-VALUED MAPS AND AN APPLICATION TO THE HOMOGENIZATION OF SPIN SYSTEMS

https://doi.org/10.1051/cocv/2021007 www.esaim-cocv.org

TWO GEOMETRIC LEMMAS FOR S

-VALUED MAPS AND AN APPLICATION TO THE HOMOGENIZATION OF SPIN SYSTEMS

Andrea Braides

and Valerio Vallocchia

Abstract. We prove two geometric lemmas forS^N−1-valued functions that allow to modify sequences of lattice spin functions on a small percentage of nodes during a discrete-to-continuum process so as to have a fixed average. This is used to simplify known formulas for the homogenization of spin systems.

Mathematics Subject Classification.35B27, 74Q05, 49J45.

Received October 27, 2020. Accepted January 9, 2021.

Dedicated to Enrique Zuazua on the occasion of his 60th birthday.

1. Introduction

A motivation for the analysis in the present work has been the study of molecular models where particles are interacting through a potential including both orientation and position variables. In particular we have in mind potentials of Gay-Berne type in models of Liquid Crystals [5, 6, 17, 19, 20]. In that context a molecule of a liquid crystal is thought of as an ellipsoid with a preferred axis, whose position is identified with a vector w ∈R³ and whose orientation is a vector u∈ S². Given α and β two such particles, the interaction energy will depend on their orientations uα,uβ and the distance vector ζαβ=wβ−wα. We will concentrate on some properties on the dependence of the energy on udue to the geometry of S²(more in general, of S^N−1).

We restrict to a lattice model where all particles are considered as occupying the sites of a regular (cubic) lattice in the reference configuration. Note that in this assumptionζαβ=β−αcan be considered as an additional parameter and not a variable. Otherwise, in general the dependence on ζ_αβ is thought to be of Lennard-Jones type (for the treatment of such energies, still widely incomplete, we refer to [8, 10,12]).

We introduce an energy density G:Z^m×Z^m× S^N−1× S^N−1→R∪ {+∞}, so that G^ξ(α, u, v) =G(α, α+ξ, u, v)

represents the free energy of two molecules oriented as u and v, occupying the sites α and β =α+ξ in the reference lattice. Note that we have included a dependence onαto allow for a microstructure at the lattice level, but the energy density is meaningful also in the homogeneous case, with G^ξ independent of α. Such energies

∗The authors acknowledge the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.

Keywords and phrases:Spin systems, maps with values on the sphere, lattice energies, discrete-to-continuum, homogenization.

Dipartimento di Matematica Universit`a di Roma “Tor Vergata”, via della ricerca scientifica 1, 00133 Rome, Italy.

*∗Corresponding author:braides@mat.uniroma2.it

Article published by EDP Sciences c EDP Sciences, SMAI 2021

are the basis for the variational analysis of complex multi-scale behaviours of spin systems (see, e.g., [9] for the derivation of energies for liquid crystals, [2] for a study of the XY-model, [14] for a very refined study of the N-clock model, [15,16] for chirality effects).

In order to understand the collective behaviour of a spin system, we introduce a small scaling parameter ε >0, so that the description of such a behaviour can be formalized as a limit asε→0. For each Lipschitz set Ω the discrete setZε(Ω) :={α∈εZ^m: (α+ [0, ε)^m)∩Ω6=∅}represents a ‘discretization’ of the set Ω at scale ε. We also let R >0 define a cut-off parameter representing the relevant range of the interactions (which we assume to be finite).

We define the family of scaled functionals E_ε(u) = X

ξ∈Z^m

|ξ|≤R

X

α∈R^ξε(Ω)

ε^mG^ξα

ε, u(α), u(α+εξ)

with domain functionsu:Zε(Ω)→ S^N−1, whereR^ξ_ε(Ω) :={α∈Zε(Ω) : α, α+εξ∈Ω}.

Extending functions defined on Zε(Ω) to piecewise-constant interpolations, we may define a discrete-to- continuum convergence of uε tou. The assumptions onGensure that utakes values in the unit ball. We can then perform an asymptotic analysis using the notation of Γ-convergence (see e.g. [7, 8,13]). Energies as E_ε, but with u taking values in general compact setsK have been previously studied by Alicandro, Cicalese and Gloria (2008) [3], who describe the limit with a two-scale homogenization formula. In the case ofS^N−1-valued functions we simplify the homogenization formula reducing to test functions usatisfying a constraint on the average. This is a non-trivial fact since this constraint is non-convex, and its proof is the main technical point of the work.

The key observation is that we can modify the sequences uεso that they satisfy an exact condition on their average. We formalize this fact in two geometrical lemmas. The first one is a simple observation that each point in the unit ball in R^N with N >1 can be written exactly as the average ofk vectors in S^N−1 for allk≥2, while the second one allows to modify sequencesuε_j satisfying an asymptotic condition on the discrete average ofuεwith a sequenceueεsatisfying a sharp one and with the same energyEεup to a negligible error. This can be done if the asymptotic average of uεhas modulus strictly less than one. In this case, most of the values of uεare not aligned; this allows to use a small percentage of these values to correct the asymptotic average to a sharp one by using the first lemma.

Optimizing on all the functions satisfying the same average condition satisfied by their limit we show that the Γ-limit of the sequence E_ε, for functions u∈L^∞(Ω, B₁^N) is a continuum functional

E₀(u) = Z

Ω

G_hom(u) dx, and the function G_hom satisfies a homogenization formula

Ghom(z) = lim

T→∞

1 T^minfn

ET(u) : 1 T^m

X

α∈Z1(Q_T)

u(α) =zo ,

where Q_T = (0, T)^m and

ET(u) = X

ξ∈Z^m

|ξ|≤R

X

β∈R^ξ₁(QT)

G^ξ(β, u(β), u(β+ξ)).

Note that the constraint in the homogenization formula involves the values of u(α), which belong to the non- convex set S^N−1. This is an improvement with respect to Theorem 5.3 in [3], where the integrand of the limit

is characterized imposing a weaker constraint on the average ofu; namely it is shown that it equals G_hom(z) = lim

η→0⁺ lim

T→∞

1 T^minfn

ET(u) :

1 T^m

X

α∈Z1(Q_T)

u(α)−z ≤ηo

. (1.1)

A formula with a sharp constraint may be useful in higher-order developments, which characterize microstruc- ture, interfaces and singularities.

The plan of the paper is as follows. In Section2we introduce the notation for discrete-to-continuum homog- enization. In Section 3 we state and prove the geometric lemmas on S^N−1-valued functions. In Section 4 we prove the homogenization formula, and finally in Section 5we give a proof of the homogenization theorem.

2. Notation and setting

Let m, n≥1, N ≥2 be fixed. We denote by {e1, . . . , em} the standard basis of R^m. Given two vectors v1, v2∈Rⁿ, by (v1, v2) we denote their scalar product. Ifv∈R^m, we use|v|for the usual euclidean norm.S^N−1 is the standard unit sphere of R^N andB^N₁ the closed unit ball of R^N. Ifx∈R, its integer part is denoted by bxc. We also set QT = (0, T)^m and B(Ω) as the family of all open subsets of Ω. If A is an open bounded set, given a functionu:A→R^N we denote its average overAas

huiA= 1

|A|

Z

A

u(x) dx.

2.1. Discrete functions

Let Ω⊂R^m be an open bounded domain with Lipschitz boundary, and letε >0 be the spacing parameter of the cubic lattice εZ^m. We define the set

Zε(Ω) :={α∈εZ^m: (α+ [0, ε)^m)∩Ω6=∅}

and we will consider discrete functionsu:Zε(Ω)→ S^N−1 defined on the lattice. Forξ∈Z^m, we define R^ξ_ε(Ω) :={α∈Z^ε(Ω) : α, α+εξ∈Ω},

while the “discrete” average of a function v:Z^ε(A)→ S^N−1 over an open bounded domainAwill be denoted by

hvi^d,ε_A = 1

#(Zε(A)) X

α∈Zε(A)

v(α).

2.2. Discrete energies

We assume that the Borel functionG:R^m×R^m× S^N−1× S^N−1→Rsatisfies the following conditions (boundedness) sup

|G(α, β, u, v)|:α, β∈R^m, u, v∈ S^N−1 <∞; (2.1) (periodicity) there existsl∈Nsuch that G(·,·, u, v) is Ql periodic; (2.2)

(lower semicontinuity)Gis lower semicontinuous. (2.3)

Given ξ∈R^m, we use the notation

G^ξ(α, u, v) =G(α, α+εξ, u, v), (2.4)

and define the functionals

X

ξ∈Z^m

|ξ|≤R

X

α∈R_ε^ξ(Ω)

ε^mG^ξα

ε, u(α), u(α+εξ)

(2.5)

foru:Zε(Ω)→ S^N−1.

2.3. Discrete-to-continuum convergence

In what follows we identify each discrete function u with its piecewise-constant extension ˜u defined by

˜

u(t) =u(α) ift∈α+ [0, ε)^m. We introduce the sets:

Aε(Ω;S^N−1) :=n

˜

u:R^m→ S^N−1 : ˜u(t)≡u(α) ift∈α+ [0, ε)^m,forα∈Zε(Ω)o .

If no confusion is possible, we will simply writeuinstead of ˜u. Ifε= 1 we will simply writeA(Ω;S^N−1) in the place of A₁(Ω;S^N−1).

Up to the identification of each function uwith its piecewise-constant extension, we can consider energies Eε:L^∞(Ω,S^N−1)→R∪ {+∞}of the following form:

E_ε(u; Ω) =









 X

ξ∈Z^m

|ξ|≤R

X

α∈R^ξε(Ω)

ε^mG^ξα

ε, u(α), u(α+εξ)

ifu∈ Aε(Ω;S^N−1),

+∞ otherwise.

(2.6)

Let εj →0 and let {uj} be a sequence of functions uj :Zε_j(Ω)→ S^N−1. We will say that {uj} converges to a function uif ˜uj is converging touweakly^∗ in L^∞. Then we will say that the functionals defined in (2.5) Γ-converge toE0 ifEεdefined in (2.6) Γ-converge toE0 with respect to that convergence.

2.4. The homogenization theorem

We will prove the following discrete-to-continuum homogenization theorem.

Theorem 2.1. LetE_ε be the energy defined in (2.5)and suppose that (2.1)–(2.3)hold. ThenE_ε Γ-converge to the functional

E0(u) = Z

Ω

Ghom(u) dx (2.7)

defined for functions u∈L^∞(Ω, B₁^N). The functionGhom is given by the following asymptotic formula Ghom(z) = lim

T→∞

1 T^minfn

E1(u;QT) :hui^d,1_Q

T =zo

. (2.8)

The treatment of the average condition in (2.8) will be performed using a geometric lemma which exploits the geometry ofS^N−1, as shown in the next section.

3. Two geometric lemmas

In this section we provide two general lemmas. The first one is a simple observation on the characterisation of sums of vectors inS^N−1, while the second one allows to satisfy conditions on the average of discrete functions with values inS^N−1.

Lemma 3.1. Let ube a vector in the ball B_k^N inR^N centered in the origin and with radiusk≥2; thenucan be written as the sum ofk vectors onS^N−1:

u=

k

X

i=1

ui ui∈ S^N−1.

Equivalently, given u∈B^N₁ andk≥2,ucan be written as the average ofk vectors onS^N−1:

u= 1 k

k

X

i=1

ui ui ∈ S^N−1.

Proof. We proceed by induction onk.

Letk= 2 and letu∈B₂^N. The set (u+S^N−1)∩ S^N−1is not empty set. If we choosev∈(u+S^N−1)∩ S^N−1 then the first induction step is proven with u1=v andu2= (u−v).

Suppose that the claim holds fork−1. Let u∈B_k^N and note that the set (u+S^N−1)∩B^N_d−1 is not empty.

Ifv∈(u+S^N−1)∩B_d−1^N by the inductive hypothesis we may write v=u₁+· · ·+u_k−1 withu_j∈ S^N−1. The claim is then proved by setting u_k =u−v.

Lemma 3.2. Let A⊂R^m be an open bounded set with Lipschitz boundary. Let δj >0 be a spacing parameter and uj :Zδ_j(A)→ S^N−1 be a sequence of discrete function. Suppose thatuj *^∗u inL^∞(A, B₁ⁿ)and that the average of uonAsatisfies |huiA|<1. Then, for all j there existu˜j such that

1. the discrete averageh˜u_ji^d_A:= 1

#Zδj(A) X

i∈Z_δj(A)

˜

u_j(i)is equal tohui_A;

2. the functionu˜j is obtained by modifying the functionuj in at most2Pj points, with P_j

#Zδ_j(A)→0.

Proof. To simplify the notation we set Zj(A) =Zδ_j(A) anduⁱ_j=uj(i).

Note that, by the weak convergence of uj,

ηj :=|huji^d− huiA|=o(1) (3.1) as j→+∞. We will treat the case thatηj 6= 0 since otherwise we simply take ˜uj=uj.

Sincehuji^d_A→ huiA, by the hypothesis that|huiA|<1 we may suppose that

|huji^d_A| ≤1−2b (3.2)

for allj, for someb∈(0,1/2).

Claim: settingB=b/(4−2b), for everyi∈Zj(A) there exist at leastB#Zj(A) indicesl∈Zj(A) such that (uⁱ_j, u^l_j)≤1−b.

Indeed, otherwise there exists at least one indexifor which the set A_b:=

l∈Zj(A) : (uⁱ_j, u^l_j)>1−b, l6=i (3.3)

is such that #Ab≥(1−B)#Zj(A) and we have

|huji^d_A| ≥(huji^d, uⁱ_j) = 1

#Zj(A) X

l∈Zj(A)

(u^l_j, uⁱ_j)

= 1

#Zj(A) X

l∈Ab

(u^l_j, uⁱ_j) + 1

#Zj(A) X

l∈Zj(A)\Ab

(u^l_j, uⁱ_j)

≥ 1

#Z^j(A)(#Ab(1−b)−(#Zj(A)−#Ab))

= 1

#Zj(A)

(2−b)#Ab−#Z^j(A)

≥(2−b)(1−B)−1 = 1−3 2b

>|huji^d_A|,

where we have used (3.2) in the last estimate. We then obtain a contradiction, thus proving the claim.

By the Claim above, there exist (B/2)#Zj(A) pairs of indices (i_s, l_s) with{i_s, l_s} ∩ {i_r, l_r}=∅ifr6=sand

(uⁱ_j^s, u^l_j^s)≤1−b. (3.4)

Sinceη_j→0, with fixedc >0 we may suppose that B#Zj(A)>2jηj

c #Zj(A)k

+ 1 (3.5)

for allj.

We now set

P_j =jη_j

c #Zj(A)k

+ 1, (3.6)

so that by (3.5) there exist pairs (is, ls) as above, with s∈Ij :={1, . . . , Pj}. Note that Pj satisfies the second claim of the theorem.

If for fixedj we define the vector

w= X

i∈Zj(A)

uⁱ_j−#(Zj(A))huiA−X

s∈Ij

(uⁱ_j^s+u^l_j^s),

then we have

|w| ≤#(Zj(A))|huji^d− huiA|+X

s∈Ij

|uⁱ_j^s+u^l_j^s|

≤#(Zj(A))η_j+X

s∈Ij

q

2 + 2(uⁱ_j^s, u^l_j^s)

≤#(Zj(A))ηj+Pj

√ 4−2b.

Since #Zj(A)ηj < c Pj by (3.6), we then have |w| ≤c Pj+Pj

√4−2b. We finally choose c >0 such that

√4−2b <2−c, so that

|w|<2Pj.

By Lemma 3.1, applied withu=−w andk= 2Pj, there exists a set of 2Pj vectors inS^N−1, that we may label as

{uⁱ_j^s, u^l_j^s :s∈Ij}, such that

X

s∈Ij

(uⁱ_j^s+u^l_j^s) =−w. (3.7)

If we now define ˜uj by setting

˜ uⁱ_j =

(uⁱ_j ifi∈ {is, l_s:s∈I_j}

uⁱ_j otherwise, (3.8)

we have

h˜u_ji^d_A = 1

#Zj(A) X

i∈Zj(A)

uⁱ_j−X

s∈Ij

(uⁱ_j^s+u^l_j^s) +X

s∈Ij

(uⁱ_j^s+u^l_j^s)

= 1

#Zj(A) X

i∈Zj(A)

uⁱ_j−X

s∈Ij

(uⁱ_j^s+u^l_j^s)−w

=huiA, and the proof is concluded.

Remark 3.3. The assumption|huiA|<1 in Lemma3.2is sharp: if|huiA|= 1, we may haveu_j*^∗u, such that uj6=uand|huji^d,δ_A ^j|= 1 at every point (for example takeuand uj constant vectors inS^N−1). In this case, in order to havehuji^d,δ_A ^j =huiA, we should change the functionuj in every point.

4. The homogenization formula

In this section we prove that the homogenization formula characterizingG_homin Theorem2.1is well defined, and derive some properties of that function.

Proposition 4.1. Let G be a function satisfying (2.1)–(2.3) and let G^ξ be defined as in (2.4). For all T >0 consider an arbitrary x_T ∈R^m, then the limit

lim

T→∞

1 T^minfn

E₁(u;x_T+Q_T) : hui^d,1_xT+QT =zo

(4.1) exists for all z∈B₁^N.

Proof. Letz∈B^N₁ be fixed. In the following we will assumeGto be 1-periodic (which means that in (2.2) we considerl= 1) andxT = 0, since the general case can be derived similarly following arguments already present

for example in [1,3] and only needing a heavier notation. Let t >0 and consider the function gt(z) = 1

t^minfn

E1(u, ζ;Qt) : hui^d,1_Q

t =zo

. (4.2)

In the rest of the proof we will drop the dependence on z. Letutbe a test function forgt such that 1

t^mE1(ut;Qt)≤gt+1

t, (4.3)

For every s > twe want to prove thatgs< gtup to a controlled error.

For fixeds, t, we introduce the following notation:

I:=n

0, . . . ,js t

k−1o^m .

We can construct a test functions forgs as us(β) =

(ut(β−ti) if β ∈ti+Qt i∈I

¯

u(β) otherwise, where ¯uis a S^N−1-valued function such thathusi^d,1_Q

s =z. We can choose such ¯uthanks to Lemma 3.1: define Z(Qs) =Z^m∩Qs, Qs,t= [

i∈I

(ti+Qt)∩Z(Qs) .

We want ¯uto be such that

X

β∈Z(Qs)

u_s(β) =z#(Z(Q_s)).

Equivalently

X

β∈Qs,t

β∈ti+Qt

u_t(β−ti) + X

β∈Z(Q_s)\Qs,t

¯

u(β) =z#(Z(Q_s)),

which means that

X

β∈Z(Qs)\Qs,t

¯

u(β) =z

#(Z(Q_s))−#(Q_s,t)

. (4.4)

On the left-hand side of (4.4) we are summing #(Z(Qs))−#(Qs,t) vectors in S^N−1 while on the right-hand side we have a vector which belongs to a ball whose radius is at most #(Z(Qs))−#(Qs,t).

If|z|<1, thanks to Lemma3.1we know that it is possible to choose the values of ¯uin such a way that the relation (4.4) is satisfied.

If|z|= 1, we simply set ¯u(β)≡z, and again (4.4) is satisfied. Moreover, we observe that R^ξ₁(Qs)⊆ [

i∈I

R^ξ₁(ti+Qt)

!

∪ R^ξ₁ Qs\[

i∈I

(ti+Qt)

!

∪

[

i∈I

(ti+ ({0, . . . , t+R}^N\{0, . . . , t−R}^N))

!

and ifβ belongs to one of the last two set of indices, then D^ξ₁ζs(β) =M(ξ/|ξ|).

Recalling now (2.1), for some ¯C >0 big enough, we have that gs≤ 1

s^mE1(us;Qs)

≤js t

k^m 1

s^mE1(ut;Qt) + 1 s^m

C¯

s^m−js t

k^m

t^m+js t

k^m

((t+R)^m−(t−R)^m) .

Using now (4.3) we get gs≤js

t km t^m

s^m

gt+1 t

+ 1

s^m C¯

s^m−js t

km

t^m+js t

km

((t+R)^m−(t−R)^m) .

Letting nows→+∞and thent→+∞, we have that lim sup

s→+∞

g_s(z)≤lim inf

t→+∞g_t(z), which concludes the proof.

Remark 4.2. Note that forz∈ S^N−1the only test function for the minimum problem in (4.1) is the constant z, so that the limit is actually an average over the period withu=z.

Proposition 4.3. The functionGhom as defined in (2.8)is convex and lower semicontinuous in B₁^N. Proof. We want to show that for every 0≤t≤1 and for everyz₁, z₂∈B^N₁ it holds:

Ghom(tz1+ (1−t)z2, M)≤t Ghom(z1, M) + (1−t)Ghom(z2, M). (4.5) Letk∈Nbe fixed; having (2.2) in mind, and thanks to Proposition4.1, it is not restrictive to takek∈lN. We define

gk(z) = 1 k^minfn

E1(u;Qk) : hui^d,1_Q

k =zo

. (4.6)

In the following we will denoteg_k¹=g_k(z₁),g²_k=g_k(z₂).

Letu¹_k andu²_k be functions such that 1

k^mE₁(u¹_k;Q_k)≤g¹_k+1

k, (4.7)

1

k^mE₁(u²_k;Q_k)≤g²_k+1

k. (4.8)

Leth > k be such thath/k∈N. Denotegh=gh(tz1+ (1−t)z2), we define the following test function forgh: uh(β) =















u¹_k(β−ki) if β∈ki+Qk i∈

0, . . . ,h k−1

^m−1

×

0, . . . , h

kt

−1

u²_k(β−ki) if β∈ki+Qk i∈

0, . . . ,h k−1

^m−1

× h

k−

h(1−t) k

, . . . ,h

k −1

¯

u(β) otherwise,

Reasoning as in Proposition4.1, thanks to Lemma 3.1we can choose the values of ¯usuch that husi^d,1_Q

s =tz1+ (1−t)z2. By (2.1) and (2.2), for some ¯C >0 we get

gh≤ 1

h^mE1(uh, ζh;Qh)

≤ 1 h^m

h k

m−1h kt

E1(u¹_k;Qk) + 1 h^m

h k

m−1h(1−t) k

E1(u²_k;Qk) + 1

h^m

C¯ h^m− h

k

^m−1h kt

+

h(1−t) k

k^m

!

+ 1 h^m

C¯ h

k ^m

((k+R)^m−(k−R)^m).

Then, thanks to (4.7) and (4.8), we can rewrite the above relation as g_h≤ k^m

h^m h

k

m−1h

kt g¹_k+1 k

+k^m

h^m h

k

m−1h(1−t)

k g_k²+1 k

+ 1 h^m

C¯ h^m− h

k

m−1 h kt

+

h(1−t) k

k^m

!

+ 1 h^m

C¯ h

k ^m

((k+R)^m−(k−R)^m).

Lettingh→+∞and thenk→+∞, we can conclude the proof of the convexity.

From the convexity and the boundedness of Ghom we deduce that it is continuous in the interior of B₁^N. Moreover, by (2.3) it is lower semicontinuous at points on the boundary ofB₁^N

Remark 4.4. Note that the functionGhom may not be continuous on B₁^N, even if it is convex. It suffices to take a nearest-neighbor energyG=G^ξ independent onα, withG(e1, e1) = 0 andG(u, v) = 1 otherwise. In this case, in particular Ghom(e1) = 0 andGhom(z) = 1 ifz∈ S^N−1,z6=e1.

5. Proof of the homogenization theorem

Thanks to the geometric lemmas in Section3, we can now easily give a proof of the homogenization theorem.

We remark that it will be sufficient to prove a lower bound, since we may resort to the homogenization result of Alicandro, Cicalese and Gloria [3] in order to give an upper bound for the homogenized functional. Indeed, by (1.1) we have Ghom≤Ghom, so that functional (2.7) is an upper bound for the homogenized energy. Note that we could directly proof the upper bound using approximation results and constructions starting from the formula forGhom, but this would be essentially a repetition of the arguments in [3].

In order to prove a lower bound we will make use of Lemma3.2and of the Fonseca-M¨uller blow-up technique [11, 18]. Let ε_j →0 be a vanishing sequence of parameters, let u∈ L^∞(Ω, B₁^N) and let u_j *^∗ u with u∈ L^∞(Ω, B₁^N). We define the measuresµ_j by setting

µj(A) = X

ξ∈Z^m

|ξ|≤R

X

α∈R^ξ_εj(Ω)

ε^m_j G^ξα εj

, uj(α), uj(α+εjξ) δ_α+εj

2ξ(A)

for allA∈ B(Rⁿ), whereδxdenotes the usual Dirac delta measure atx. Since the measures are equibounded, by the weak* compactness of measures there exists a limit measureµon Ω such that, up to subsequences,µj*^∗µ.

We consider the Radon-Nikodym decomposition of the limit measure µ with respect to the m-dimensional Lebesgue measureL^m:

µ= dµ

dxdL^m+µ^s, µ^s⊥ L^m.

Besicovitch Derivation Theorem [4] states that almost every point in Ω with respect toL^mis a Lebesgue point for µ. So, we may suppose that x0 ∈ Zε_j(Ω) be a Lebesgue point both for u and for µ and let Qρ(x0) = x0+ (−ρ/2, ρ/2)^m. We then have

dµ

dx(x₀) = lim

ρ→0⁺

µ(Qρ(x0))

L^m(Q_ρ(x₀)) = lim

ρ→0⁺

1

ρ^mµ(Q_ρ(x₀)). (5.1)

Recalling that

µ(Qρ(x0)) = lim

j→+∞µj(Qρ(x0)) (5.2)

except for a countable set ofρ, by a diagonalization argument on (5.1) and (5.2) we can extract a subsequence {ρj}such that

dµ

dx(x0) = lim

j→+∞

1

ρ^m_j µj(Qρ_j(x0)) holds. This means that

dµ

dx(x₀) = lim

j→+∞

ε_j ρj

^m X

ξ∈Z^m

|ξ|≤R

X

α∈R^ξ_εj(Ω)

G^ξα εj

, u_j(α), u_j(α+ε_jξ) δ_α+εj

2ξ(Q_ρj(x₀)). (5.3)

Also note that, by the weak^∗ convergence ofu_j tou, we have huiQ_ρ(x₀)= lim

j huji^ε_Q^j

ρ(x0)= lim

j huji^d,ε_Q ^j

ρ(x0), and that for almost everyx0 we have

lim

ρ→0⁺huiQ_ρ(x₀)=u(x0), so that we may assume that

lim

j huji^d,ε_Q ^j

ρj(x0)=u(x₀).

We can parameterizing functions on a common unit cube, by setting v_j(γ) =u_j(x₀+ρ_jγ) and δ_j= ε_j

ρj

.

With this parameterization, (5.3) reads dµ

dx(x₀) = lim

j→+∞δ^m_j X

ξ∈Z^m

|ξ|≤R

X

γ+^x0

ρj∈R^ξ_δj(¹

ρjΩ)

G^ξx0

ε_j + γ

δ_j, v_j(γ), v_j(γ+δ_jξ)

δγ−^x_ρj⁰+^δj₂ξ(Q₁(0)).

Note that we have limjhvji^d,δ_Q ^j

1(0) =u(x0). We can then apply Lemma 3.2 with vj in the place of uj and A=Q1(0). We obtain a familyevj with

hevji^d,δ_Q ^j

1(0)=u(x0), (5.4)

and such thatevj(γ) =vj(γ) except for a setPj ofγwith #Pj=o(δ^−m_j ). From this, the finiteness of the range of interactions and the boundedness ofG^ξ, we further rewrite as

dµ

dx(x₀) = lim

j→+∞δ^m_j X

ξ∈Z^m

|ξ|≤R

X

γ+^x_ρj⁰∈R^ξ_δj(_ρj¹Ω)

G^ξx₀ εj

+ γ δj

,ev_j(γ),ev_j(γ+δ_jξ)

δγ−^x_ρj⁰+^δj₂ξ(Q₁(0)).

We now set

T_j= ρ_j εj

=δ_j⁻¹, x_Tj =x₀ εj

,

so that

dµ dx(x0)

= lim

j→+∞

1 T_j^m

X

ξ∈Z^m

|ξ|≤R

X

γ+^xT_Tj^j∈R^ξ

T−1 j

(_ρj¹Ω)

G^ξ

xT_j+Tjγ,vej(γ),evj

γ+ ξ

Tj

δγ−^xT_Tj^j+₂¹_Tjξ(Q1(0))

= lim

j→+∞

1 T_j^m

X

ξ∈Z^m

|ξ|≤R

X

η+x_Tj∈R₁^ξ(¹

εjΩ)

G^ξ

xT_j+η,evj(η T_j),vej

η+ξ T_j

δ_η−x

Tj+¹₂ξ(QT_j(0)).

We now set wj(η) =vj(_T^η

j) and use the boundedness ofG^ξ andR to deduce that dµ

dx(x₀)≥ lim

j→+∞

1 T_j^m

X

ξ∈Z^m

|ξ|≤R

X

η+x_Tj∈R^ξ₁(Q_Tj)

G^ξ x_Tj +η,we_j(η),ev_j(η+ξ) δ_η−x

Tj+¹₂ξ(Q_Tj(0)).

Note indeed that by considering interactions in R^ξ₁(QT_j) we neglect a contribution of a number of interactions of order O(T_j^m−1);i.e., an energy contribution of orderO(T_j⁻¹). Noting thatwj satisfies the constraint

hweji^d,1_x

Tj+Q_Tj(0)=u(x0),

thanks to (5.4), by Proposition4.1we finally deduce that dµ

dx(x₀)≥ lim

j→∞

1 T_j^minfn

E₁(w;x_Tj+Q_Tj) : hwi^d,1_x

Tj+Q_Tj =u(x₀)o

=G_hom(u(x₀)).

Since this holds for almost all x0 ∈Ω, we have proved the desired lower bound, and hence the convergence result.

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