. 3) (as seen in [4, 11, 17] in a slightly different context). This convergence can be re-obtained using the results in [19], where transportation maps are used to transform discrete energies in convolution functionals.
section_name: W(x, y)|v(x) -v(y)| dx dy

section_content: We denote by Let L ⊂ ℝ d be a discrete periodic set. We can suppose without loss of generality that it is periodic in the coordinate directions with period 1; i. e., L + e i = L for all i ∈ {1,. . ., d}. The Voronoi cells of L are defined as

By the periodicity of L, there exists a constant

Each spin function u : εL → {0, 1} is identified with its piecewise-constant interpolation, which is the L ∞ function defined by

Note that, by (2. 1), the L 1 norm of such u is equivalent to ε d #{i : u i ̸ = 0}. In this section, we prove coerciveness properties for energies E ε defined on spin functions u : εL → {0, 1} by

where we denote u i = u(εi), and η = η ε are such that

Lemma 1 (Compactness). Let u ε be spin functions such that E ε (u ε ) is equibounded. Then, up to subsequences, the corresponding piecewise-constant interpolations, still denoted by u ε, converge in L 1 loc (ℝ d ) to u = χ A, where A is a set of finite perimeter. 

Proof. The idea of the proof is to subdivide the set of indices L into disjoint cubes of side length η 4ε. The factor 4 is chosen so that if we consider i, j indices belonging to two neighboring cubes with this side length, respectively, then ij ∈ Q η/ε so that they interact in energy E ε. In such a way, we can associate to each u ε and each such smaller cube the value 0 or 1 of the "majority phase" if such majority phase is sufficiently close to 0 and 1, respectively, while we prove that the remaining cubes can be neglected. In this way, we will construct coarse-grained functions for which the energy E ε can be viewed as a standard nearest-neighbor ferromagnetic energy, and the compactness then follows by interpreting spin functions as sets of finite perimeter. 

For any k ∈ ℤ d, we set

For u : εL → {0, 1}, we define

. 

Note that D(ε, k) (u) measures how close the function u is to its majority phase; more precisely,

will be considered as the cubes where u is not close to a phase 1 or 0. We will first show that such cubes are negligible. Indeed, note that, thanks to the first inequality in (2. 1), the number of pairs of indices i, j within Q ε k are of order ( η ε ) 2d, and hence, there exists C δ > 0 such that if k ∈ B ε (u), then the number of "interactions within the cube"

Hence, if u ε are as in the hypotheses of the lemma, that is,

As for the indices such that D(ε, k) (u) ≥ 1 -δ, we subdivide them into the sets

and define

εQ ε k for j = 0, 1. 

In order to estimate the measure of the boundary of K ε 1 (u), we estimate the number of cubes

To that end, note that if

Hence,

For the functions u ε, we then obtain

so that

where C δ is a positive constant depending only on d, c and δ. By the compactness of sets of equibounded perimeter, this shows that the characteristic functions of the sets

Moreover, if we denote a limit of the sets K ε j (u ε ) by K j, then we have

by (2. 4). We highlight the possible dependence of the sets obtained by this procedure on δ by renaming them

Since in both cases (2. 5) holds, we must have

and K δ 0 = K δ 0 so that these sets are independent of δ and we may go back to denoting them by K 1 and K 0. We can now prove the convergence of u ε. Fixed δ < 1 as above, we write

. By (2. 4), the last term converges to 0 in L 1 (ℝ d ). As for the other two terms, we localize the convergence by restricting to a cube Q R. Note that, for k ∈ A ε 1 (u ε ), we have

, where C denotes a positive constant not depending on δ. Analogously, for k ∈ A ε 0 (u ε ), we have

so that, by the arbitrariness of δ, u ε converge locally to χ K 1. 

Remark 2. The proof of Lemma 1 works exactly in the same way if we suppose that "almost all" pairs of indices of L within Q η/ε interact; namely, if in place of energy (2. 2), we consider

with the requirement that there exists c > 0 such that

uniformly in x ∈ ℝ d. Condition (2. 6) is trivially satisfied by energies (2. 2) for c = 1. Note that condition (2. 6) cannot be relaxed to "having a proportion" of pairs of indices of L within Q η/ε interacting, however large this proportion may be below 1; i. e., it is not sufficient that

for any λ < 1. To check this, we may consider the following example: choose L = ℝ d, fix N ∈ ℕ, and define

2, but if we define

this shows that Lemma 1 does not hold. 

In this example, the subset Nℤ d of ℤ d can be considered as a "perforation" of the domain and can be treated as such, considering convergence only of the restriction of the functions to ℤ d \ Nℤ d (see [10, Section 3] ). However, the situation can be more complicated if we take

that can be regarded as representing a "high-contrast medium", for which the effect of the "perforation" cannot be neglected and for some values of c ε may give a "double porosity" effect [10].
section_name: A Compactness Result

section_content: Let a : ℝ d → [0, +∞) be such that a(ξ)|ξ| is Riemann integrable on bounded sets and such that

for some c 0, r 0 > 0. Given ε, η = η ε satisfying (2. 3), we define the coefficients

for i, j ∈ ℤ d, and the energies

By hypothesis (3. 1), we may apply the compactness lemma (Lemma 1), obtaining that the family {F ε } is coercive with respect to this convergence. 

Proposition 4. Let {u ε } be such that sup ε F ε (u ε ) < +∞. Then, up to subsequences, there exists a set of finite perimeter A such that u ε converge to A in the sense of Definition 3. 

This coerciveness property justifies the computation of the Γ-limit of F ε with respect to the convergence in Definition 3. We use standard notation in the theory of sets of finite perimeter (see, e. g., [6, 22] ). 

Theorem 5 (Homogenization). The functionals defined in (3. 2) Γ-converge with respect to the convergence in Definition 3 to the functional F defined on sets of finite perimeter by

where ∂ * A denotes the reduced boundary of A, ν the outer normal to A and φ a is given by

Proof. In order to better illustrate the proof in the general d-dimensional case, we first deal with the one-dimensional case, in which we may highlight the coarse-graining procedure without the technical complexities of the higher-order geometry. In this case, we may rewrite the energies as

The relevant computation is that of the lower bound for the target A = [0, +∞). Let u ε converge to A. For each ξ ∈ ℤ \ {0} and i ∈ {1,. . ., |ξ|}, we consider the function u ε restricted to εi + εξ ℤ. By the L 1 loc convergence, we may suppose that each such restriction changes value; i. e., there exists some k i,ξ ∈ ℤ such that

The set of ξ and i for which this does not hold is negligible for ε → 0; the precise proof is directly given for the d-dimensional functionals below. For each ξ, we then have

the latter being a Riemann sum giving the integral ∫ ℝ a(ξ)|ξ| dξ, which is F(A). We now deal with the d-dimensional case. The proof of the lower bound follows the argument above, but is more complex since we must take into account the direction of the interaction vectors ξ. We prove the inequality by applying the blow-up technique (see [13, 18], and for instance [14, 15, 23] for the discrete setting). 

We assume that the sequence {F ε (u ε )} is equibounded and that u ε converge in L 1 loc (ℝ d ) to u = χ A, where A is a set of finite perimeter. Up to subsequences, we can assume that lim inf ε→0 F ε (u ε ) = lim ε→0 F ε (u ε ). We define the localized energy on an open set U by

and define the measures μ ε (U) = F ε (u ε ; U); since the family {μ ε } is equibounded, we can assume that μ ε * ⇀ μ up to subsequences. Now, let λ = H d-1 ∂ * A; the lower bound inequality follows if we show that, for

where dμ dλ denotes the Radon-Nikodym derivative of μ with respect to the Hausdorff d -1-dimensional measure λ. By the Besicovitch derivation theorem, for H d-1 -a. a. x ∈ ∂ * A, we have

, where λ is the measure H d-1 ∂ * A, ν is the normal vector to ∂ * A at x and Q ν ϱ (x) is a cube centered in x with side length ϱ and a face orthogonal to ν. We can fix x = 0 and denote Q ν ϱ (0) by Q ν ϱ. Hence, the lower bound follows if we show that lim

We may therefore assume that ϱ = ϱ ε be such that ε ϱ → 0 and the scaled functions u ε ( ε ϱ i) interpolated on the lattice ε ϱ ℤ d converge to the characteristic function of the half space H ν = {x : ⟨x, ν⟩ < 0} on Q 1 ν. We define

We begin by estimating

With fixed α ∈ (0, 1) for each ξ ∈ ℤ d satisfying

we define

which is not empty by (3. 4), and

(see Figure 1 ). Furthermore, we fix β with

Since we will restrict our arguments to sets P α,ξ and R α,ξ above with ξ satisfying

we omit the dependence of the sets P α,ξ and R α,ξ on ν since the estimates we will obtain will be independent on ν. As in the one-dimensional case, we consider the functions restricted to the discrete lines ε ϱ i + ε ϱ ξ ℤ. The parameter α is introduced so as to estimate the number of sites of such discrete lines inside Q 1 ν. We then set

We can estimate the number of "discrete lines" intersecting R α,ξ as

where the last term is an error term accounting for the cubes intersecting the boundary of R α,ξ. Note that to every element of the complement of Bα,ξ ε/ϱ there correspond at least ⌊ α ε ϱ |⟨ξ,ν⟩| ⌋ points in ε ϱ ℤ d ∩ A ε so that, for ε sufficiently small, we get

, with C α again a positive constant accounting for boundary cubes, and hence,

(see also Figure 1 ), and hence, upon fixing R > 0 and introducing the set

Note that, by (3. 5), we may let first α → 0 and then β → 0. We eventually obtain lim inf The upper bound is obtained by a density argument (see [7, Section 1. 7] ). Hence, it suffices to treat the case of A polyhedral. In this case, it suffices to take (the interpolations) u ε i = χ A (εi) for i ∈ ℤ d. Indeed, we write ∂A as a union of N d -1-dimensional polytopes Σ k, and we denote by ν k the outer normal to Σ k and by K the d -2-dimensional skeleton of A. 

We note that there exists a constant C depending only on A such that, for any η, R > 0, after removing the closed neighborhood K + B CηR from ∂A, we obtain a disjoint collection Σ1,. . ., ΣN with Σk ⊂ Σ k such that ( Σk + B ηR ) ∩ ( Σk + B ηR ) = 0 for any k ̸ = k (see Figure 2 ). Hence, for any ξ ∈ ℤ d with |εξ| ≤ ηR, k ∈ {1,. . ., N}, and j ∈ ℤ d such that the line εj + εξ ℝ intersects Σk, the values u ε i change only once on the points of the discrete lines εj + εξ ℤ which lie in an ηR neighborhood of Σk. We note that, for lines intersecting Σ k at a point of distance not larger than CηR from K, such changes of value are at most N; then, repeating the counting argument used in the lower bound, we obtain lim sup Since this term vanishes as R → +∞, the upper bound follows. 

Example 6. If a is radially symmetric, then we have

where σ is given by

In particular, we may take a = χ B 1 the characteristic function of the unit ball in ℝ d. In this case, the limit of

Then the Γ-limit is

with minor modifications in the proof.
section_name: Homogenization of Long-Range Lattice Systems

section_name: Acknowledgment:

section_content: We thank the anonymous referee of [8], who drew our attention to the problem in this paper.
section_name: Acknowledgment:

section_content: Funding: Andrea Braides acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006. Margherita Solci acknowledges the project "Fondo di Ateneo per la ricerca 2019", funded by the University of Sassari.