François Alouges
email: francois.alouges@polytechnique.edu

Giovanni Di Fratta
email: difratta@cmap.polytechnique.fr

Homogenization of Composite Ferromagnetic Materials

Nowadays, nonhomogeneous and periodic ferromagnetic materials are the subject of a growing interest. Actually such periodic configurations often combine the attributes of the constituent materials, while sometimes, their properties can be strikingly different from the properties of the different constituents. These periodic configurations can be therefore used to achieve physical and chemical properties difficult to achieve with homogeneous materials. To predict the magnetic behavior of such composite materials is of prime importance for applications.

The main objective of this paper is to perform, by means of Γ-convergence and two-scale convergence, a rigorous derivation of the homogenized Gibbs-Landau free energy functional associated to a composite periodic ferromagnetic material, i.e. a ferromagnetic material in which the heterogeneities are periodically distributed inside the ferromagnetic media. We thus describe the Γ-limit of the Gibbs-Landau free energy functional, as the period over which the heterogeneities are distributed inside the ferromagnetic body shrinks to zero.

Introduction

Composite materials are an important class of natural or engineered heterogeneous media, composed of a mixture of two or more constituents with significantly different physical or chemical properties, firmly bonded together, which remain separate and distinct within the finished structure. Finding a model which considers the composite as a bulk and whose coefficients and terms are computed from suitable averages of those of its constituents and the geometry of the microstructure is the aim of homogenization theory. The study of composites and their homogenization is a subject with a long history, which has attracted the interest and the efforts of some of the most illustrious names in science [START_REF] Markov | Heterogeneous media: micromechanics modeling methods and simulations[END_REF]: In 1824, Poisson, in his first Mémoire sur la théorie du magnétisme [START_REF] Poisson | Mémoire sur la théorie du magnétisme[END_REF], put the basis of the theory of induced magnetism assuming a model in which the body is composed of conducting spheres embedded in a nonconducting material. This paper is the origin of the basic models and ideas that prevailed in the theory of heterogeneous media in almost all domains of continuum mechanics, for almost a century after its appearance [START_REF] Markov | Heterogeneous media: micromechanics modeling methods and simulations[END_REF]. We refer the reader to the papers of Markov [START_REF] Markov | Heterogeneous media: micromechanics modeling methods and simulations[END_REF] and Landauer [START_REF] Landauer | Electrical conductivity in inhomogeneous media AIP Conference Proceedings[END_REF] for more historical details.

Nowadays, nonhomogeneous and periodic ferromagnetic materials are the subject of a growing interest. Actually such periodic configurations often combine the attributes of the constituent materials, while sometimes, their properties can be strikingly different from the properties of the different constituents [START_REF] Milton | The Theory of Composites[END_REF]. These periodic configurations can be therefore used to achieve physical and chemical properties difficult to achieve with homogeneous materials. To predict the magnetic behavior of such composite materials is of prime importance for applications [START_REF] Milton | The Theory of Composites[END_REF]. From a mathematical point of view, the study of composite materials, and more generally of media which involve microstructures, is the main source of inspiration for the Mathematical Theory of Homogenization which, roughly speaking, is a mathematical procedure which aims at understanding heterogeneous materials with highly oscillating heterogeneities (at the microscopic level) via an effective model [START_REF] Nandakumaran | An overview of Homogenization[END_REF].

The main objective of this paper is to perform, in the framework of De Giorgi's notion of Γ-convergence [START_REF] De Giorgi | Su un tipo di convergenza variazionale[END_REF] and Allaire's notion of two-scale convergence [START_REF] Allaire | Homogenization and two-scale convergence[END_REF] (see also the paper by Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF]), a mathematical homogenization study of the Gibbs-Landau free energy functional associated to a composite periodic ferromagnetic material, i.e. a ferromagnetic material in which the heterogeneities are periodically distributed inside the ferromagnetic media. Compared to earlier works related to the subject (see for instance [START_REF] Simone | Energy minimizers for large ferromagnetic bodies[END_REF][START_REF] Simone | Hysteresis and imperfections sensitivity in small ferromagnetic particles[END_REF][START_REF] Haddar | Homogenized model for a laminar ferromagnetic medium[END_REF][START_REF] Santugini-Repiquet | Homogenization of the demagnetization field operator in periodically perforated domains[END_REF]) we consider here the full Gibbs-Landau functional for mixtures of different materials in the three-dimensional space.

1.1 The Landau-Lifshitz micromagnetic theory of single-crystal ferromagnetic materials

According to Landau and Lifshitz micromagnetic theory of ferromagnetic materials (see [START_REF] Bertotti | Hysteresis in magnetism: for physicists, materials scientists, and engineers[END_REF][START_REF] Brown | The fundamental theorem of the theory of fine ferromagnetic particles[END_REF][START_REF] Landau | On the theory of the dispersion of magnetic permeability in ferromagnetic bodies[END_REF][START_REF] Mayergoyz | Nonlinear magnetization dynamics in nanosystems[END_REF]), the states of a rigid single-crystal ferromagnet, occupying a region Ω ⊆ R 3 , and subject to a given external magnetic field h a , are described by a vector field, the magnetization M, verifying the so-called fundamental constraint of micromagnetic theory: A ferromagnetic body is always locally saturated, i.e. there exists a positive constant M s such that |M| = M s (T ) a.e. in Ω.

(

The saturation magnetization M s depends on the specific material and on the temperature T , and vanishes above a temperature (characteristic of each crystal type) known as the Curie point. Since we will assume that the specimen is at a fixed temperature below the Curie point of the material, the value M s will be regarded as a material dependent function, and therefore as a constant function when working on singlecrystal ferromagnets. Due to the constraint [START_REF] Acerbi | Existence and regularity for mixtures of micromagnetic materials[END_REF] in the sequel we express the magnetization M under the form M := M s (T )m where m : Ω → S 2 is a vector field which takes its values on the unit sphere S 2 of R 3 .

Even though the magnitude of the magnetization vector is constant in space, in general it is not the case for its direction, and the observable states can be mathematically characterized as local minimizers of the Gibbs-Landau free energy functional associated to the single-crystal ferromagnetic particle (using the notation of [START_REF] Bertotti | Hysteresis in magnetism: for physicists, materials scientists, and engineers[END_REF][START_REF] Mayergoyz | Nonlinear magnetization dynamics in nanosystems[END_REF])

G L (m) := Ω a ex |∇m| 2 dτ =:E(m) + Ω ϕ an (m) dτ =:A(m) - µ 0 2 Ω h d [M s m] • M s m dτ =:W(m) -µ 0 Ω h a • M s m dτ =:Z(m)
.

(

) 2
The first term, E(m), called exchange energy, penalizes spatial variations of m. The factor a ex in the term is a phenomenological positive material constant which summarizes the effect of (usually very) short-range exchange interactions. The second term, A(m), or the anisotropy energy, models the existence of preferred directions for the magnetization (the so-called easy axes), which usually depend on the crystallographic structure of the material. The anisotropy energy density ϕ an : S 2 → R + is assumed to be a non-negative even and globally lipschitz continuous function, that vanishes only on a finite set of unit vectors (the easy axes).

The third term, W(m), is called the magnetostatic self-energy, and is the energy due to the (dipolar) magnetic field, also known in literature as the stray field, h d

∆u m = -div(mχ Ω ) in D R 3 . (3)
In (3) we have indicated with mχ Ω the extension of m to R 3 that vanishes outside Ω. Lax-Milgram theorem guarantees that equation (3) possesses a unique solution in the Beppo-Levi space:

BL 1 R 3 = u ∈ D (R 3 ) : u(•) 1 + | • | 2 ∈ L 2 (R 3 ), ∇u ∈ L 2 (R 3 , R 3 ) . (4)
Eventually, the fourth term Z(m), is called the interaction energy (or Zeeman energy), and models the tendency of a specimen to have its magnetization aligned with the external field h a , assumed to be unaffected by variations of m. The competition of those four terms explain most of the striking pictures of the magnetization that ones can see in most ferromagnetic material [START_REF] Hubert | Magnetic Domains: The Analysis of Magnetic Microstructures[END_REF], in particular the so-called domain structure, that is large regions of uniform or slowly varying magnetization (the magnetic domains) separated by very thin transition layers (the domain walls).

The Gibbs-Landau energy functional associated to composite ferromagnetic materials

Physically speaking, when considering a ferromagnetic body composed of several magnetic materials (i.e. a non single-crystal ferromagnet) a new mathematical model has to be introduced. In fact, as far as the ferromagnet is no more a single crystal, the material depending functions a ex , M s (T ) and ϕ an are no longer Figure 1: If we assume that the heterogeneities are evenly distributed inside the ferromagnetic media Ω, we can model the material as periodic. As illustrated in the figure, this means that we can think of the material as being built up of small identical cubes Q ε , the side length of which we call ε.

constant on the region Ω occupied by the ferromagnet. Moreover one has to describe the local interactions of two grains with different magnetic properties at their touching interface [START_REF] Acerbi | Existence and regularity for mixtures of micromagnetic materials[END_REF].

From a mathematical point of view, this latter requirement is usually taken into account in two different ways. Either one adds to the model a surface energy term which penalizes jumps of the magnetization direction m at the interface of both grains, or, and we stick on this later on, one simply considers a strong coupling, meaning that the direction of the magnetization does not jump through an interface. We insist on the fact that only the direction is continuous at an interface while the magnitude M s is obviously discontinuous . Therefore, the natural mathematical setting for the problem turns out to be characterized by the assumption that the magnetization direction m is in the "weak" Sobolev metric space H 1 (Ω, S 2 ), d L 2 (Ω,S 2 ) , i.e. on the metric subspace of H 1 (Ω, R 3 ) constituted by the functions constrained to take values on the unit sphere of R 3 and endowed with the L 2 (Ω) metric. It is in this framework that we will conduct our work from now on. We start by recalling the basic idea of the mathematical theory of homogenization. Let Ω ⊂ R 3 be the region occupied by the composite material. If we assume that the heterogeneities are regularly distributed, we can model the material as periodic. As illustrated in Fig. 1, this means that we can think of the material as being built up of small identical cubes, the side length of which being called ε. Let Q = [0, 1] 3 be the unit cube of R 3 . We let for y ∈ Q, a ex (y), M s (y), ϕ an (y, m) be the periodic repetitions of the functions that describe how the exchange constant a ex , the saturation magnetization M s and the anisotropy density energy ϕ an (y, m) vary over the representative cell Q (see Fig. 1). Substituting x/ε for y, we obtain the two-scale functions a ε (x) := a ex (x/ε), M ε (x) := M s (x/ε) and ϕ ε (x, m) := ϕ an (x/ε, m) that oscillate periodically with period ε as the variable x runs through Ω, describing the oscillations of the material dependent parameters of the composite. At every scale ε, the energy associated to the ε-heterogeneous ferromagnet, will be given by the following generalized Gibbs-Landau energy functional

G ε L (m) := Ω a ε |∇m| 2 dτ =:Eε(m) + Ω ϕ ε (•, m) dτ =:Aε(m) - µ 0 2 Ω h d [M ε m] • M ε m dτ =:Wε(m) -µ 0 Ω h a • M ε m dτ =:Zε(m) . (5)
The asymptotic Γ-convergence analysis of the family of functionals (G ε L ) ε∈R + as ε tends to 0, is the object of the present paper.

Statement of the main result

The main purpose of this paper is to analyze, by the means of both Γ-convergence and two-scale convergence techniques, the asymptotic behavior, as ε → 0, of the family of Gibbs-Landau free energy functionals (G ε L ) ε∈R + expressed by [START_REF] Babadjian | Homogenization of variational problems in manifold valued Sobolev spaces[END_REF]. Let us make the statement more precise.

We consider the unit sphere S 2 of R 3 and, for every s ∈ S 2 , the tangent space of S 2 at a point s will be denoted by T s S 2 . The class of admissible maps we are interested in is defined as

H 1 (Ω, S 2 ) := m ∈ H 1 (Ω, R 3 ) : m(x) ∈ S 2 for τ -a.e. x ∈ Ω ,
where we have denoted by τ the Lebesgue measure on R 3 . We consider H 1 (Ω, S 2 ) as a metric space endowed with the metric structure induced by the classical L 2 (Ω, R 3 ) metric. For every positive real number t > 0, we set Q t := [0, t] 3 and Q := Q 1 = [0, 1] 3 . We recall that a function u : R 3 → R is said to be Q-periodic if u(•) = u(• + e i ) for every e i in the canonical basis (e 1 , e 2 , e 3 ) of R 3 .

For the energy densities appearing in the family (G ε L ) ε∈R + we assume the following hypotheses:

[H 1 ]
The exchange parameter a ex is supposed to be a Q-periodic measurable function belonging to L ∞ (Q) which is bounded from below and above by two positive constants c ex > 0, C ex > 0, i.e. 0 < c ex a ex (y) C ex for τ -a.e. y ∈ Q. In the setting of classical Calculus of Variations, this hypothesis guarantees that the exchange energy density, which has the form g(y, ξ) := a ex (y)|ξ| 2 , ξ ∈ R 3×3 , is a Carathéodory integrand satisfying the following quadratic growth condition for τ -a.e. y ∈ Q

∀ξ ∈ R 3×3 c ex |ξ| 2 g(y, ξ) C ex (1 + |ξ| 2 ). ( 6
)
Then we set a ε (x) := a ex (x/ε).

[H 2 ] The anisotropy density energy ϕ an : R 3 × S 2 → R + is supposed to be a Q-periodic measurable function belonging to L ∞ (Q) with respect to the first variable, and globally lipschitz with respect to the second one (uniformly with respect to the first variable), i.e. ∃κ L > 0 such that ess sup

y∈Q |ϕ an (y, s 1 ) -ϕ an (y, s 2 )| κ L |s 1 -s 2 | ∀s 1 , s 2 ∈ S 2 . ( 7
)
We then set ϕ ε (x, s) := ϕ an (x/ε, s). The hypotheses assumed on ϕ an are sufficiently general to treat the most common classes of crystal anisotropy energy densities arising in applications. As a sake of example, for uniaxial anisotropy, the energy density reads as

ϕ an (y, m(x)) = κ(y)[1 -(u(y) • m(x)) 2 ], (8)
the spatially dependent unit vector u(•) being the easy axis of the crystal. For cubic type anisotropy, the energy density reads as:

ϕ an (y, m(x)) = κ(y) 3 i=1 [(u i (y) • m(x)) 2 -(u i (y) • m(x)) 4 ], (9)
the mutually orthogonal unit vectors u i (•) being the three easy-axes of the cubic crystal. Note that the anisotropy depends on the material both in strength (κ(y)) and in direction (u i (y)).

[H 3 ] The saturation magnetization M s is supposed to be a Q-periodic measurable function belonging to L ∞ (Q), and we set

M ε (•) = M s (•/ε).
The main result of this paper is the following:

Theorem 1.1 Let (G ε L )
ε∈R + be a family of Gibbs-Landau free energy functionals satisfying the hypotheses

[H 1 ], [H 2 ] and [H 3 ]. Then (G ε L ) ε∈R + is equicoercive in the metric space H 1 (Ω, S 2 ), d L 2 (Ω,S 2 ) . Moreover (G ε L ) ε∈R + Γ-converges in H 1 (Ω, S 2 ), d L 2 (Ω,S 2 ) to the functional G hom : H 1 (Ω, S 2 ) → R + defined by G hom (m) := E hom (m) + A hom (m) + µ 0 2 W hom (m) + µ 0 Z hom (m). ( 10
)
The four terms that appear in [START_REF] Simone | Energy minimizers for large ferromagnetic bodies[END_REF] have the following expressions: The homogenized exchange energy is given by

E hom (m) := Ω A hom ∇m(x) : ∇m(x) dx, (11)
where A hom is the classical homogenized tensor A hom given by the average

A hom := a ex (y)(I + ∇ϕ(y)) T (I + ∇ϕ(y)) Q , ϕ := (ϕ 1 , ϕ 2 , ϕ 3 ), (12)
where for every j ∈ N 3 the component ϕ j is the unique (up to a constant) solution of the following scalar unit cell problem

ϕ j := argmin ϕ∈H 1 # (Q) Q a ex (y)[|∇ϕ(y) + e j | 2 ] dy. ( 13
)
The homogenized anisotropy energy is given by

A hom (m) := Ω×Q ϕ an (y, m(x)) dy dx, (14)
while the homogenized magnetostatic self-energy is given by

W hom (m) := -M s 2 Q Ω h d [m] • m dτ + Ω×Q |∇ y v m (x, y)| 2 dx dy, (15)
where, for every x ∈ Ω, the scalar function v m : Ω × Q → R, is the unique solution of the cell problem:

m(x) • Q M s (y)∇ y ψ(y) dy = - Q ∇ y v m (x, y) • ∇ y ψ(y) dy, Q v m (x, y) dy = 0 ( 16
)
for all ψ ∈ H 1 # (Q). Finally, the homogenized interaction energy is given by

Z hom (m) = -M s Q Ω h a • m dτ. ( 17
)
The paper is organized as follows: In Section 2 we give a brief survey of the main mathematical concepts and results used throughout the paper. The equicoercivity of the family (G ε L ) ε∈R + is established in Section 3; the Γ-limit of the exchange energy family of functionals (E ε ) ε∈R + is computed in Section 4; in Section 5 it is shown that the family of magnetostatic self-energies (W ε ) ε∈R + continuously converges to W hom , while in Section 6 it is established the continuous convergence of the family of anisotropy energies (A ε ) ε∈R + to A hom and the continuous convergence of the family of interaction energies (Z ε ) ε∈R + to the functional Z hom . Eventually, the proof of mht (Theorem 1.1) is completed in Section 7, and some well-known results in homogenization theory, though somewhat difficult to find in the literature are given in the appendix.

Mathematical Preliminaries

The purpose of this section is to fix some notations and to give a survey of the concepts and results that are used throughout this work. All results are stated without proof as they can be readily found in the references given below.

Γ-convergence of a family of functionals

We start by recalling De Giorgi's notion of Γ-convergence and some of its basic properties (see [START_REF] De Giorgi | Su un tipo di convergenza variazionale[END_REF][START_REF] Maso | An introduction to Γ-convergence[END_REF]). Throughout this part we indicate with (X, d) a metric space and, for every m ∈ X, with C d (m) the subset of all sequences of elements of X which converge to m. Definition 2.1 (Γ-convergence of a family of functionals) Let (F n ) n∈N be a sequence of functionals defined on X with values on R. The functional F : X → R is said to be the Γ-lim of (F n ) n∈N with respect to the metric d, if for every m ∈ X we have:

∀(m n ) ∈ C d (m) F(m) lim inf n→∞ F n (m n ) ( 18
)
and

∃( mn ) ∈ C d (m) F(m) = lim n→∞ F n ( mn ). ( 19
)
In this case we write

F = Γ-lim n→∞ F n . If (F ε ) ε∈R + is a family of functionals, we say that F : X → R is the Γ-lim of (F ε ) ε∈R + as ε → 0, if for every ε n ↓ 0 one has F = Γ-lim n→∞ F εn . In this case we write F = Γ-lim ε→0 F ε .
The condition [START_REF] Landauer | Electrical conductivity in inhomogeneous media AIP Conference Proceedings[END_REF] is sometimes referred to in literature as the existence of a recovery sequence.

One of the most important properties of Γ-convergence, and the reason why this kind of variational convergence is so important in the asymptotic analysis of variational problems, is that under appropriate compactness hypotheses it implies the convergence of (almost) minimizers of a family of equicoercive functionals to the minimum of the Γ-limit functional. More precisely, the following result holds:

Theorem 2.2 (Fundamental Theorem of Γ-convergence) If (F ε ) ε∈R + is
a family of equicoercive functionals Γ-converging on X to the functional F. Then F is coercive and lower semicontinuous (therefore there exists a minimizer for F on X) and we have the convergence of minima values

min m∈X F(m) = lim ε→0 inf m∈X F ε (m). (20)
Moreover, given ε n ↓ 0 and (m n ) n∈N a converging sequence such that

lim n→∞ F εn (m n ) = lim n→∞ ( inf m∈X F εn (m)), (21)
its limit is a minimizer for F on X. If (21) holds, the sequence (m n ) n∈N is said to be a sequence of almostminimizers for F.

Let us recall now that given two families of functional (F ε ) ε∈R + and (G ε ) ε∈R + Γ-converging respectively to F and G, it is in general not the case (see [START_REF] Maso | An introduction to Γ-convergence[END_REF]) that Γ-lim ε→0 (F ε + G ε ) = F + G . A sufficient condition for that property to hold is that at least one of the two families of functionals satisfies a stronger type of convergence:

Definition 2.3 We say that a family of functionals (G ε ) ε∈R + is continuously convergent in X to a functional G : X → R, and we will write G ε Γcont ---→ G, if for every m 0 ∈ X lim (m,ε)→(m0,0) G ε (m) = G(m 0 ).
We then have (see [START_REF] Maso | An introduction to Γ-convergence[END_REF] for a proof):

Proposition 2.4 Let F = Γ-lim ε→0 F ε .
Suppose that the family of functionals (G ε ) ε∈R + continuously converges to G, and that G ε and G are everywhere finite on X.

Then G = Γ-lim ε→0 G ε and Γ-lim ε→0 (F ε + G ε ) = F + G. In particular if Z : X → R is a continuous functional then Γ-lim ε→0 (F ε + Z) = F + Z and Z is called a continuous perturbation of the Γ-limit.

Two-scale convergence

The aim of this section is to present in a schematic way the main properties of two-scale convergence, a notion that is first due to Nguetseng [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF], developed as a methodology by Allaire [START_REF] Allaire | Homogenization and two-scale convergence[END_REF] and further investigated by many others (see [START_REF] Allaire | Multiscale convergence and reiterated homogenisation[END_REF] and references therein for instance).

We denote by C ∞ # (Q) the set of infinitely differentiable real functions over R 3 that are Q-periodic and define

H 1 # (Q) as the closure of C ∞ # (Q) in H 1 loc (Ω).
Obviously any element of H 1 # (Q) has the same trace on the opposite faces of Q.

A generalized version of the Riemann-Lebesgue Lemma holds for the weak limit of rapidly oscillating functions. For a proof we refer the reader to [START_REF] Cioranescu | An introduction to homogenization[END_REF].

Proposition 2.5 Let Ω ⊂ R 3 be any open set. Let 1 p < ∞ and t > 0 be a positive real number. Let u ∈ L p (Q t ) be a Q t -periodic function. Set u ε (x) := u(x/ε) τ -a.e. on Ω. Then, if p < ∞, as ε → 0 u ε u Qt := 1 |Q t | Qt u dτ weakly in L p (Ω). If p = ∞, one has u ε u Qt := 1 |Q t | Qt u dτ weakly * in L ∞ (Ω).
Definition 2.6 Let Ω be an open set Ω ⊂ R 3 , and let (ε k ) k∈N be a fixed sequence of positive real numbers (when it is clear from the context we will omit the subscript k) converging to 0. The sequence of functions

(u ε ) ∈ L 2 (Ω) is said to two-scale converge to a limit u ∈ L 2 (Ω × Q), if for any function ϕ ∈ D[Ω; C ∞ # (Q)] we have lim ε→0 Ω u ε (x)ϕ(x, x/ε) dx = Ω×Q u(x, y)ϕ(x, y) dy dx (22)
In this case we write u ε u. We say that (u ε ) in L 2 (Ω) strongly two-scale converges to a limit u ∈

L 2 (Ω × Q) if u ε u and moreover u Ω×Q = lim ε→0 u ε Ω .
The importance of this new notion of convergence relies on the following compactness results.

Proposition 2.7 For each bounded sequence (u ε ) ∈ L 2 (Ω), there exists an u ∈ L 2 (Ω × Q) such that, up to a subsequence, u ε u.

Moreover, for bounded sequences in H 1 (Ω) we have the following result:

Proposition 2.8 Let (u ε ) be a sequence in H 1 (Ω) that converges weakly to a limit u ∈ H 1 (Ω). Then u ε u and there exists a function v ∈ L 2 [Ω; H 1 # (Q)/R] such that, up to a subsequence:

∇u ε ∇u + ∇ y v.
Next we recall that if the sequence (u ε ) is bounded in L 2 (Ω), it is possible to enlarge the class of test functions used in the definition of two-scale convergence.

Proposition 2.9 Let (u ε ) be a bounded sequence in L 2 (Ω) which two-scale converges to u ∈ L 2 (Ω × Q). Then lim ε→0 Ω u ε (x)ϕ(x, x/ε) dx = Ω×Q u(x, y)ϕ(x, y) dy dx for every ϕ ∈ L 2 [Ω; C # (Q)].
Finally we recall a simple criteria that permits to bypass the problem concerning the convergence of the product of two L 2 (Ω)-weakly convergence sequences (cfr. [START_REF] Allaire | Homogenization and two-scale convergence[END_REF][START_REF] Lukkassen | Two-scale convergence[END_REF]).

Proposition 2.10 Let (u ε ) and (v ε ) be sequences in L 2 (Ω) that respectively two-scale converge to u and v in L 2 (Ω × Q). If at least one of them strongly two-scale converges, then

u ε v ε uv.
In particular, if (u ε v ε ) is bounded in L 2 (Ω), from the previous proposition, we have

Ω u ε (x)v ε (x)ϕ(x, x/ε) dx = Ω×Q u(x, y)v(x, y)ϕ(x, y) dy dx for every ϕ ∈ L 2 [Ω; C # (Q)].
3 The equicoercivity of the composite Gibbs-Landau free energy functionals

This section is devoted to the proof of the equicoercivity of the family of Gibbs-Landau free energy functionals (G ε L ) ε∈R + expressed by [START_REF] Babadjian | Homogenization of variational problems in manifold valued Sobolev spaces[END_REF]. Equicoercivity has an important role in homogenization theory. In fact, the metric space in which to work, must be able to guarantee the equicoercivity of the family of functionals under consideration, i.e. the validity of the Fundamental Theorem of Γ-convergence.

Proposition 3.1 The family (G ε L ) ε∈R + of Gibbs-Landau energy functionals is equicoercive on the metric space H 1 Ω, S 2 , d L 2 (Ω,S 2 ) . Proof 3.2 According to the hypotheses [H 1 ], [H 2 ] and [H 3 ], there exist positive constants c ex , C ex , C s , C an ∈ R + such that for every y ∈ Q 0 < c ex a ex (y) C ex , 0 M s (y) C s , 0 ϕ(y, m) C an .
Therefore, for every ε > 0 on has

c ex ∇m 2 Ω G ε L (m) C (1 + ∇m 2 Ω ) (23)
with

C = max C ex , µ0 2 C 2 s |Ω|, C s |Ω| 1/2 h a Ω , C
an |Ω| . Now observe that for every constant in space magnetization u and for every ε > 0 on has G ε L (u) C ; thus inf

m∈H 1 (Ω,S 2 ) G ε L = inf m∈K G ε L . (24)
To finish, we simply observe that, due to Rellich-Kondrachov theorem, K is a compact subset of H 1 (Ω, S 2 ), d L 2 (Ω,S 2 ) .

4

The Γ-limit of exchange energy functionals E ε

The fundamental constraint of micromagnetic theory, i.e. the fact that the domain of definition of the family E ε is a manifold value Sobolev space, plays a fundamental role in the homogenization process. In fact, although the unconstrained problem has been fully investigated (see [START_REF] Braides | Homogenization of Multiple Integrals[END_REF][START_REF] Marcellini | Periodic solutions and homogenization of non linear variational problems[END_REF][START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF]), it is not possible to get full information about the manifold constrained Γ-limit by just looking at the unconstrained one. This is due to the well-known fact (see [START_REF] Maso | An introduction to Γ-convergence[END_REF]) that if (F ε ) is a family of functionals defined in the metric space X and Γ-converging to F, and

G = Γ-lim[G ε := F ε|Y ]
where (G ε ) := (F ε|Y ) represents the family of functionals obtained as the restriction of (F ε ) to some (metric) subspace Y of X, then F |Y G. Thus the identification of the manifold constrained Γ-limit requires more effort.

The tangential homogenization theorem

In what follows we make use of the following theorem due to Babadjian and Millot (see [START_REF] Babadjian | Homogenization of variational problems in manifold valued Sobolev spaces[END_REF]) in which the dependence of the Γ-limit from the tangent bundle of the manifold is taken into account via the so-called tangentially homogenized energy density. We state the tangential homogenization theorem (thm) in a bit less general form which is adequate for our purposes.

Proposition 4.1 (thm) [START_REF] Babadjian | Homogenization of variational problems in manifold valued Sobolev spaces[END_REF] Let M be a connected smooth submanifold of R 3 without boundary and g : R 3 × R 3×3 → R + be a Carathéodory function such that 1. For every ξ ∈ R 3×3 the function g(•, ξ) is Q-periodic, i.e. such that if (e 1 , e 2 , e 3 ) denotes the canonical basis of R 3 , one has ∀i ∈ N 3 , ∀y ∈ R 3 , ∀ξ ∈ R 3×3 g(y + e i , ξ) = g(y, ξ).

2. There exist 0 < α β < ∞ such that α|ξ| 2 g(y, ξ) β(1 + |ξ| 2 ) for a.e. y ∈ R 3 and all ξ ∈ R 3×3 .

Then the family

E ε (m) := Ω g(x/ε, ∇m)dτ (25)
defined in the metric space (H 1 (Ω, M), d L 2 (Ω,M) ) Γ-converges to the functional

E hom (m) := Ω Tg hom (m, ∇m)dτ, (26)
where for every s ∈ M and ξ

∈ [T s (M)] 3 , Q t := [0, t] 3 , Tg hom (s, ξ) = lim t→∞ inf ϕ∈W 1,∞ 0 (Qt,Ts(M)) 1 |Q t | Qt g(y, ξ + ∇ϕ(y))dy , (27)
is the tangentially homogenized energy density.

We refer the reader to [START_REF] Babadjian | Homogenization of variational problems in manifold valued Sobolev spaces[END_REF] for a more general version and the proof.

The role of tangent bundle.

Let us emphasize why the tangent bundle [T (M)] 3 := s∈M [T s (M)] 3 plays a role. In order to understand this, it is convenient to develop a minimizer m ε of E ε under the so-called multiscale expansion

m ε (x) = m 0 (x) + εm 1 x, x ε + o(ε), (28)
where m 0 , m 1 are respectively a minimizer of the Γ-limit of E ε and the null average first order corrector.

Clearly, due to the constraint m ε (x) ∈ M for a.e. x ∈ Ω, we get

0 ≡ n[m ε ] • ∇m ε , (29)
where we have denoted by n the local normal field defined around m ε (x) ∈ M. By passing to the two-scale limit in both terms of (29), we formally reach the equality 0 To this end, let us denote by n(m) the normal vector at m ∈ M and observe that it is sufficient to prove that the scalar function n(m(x)) • v(x, y) does not depend on the y variable, i.e. that in the language of distributions on Ω × Y one has

≡ n[m 0 ] • (∇m 0 + ∇ y m 1 ) ≡ n[m 0 ] • ∇ y m 1 , which shows that n[m 0 (x)] • m 1 (x,
Ω×Q [n(m(x)) • v(x, y)] div y ϕ(x, y)dydx = 0 ∀ϕ ∈ D[Ω; C ∞ # (Q)]. (30)
Indeed, as far as n(m(x)) • v(x, y) is independent from the y variable, since by assumption v(x,

•) Q = 0 for a.e. x ∈ Ω, one has n(m(x)) • v(x, y) = n(m(x)) • v(x, •) Q = 0 and therefore v(x, y) ∈ T m(x) (M) for a.e. (x, y) ∈ Ω × Q.
To prove [START_REF] Santugini-Repiquet | Homogenization of the demagnetization field operator in periodically perforated domains[END_REF] we note that since m ε → m in L 2 (Ω), one also has n(m ε ) → n(m) in L 2 (Ω). Therefore the family n(m ε ) strongly two-scale converges to n (m) and moreover 0

= (n(m ε ) • ∇, m ε ) Ω → (n(m) • ∇, m) Ω .
Hence, due to Proposition 2.10 we get

0 = lim ε→0 Ω [m ε (x) • ∇m ε (x)] • ϕ(x, x/ε)dx = Ω×Q m(x) • [∇m(x) + ∇ y v(x, y)] • ϕ(x, y) dydx = Ω×Q [m(x) • ∇ y v(x, y)] • ϕ(x, y) dydx = - Ω×Q [m(x) • v(x, y)] div y ϕ(x, y) dx,
i.e. the desired relation [START_REF] Santugini-Repiquet | Homogenization of the demagnetization field operator in periodically perforated domains[END_REF].

The role of tangentially homogenized energy density

Although the following considerations are not completely rigorous, they give a very good explanation of the idea behind the expression of the tangentially homogenized energy density [START_REF] Nguetseng | A general convergence result for a functional related to the theory of homogenization[END_REF]; they are mainly based on the notion of two-scale convergence while [START_REF] Babadjian | Homogenization of variational problems in manifold valued Sobolev spaces[END_REF] relies on Γ-convergence.

Let us denote by (E ε M ) the family of functionals

m ∈ (H 1 (Ω, M), d L 2 (Ω,M) ) → Ω a ex x ε |∇m(x)| 2 dx, (31)
and let us denote by E M its Γ-limit in the metric space (H

1 (Ω, M), d L 2 (Ω,M) ). Since H 1 (Ω, M) is a metric subspace of H 1 Ω, R 3
, from the properties of Γ-convergence under restriction to subspaces (see [START_REF] Maso | An introduction to Γ-convergence[END_REF]), we know that for every m 0 ∈ H 1 (Ω, M) and for every (

m ε ) ∈ C d L 2 (Ω,M) (m 0 ) E R 3 (m 0 ) E M (m 0 ) lim inf ε→0 E ε M (m ε ). ( 32
)
Now, let M be a convex smooth manifold, i.e. a smooth manifold lying on some subset of the boundary of a convex bounded domain Θ M . Next, define the tangent cone of M at s ∈ M by the position

T s (M) := v ∈ R 3 : ∀λ 0, s + λv / ∈ Θ M ,
and denote by Π M the nearest point projection on Θ M . Since Π M is a (Lipschitz) non-expansive map, one has Π M [u] ∈ H 1 (Ω, M) for every u ∈ H 1 (Ω, Θ c M ), moreover (see [START_REF] Alouges | A new algorithm for computing liquid crystal stable configurations: the harmonic mapping case[END_REF])

|∇Π M [u]| |∇u| τ -a.e. in Ω. ( 33
)
Let us now suppose that m 0 is sufficiently smooth so that for every test function

m 1 ∈ D[Ω; C ∞ # (Q, T m0 (M))], the family m ε (x) := m 0 (x) + εm 1 (x, x/ε) belongs to H 1 (Ω, Θ c M ). In this hypothesis one has Π M [m ε ] → m 0 in (H 1 (Ω, M), d L 2 (Ω,M) ).
Therefore, taking into account the estimates (33), [START_REF] Skomski | Giant energy product in nanostructured two-phase magnets[END_REF] and the fact that ∇ y m 1 is an admissible test function (see [START_REF] Allaire | Homogenization and two-scale convergence[END_REF], Remark 1.11), we get (passing to the two-scale limit):

E R 3 (m 0 ) E M (m 0 ) lim inf ε→0 E ε M (Π M [m ε ]) lim ε→0 Ω a ex x ε ∇m 0 (x) + ∇ y m 1 x, x ε 2 dx = Ω Q a ex (y)|∇m 0 (x) + ∇ y m 1 (x, y)| 2 dy dx. Since m 1 ∈ D[Ω; C ∞ # (Q, T m0 (M))
] is an arbitrary test function, passing to the infimum we finish with the following upper and lower bound for the manifold constrained homogenized functional:

E R 3 (m 0 ) E M (m 0 ) Ω inf ϕ∈C ∞ # (Q,Ts(M)) I[ξ, ϕ] (s,ξ):=(m0(x),∇m0(x)) dx, (34)
with

I[ξ, ϕ] := 1 |Q| Q a ex (y)|ξ + ∇ϕ(y)| 2 dy. ( 35
)
Since the functional

I[ξ, •] : C ∞ # (Q, T s (M)) → R + is continuous with respect to the H 1 # (Q, T s (M)) norm and C ∞ # (Q, T s (M)) is dense in H 1 # (Q, T s (M)
), the infimum in (34) can be taken over

H 1 # (Q, T s (M)).
Remark 4.4 Quite remarkably, as we prove below, when M := S 2 the infimum appearing in the right-hand side of (34) does not depend on the s-variable and coincides with the infimum in the left-hand side of (34). Thus for M := S 2 also the lower bound to E M is sharp and

E R 3 ≡ E S 2 .

The tangentially homogenized Exchange Energy E hom

Let us go back to the application of the tangentially homogenization result in our setting. We consider the family of exchange energy functionals, all defined in H 1 (Ω, S 2 ), given by (E ε ) ε∈R + . Since [H 1 ] holds, Proposition 4.1 ensures that the family (E ε ) ε∈R + Γ-converges in the metric space H 1 (Ω; S 2 ), d L 2 (Ω,S 2 ) , i.e. with respect to the topology induces on H 1 (Ω, S 2 ) by the strong L 2 (Ω, R 3 ) topology, to the functional

E hom : H 1 Ω, S 2 → R + , m → E hom (m) = Ω Tg hom (m, ∇m)dτ, (36)
where for every s ∈ S 2 and every ξ ∈ [T s (S 2 )] 3 ,

Tg hom (s, ξ) = lim t→∞ inf ϕ∈W 1,∞ 0 (Qt,Ts(S 2 )) I t [ξ, ϕ] (37)
with

I t [ξ, ϕ] := 1 |Q t | Qt a ex (y)|ξ + ∇ϕ(y)| 2 dy. Equivalently, since I t [ξ, •] : H 1 0 (Q t , T s (S 2 )) → R + is a continuous functional, and the subspace W 1,∞ 0 (Q t , T s (S 2 )) is dense in H 1 0 (Q t , T s (S 2
)), we have for every s ∈ S 2 and every ξ

∈ [T s (S 2 )] 3 inf ϕ∈W 1,∞ 0 (Qt,Ts(S 2 )) I t [ξ, ϕ] = inf ϕ∈H 1 0 (Qt,Ts(S 2 )) I t [ξ, ϕ], (38)
and hence

Tg hom (s, ξ) = lim t→∞ inf ϕ∈H 1 0 (Qt,Ts(S 2 )) I t [ξ, ϕ] . (39)
This latter formulation is a more convenient one. Indeed, it can be easily verified, by Lax-Milgram theorem, that for every t ∈ N, every s ∈ S 2 and every ξ ∈ [T s (S 2 )] 3 , there exists a unique solution ϕ t (s, ξ) ∈ H 1 0 (Q t , T s (S 2 )) of the problem ϕ t (s, ξ) := argmin ϕ∈H 1 0 (Qt,Ts(S 2 ))

I t [ξ, ϕ]. (40)
Moreover, for every t ∈ N and every s ∈ S 2 , the function

ξ ∈ [T s (S 2 )] 3 → ϕ t (s, ξ) ∈ W 1,∞ 0 (Q t , T s (S 2 )), (41)
is a (continuous) linear map.

Let us now consider the problem that defines the classical homogenization problem (see [START_REF] Braides | Homogenization of Multiple Integrals[END_REF][START_REF] Marcellini | Periodic solutions and homogenization of non linear variational problems[END_REF][START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF]), namely

g hom (ξ) := lim t→∞ inf ϕ∈H 1 0 (Qt,R 3 ) I t [ξ, ϕ] . ( 42
)
Our aim is now to prove that the natural extension of g hom to the tangent bundle [T (S 2 )] 3 := s∈S 2 [T s (S 2 )] 3 coincides with the tangentially homogenized energy density Tg hom (s, ξ). To this end we observe that in the classical problem (42), the function space among which the minimization takes place is bigger than the one involved in the original problem (39), so that g hom (ξ) Tg hom (s, ξ) for every (s, ξ) ∈ [T (S 2 )] 3 .

To prove that g hom (ξ) ≡ Tg hom (s, ξ) on [T (S 2 )] 3 it is thus sufficient to show that for every t ∈ N and for every (s, ξ) ∈ [T (S 2 )] 3 there exists a function ψ t,ξ ∈ H 1 0 (Q t , T s (S 2 )) such that I t [ξ, ψ t,ξ ] g hom (ξ). Having this goal in mind, we observe that if ϕ t,ξ is the unique solution of the minimization problem arising in (42), denoting by ψ t,ξ := ϕ t,ξ -(ϕ t,ξ • s)s the nearest point projection of ϕ t,ξ on [T s (S 2 )] 3 , one has

ψ t,ξ ∈ H 1 0 (Q t , T s (S 2
)) and

|ξ + ∇ϕ t,ξ | 2 = |ξ + ∇ψ t,ξ | 2 + |s ⊗ ∇(ϕ t,ξ • s)| 2 + 2(ξ + ∇ψ t,ξ ) : s ⊗ ∇(ϕ t,ξ • s) = |ξ + ∇ψ t,ξ | 2 + |∇(ϕ t,ξ • s)| 2 + 2(ξ + ∇ψ t,ξ )∇(ϕ t,ξ • s) • s = |ξ + ∇ψ t,ξ | 2 + |∇(ϕ t,ξ • s)| 2
where, in deriving the previous equalities, we have used the relations |a⊗b| 2 = |a| 2 |b| 2 , (A : B(a⊗b)) = Ab•Ba, and the orthogonality relations [∇ψ t,ξ ] T s = 0, ξ T s = 0. Thus we have

I t [ξ, ψ t,ξ ] = 1 |Q t | Qt a ex (y)|ξ + ∇ϕ t,ξ (y)| 2 dy - 1 |Q t | Qt a ex (y)|∇(ϕ t,ξ • s)(y)| 2 dy = I t [ξ, ϕ t,ξ ] - 1 |Q t | Qt a ex (y)|∇(ϕ t,ξ • s)(y)| 2 dy I t [ξ, ϕ t,ξ ],
and therefore, passing to the limit for t → ∞ we get g hom (ξ) ≡ Tg hom (s, ξ). In particular Tg hom (s, ξ) does not depend on s, and is given by (42).

Remark 4.5 The fact that tangential homogenization energy density Tg hom (s, ξ) reduces to the classical one (i.e. g hom ), which does not depend from the s-variable, is a quite remarkable fact. Indeed, it is possible to build elementary examples where the dependence on the s-variable in the tangential homogenization energy density is explicit (cfr. [START_REF] Babadjian | Homogenization of variational problems in manifold valued Sobolev spaces[END_REF]). The independence from the s-variable in our framework, is mainly due to the very particular situation that for every y ∈ R 3 , the Carathéodory function g(y, •) = a ex (y)| • | 2 is invariant under the rotation group of the manifold under consideration (the 2-sphere S 2 ).

To complete the proof concerning the exchange energy part stated in Theorem 1.1, it is sufficient to recall that g hom is a quadratic form in ξ (see [START_REF] Braides | Homogenization of Multiple Integrals[END_REF][START_REF] Marcellini | Periodic solutions and homogenization of non linear variational problems[END_REF]), i.e. that there exists a symmetric and positive definite matrix

A hom ∈ R 3×3 such that for every ξ ∈ R 3×3 g hom (ξ) = A hom ξ : ξ. ( 43
)
Precisely, it is possible to show that the classical problem (42) may be equivalently reformulated by replacing homogeneous boundary conditions by periodic boundary conditions (see [START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF] and Appendix A), so that for every (s, ξ)

∈ [T (S 2 )] 3 Tg hom (s, ξ) = g hom (ξ) = lim t→∞ inf ϕ∈H 1 # (Qt,R 3 ) I t [ξ, ϕ] . ( 44
)
Furthermore, from the convexity of the integrand, it is routinely seen that we can replace the limit for t → ∞ by the computation on the unit cell (though still with periodic boundary conditions). We are therefore left (see [START_REF] Braides | Homogenization of Multiple Integrals[END_REF] and Appendix A) for every (s, ξ) ∈ [T (S 2 )] 3 with

Tg hom (s, ξ) = g hom (ξ) = inf ϕ∈H 1 # (Q,R 3 ) Q a ex (y)|ξ + ∇ϕ(y)| 2 dy. ( 45
)
Now, Lax-Milgram theorem shows that for every ξ ∈ [T s (S 2 )] 3 there exists a unique solution ϕ ξ of this latter problem, up to an additive constant that we may fix by requiring that ϕ ξ (y)

Q = 0. Moreover, the map ξ ∈ [T s (S 2 )] 3 → ϕ ξ ∈ H 1 # (Q, R 3
) is a (continuous) linear map, and a direct computation, shows that A hom can be expressed as:

A hom := a ex (y)(I + ∇ϕ(y)) T (I + ∇ϕ(y)) Q , ϕ := (ϕ 1 , ϕ 2 , ϕ 3 ) (46)
where for every j ∈ N 3 the component ϕ j is the unique (up to a constant) solution of the scalar unit cell problem

ϕ j := argmin ϕ∈W 1,∞ # (Q,R 3 ) Q a ex (y)[|∇ϕ(y) + e j | 2 ]dx. ( 47
)
Remark 4.6 By repeating the same argument followed above to deduce the equality Tg hom (s, ξ) = g hom (ξ), it is now simple to show that the tangentially homogenized exchange energy density can be equivalently characterized by the position

Tg hom (s, ξ) = inf ϕ∈H 1 # (Q,Ts(S 2 )) Q a ex (y)|ξ + ∇ϕ(y)| 2 dy,
which is exactly the one deduced in equation (34).

The periodic homogenization of the demagnetizing field

This Section is devoted to show that the family of magnetostatic self-energies (W ε ) ε∈R + continuously converges to W hom . To this end, let us first recall some essential facts concerning the demagnetizing field operator.

The Beppo-Levi space and the variational formulation for the demagnetizing field

From the mathematical point of view, assuming Ω to be open, bounded and with a Lipschitz boundary, a given magnetization m ∈ L 2 (Ω, R 3 ) generates the stray field h d [m] = ∇u m where the potential u m solves:

∆u m = -div(mχ Ω ) in D R 3 . (48)
In (48) we have denoted by mχ Ω the extension of m to R 3 that vanishes outside Ω.

Once introduced the weight ω(x) = (1 + |x| 2 ) -1/2 , and the weighted Lebesgue space

L 2 ω (Ω) = u ∈ D (Ω) : uω ∈ L 2 (Ω) ,
we define the Beppo-Levi space

BL 1 (Ω) = u ∈ D (Ω) : u ∈ L 2 ω (Ω) and ∇u ∈ L 2 Ω, R 3 , (49)
which is an Hilbert space when endowed with the scalar product (u, v) BL 1 (R 3 ) := (∇u, ∇v) Ω . It is straightforward to show, by the means of Lax-Milgram theorem, the existence and uniqueness of the solution of the variational formulation associated to equation (48): namely to find a potential

u m ∈ BL 1 (R 3 ) such that for all ϕ ∈ BL 1 (R 3 ) (u m , ϕ) BL 1 (R 3 ) := R 3 ∇u m • ∇ϕdτ = - R 3 m • ∇ϕdτ =: F m [ϕ]. (50)
Thus, for every m ∈ L 2 (R 3 ) there exists a unique u m ∈ BL 1 (R 3 ) such that (50) holds, and moreover, the following stability estimate holds:

u m BL 1 (R 3 ) ≡ ∇u m 2 Ω sup ϕ∈BL 1 (R 3 ) ∇ϕ Ω =1 |F m [ϕ]| m L 2 (R 3 ) . (51)
The quantity h d [m] := ∇u m is what is referred to as the demagnetizing field, and it is a linear and continuous operator from

L 2 (R 3 , R 3 ) into L 2 (R 3 , R 3
). In particular mχ Ω ∈ L 2 (R 3 ) for every m ∈ L 2 (Ω) and therefore h d is also a linear and continuous operator from

L 2 (Ω, R 3 ) into L 2 (R 3 , R 3 ).
Exactly with the same diagonal argument, it is possible to prove the weighted variant of the compactness result stated in Proposition 2.8.

Proposition 5.3 Let (u ε ) be bounded sequence in BL 1 (R 3 ) weakly convergent to u ∞ . Then u ε L 2 ω -two-scale converges to u ∞ ∈ L 2 ω (R 3 ) and there exists a function v ∈ L 2 [R 3 ; H 1 # (Q)/R] such that, up to the extraction of a subsequence: ∇u ε ∇u ∞ + ∇ y v. (62)
Proof 5.4 We start by observing that since

u ε(n) u ∞ weakly in BL 1 (R 3 ), u ε(n) u ∞ in L 2 ω (R 3 )
, and therefore, according to the previous proposition, there exists a function u

∈ L 2 ω (R 3 × Q) such that, up to a subsequence, u ε(n) u(x, y) in L 2 ω (R 3 × Q) u ε(n) u ∞ (x) ≡ u(x, y) Q in L 2 ω (R 3 ). (63)
We now consider a sequence of bounded domain (Ω i ) i∈N . Proceeding as in the proof of the previous Proposition 5.1, one proves that for every i ∈ N there exists a subsequence u ε(n k i ) such that

n ki ⊆ n ki-1 and u ε(n k i ) u ∞ (x) ≡ u(x, y) Q ≡ u(x, y) in L 2 (Ω i × Q). (64)
We then define the diagonal sequence of indices defined by the position n k∞(i) = n ki(i) . From (64) we get that for every i ∈ N, up to the first i -1 terms, the sequence of indices n k∞ is included in n ki , and this means that for every i

∈ N u ε(n k∞ ) u ∞ (x) ≡ u(x, y) Q ≡ u(x, y) in L 2 (Ω i × Q). ( 65
) Thus u ≡ u ∞ ∈ L 2 ω (R 3 ) in R 3 . Next, we observe that since u ε(n) u ∞ weakly in BL 1 (R 3 ) we have ∇u ε(n) ∇u ∞ and (∇u ε(n) ) bounded in L 2 (R 3
). Thus, according to the classical two-scale compactness result (see Proposition 2.8) there exists a function

κ ∞ ∈ L 2 (R 3 × Q) such that, up to a subsequence, ∇u ε κ ∞ in L 2 (R 3 , R 3 ). (66)
Thus, for any test function

[ϕ ⊗ ψ # ](x, y) := ϕ(x)ψ # (y) ∈ D[R 3 ; C ∞ # (Q)] with div y ψ # (y) = 0 one has R 3 u ε (x) div x [ϕ ⊗ ψ # ](x, x/ε)dx = - R 3 ∇u ε (x) • [ϕ ⊗ ψ # ](x, x/ε)dx. (67)
Passing to the two-scale convergence on both sides we get that for a.e. x ∈ R 3 and for every

ψ # ∈ C ∞ # (Q) such that div y ψ # (y) = 0 in Q. Q [κ ∞ (x, y) -∇u ∞ (x)] • ψ # (y)dy = 0. ( 68
)
Since the orthogonal complement of the divergence-free functions is the space of gradients, for a.e. x ∈ R 3 there exists a unique function v(x,

•) ∈ H 1 # (Q)/R such that ∇ y v(x, y) ≡ κ ∞ (x, y) -∇u ∞ (x). Thus ∇ y v ∈ L 2 (R 3 × Q) and v(x, •) ∈ H 1 # (Q)/R, i.e. v ∈ L 2 [R 3 , H 1 # (Q)/R].
This completes the proof. We are now ready to prove the two-scale convergence of the demagnetizing field operator (for this we follow the lines of [START_REF] Santugini-Repiquet | Homogenization of the demagnetization field operator in periodically perforated domains[END_REF]): Proposition 5.5 Let (m ε ) ε∈R + be a bounded family in L 2 (R 3 , R 3 ) that two-scale converges to m(x, y). Then the two-scale limit of (h d [m ε ]) ε∈R + ∈ L 2 (R 3 , R 3 ) exists and is given by

h d (x, y) = h d [ m(x, •) Q ] + ∇ y v m (x, y) (69)
where for every x ∈ R 3 the scalar function v m (x, •) is the unique solution in

H 1 # (Q) of the cell problem ∆ y v m (x, y) = -div y m(x, y) in H 1 # (Q)/R (70)
and therefore of the variational cell problem

Q m(x, y) • ∇ y ψ(y)dy = - Q ∇ y v m (x, y) • ∇ y ψ(y)dy, Q v m (x, y)dy = 0 (71) for all ψ ∈ H 1 # (Q).
Proof 5.6 Since (m ε ) is bounded in L 2 (R 3 ), due to the stability estimate (51), the sequence of magnetostatic potentials (u ε m ) solution of the problem ∆u ε m = -div(m ε ), remains bounded in BL 1 (R 3 ). This means that, up to a subsequence, (u ε m ) u m weakly in BL 1 (R 3 ) for some suitable u m ∈ BL 1 (R 3 ). Thus, according to Proposition 5.3, there exist functions

u m ∈ BL 1 (R 3 ) and v m ∈ L 2 [R 3 ; H 1 # (Q)/R] such that (u ε m ) u m in L 2 ω and ∇u ε m (x) ∇u m (x) + ∇ y v m (x, y). (72)
In view of the previous limit relations, u ε m is expected to behave as u m (x) + εv m (x, x/ε). This suggest to use, in the variational formulation of the magnetostatic problem expressed by equation (50), test functions having the form ϕ(x) + εψ(x, x/ε), with ϕ ∈ D(R 3 ) and

ψ ∈ D[R 3 ; C ∞ # (Q)]. This yields R 3 ∇u ε m • ∇ϕ + ε∇ψ x, x ε + ∇ y ψ x, x ε dx = - R 3 m ε • ∇ϕ + ε∇ψ x, x ε + ∇ y ψ x, x ε dx.
From the second of the two limit relations (72), we get

R 3 ×Q (∇u m (x) + ∇ y v m (x, y)) • (∇ϕ + ∇ y ψ(x, y))dydx = - R 3 ×Q m(x, y) • (∇ϕ + ∇ y ψ(x, y))dydx. (73)
In particular, by choosing ψ ≡ 0 we get

- R 3 m(x, y) Q • ∇ϕ(x)dx = R 3 ×Q (∇u m (x) + ∇ y v m (x, y)) • ∇ϕdydx = R 3 ∇u m (x) • ∇ϕ(x), (74)
where the last equality follows from the fact that ∇ y v m (x, y) Q = 0. Thus, we reach the conclusion that the weak limit u m satisfies the variational formulation (74), i.e. is a solution of the homogenized equation

u m (x) = -div m(x, y) Q in BL 1 (R 3 ). ( 75
)
On the other hand by choosing ϕ ≡ 0 and ψ(x, y) = ψ 1 (x)ψ 2 (y) in (73) we get

R 3 (∇u m (x) + ∇ y v m (x, y)) • ∇ y ψ 2 (y) Q ψ 1 (x)dx = - R 3 m(x, y) • ∇ y ψ 2 (y) Q ψ 1 (x)dx ,
and hence the so-called cell problem

- Q m(x, y) • ∇ y ψ 2 (y)dy = Q (∇u m (x) + ∇ y v m (x, y)) • ∇ y ψ 2 (y)dy = Q ∇ y v m (x, y) • ∇ y ψ 2 (y)dy , (76)
where, again, the last equality follows from the fact that ∇ y ψ 2 (y) Q = 0. Note that the variational formulation (76) can be more concisely expressed in the form

∆ y v m (x, y) = -div y m(x, y) in H 1 # (Q)/R, (77)
and the well-posedness of the previous variational problem is again a direct consequence of Lax-Milgram theorem.

The continuous limit of magnetostatic self-energy functionals W ε

In what follows we will make use of Proposition 5.5, to prove the following Proposition 5.7 The family of magnetostatic self-energies

W ε : m ∈ L 2 (Ω, S 2 ) → -(h d [M ε m], M ε m) Ω (78)
continuously converges to the functional

W hom : m ∈ L 2 (Ω, S 2 ) → -M s 2 Q (h d [m], m) Ω + ∇ y v m 2 Ω×Q (79)
where for every x ∈ Ω the scalar function v m : Ω × Q → R is the unique solution of the following variational cell problem:

m(x) • Q M s (y)∇ y ψ(y)dy = - Q ∇ y v m (x, y) • ∇ y ψ(y)dy, (80)
Q v m (x, y)dy = 0, (81)
for all ψ ∈ H 1 # (Q).

Proof 5. [START_REF] Brown | The fundamental theorem of the theory of fine ferromagnetic particles[END_REF] We know (see Proposition 2.5) that

|M ε m| 2 ≡ |M ε | 2 |M ε | 2 Q weakly * in L ∞ (Ω).
In particular, by choosing |m| 2 ∈ L 1 (Ω) as a test function we get

M ε 2 Ω = (|M ε m| 2 , |m| 2 ) Ω → ( |M ε | 2 Q , |m| 2 ) Ω = |Ω| |M ε | 2 Q = M s (y)m(x) 2 Ω×Q and therefore M ε (x)m(x) M s (y)m(x) strongly. Next, since h d [M ε m] • M ε m is bounded in L 2 (Ω)
, from Proposition 2.10 and Proposition 5.5, we get

lim ε→0 -(h d [M ε m], M ε m) Ω = -M s 2 Q (h d [m], m) Ω - Ω×Q ∇ y v m (x, y) • M s (y)m(x)dxdy. (82)
-(h d [M ε m], M ε m) Ω = -M s 2 Q (h d [m], m) Ω + ∇ y v m 2 Ω×Q =: W hom (m). ( 84
)
Now we show that the family W ε continuously converges to W hom . This amounts to prove that for every m ∈ L 2 (Ω, S 2 ) and every ε ∈ R + , ε < ε 0 and mm 0 Ω < δ implies |W ε (m) -W hom (m 0 )| < η. To this end, for every m, m 0 ∈ L 2 Ω, S 2 we split:

|W ε (m) -W hom (m 0 )| |W ε (m) -W hom (m)| + |W hom (m) -W hom (m 0 )|. (85)
From (84) it follows the existence of a sufficiently small ε 0 such that

∀ε < ε 0 |W ε (m) -W hom (m)| < η 2 . ( 86
)
On the other hand, since m → ∇ y v m is a linear and continuous map from

L 2 Ω, R 3 into L 2 Ω × Q, R 3 with ∇ y v m Ω×Q c M m Ω (and c M := M s Q ) one has |W hom (m) -W hom (m 0 )| M s 2 Q |(h d [m], m) Ω -(h d [m 0 ], m 0 ) Ω | +| ∇ y v m 2 Ω×Q -∇ y v m0 2 Ω×Q | M s 2 Q |(h d [m + m 0 ], m -m 0 ) Ω | +|(∇ y v m+m0 , ∇ y v m-m0 ) Ω×Q | 2|Ω| 1/2 ( M s 2 Q + c 2 M ) m -m 0 Ω . (87)
and the previous estimate together with (86) clearly concludes the proof.

The homogenized anisotropy and interaction energies

This section is devoted to the proof of the continuous convergence of the family of anisotropy energy functionals A ε and of the family of interaction energy functionals Z ε , respectively to A hom and Z hom , whose expression is given by ( 14) and ( 17).

6.1 The continuous limit of the anisotropy energy functionals A ε Proposition 6.1 If the anisotropy energy density ϕ an : R 3 × S 2 → R + is Q-periodic with respect to the first variable and globally lipschitz with respect to the second one (uniformly with respect to the first variable) then the family A ε of anisotropy energies continuously converges to the homogenized anisotropy energy

A hom : m ∈ L 2 Ω, S 2 → Ω×Q ϕ an (y, m(x))dydx. ( 88
)
Proof 6.2 Again, we have to prove that for every m 0 ∈ L 2 (Ω, S 2 ) one has lim

(ε,m)→(0,m0) A ε (m) = A hom (m 0 ). For every m, m 0 ∈ L 2 Ω, S 2 we split Ω×Q ϕ(x/ε, m(x)) -ϕ(y, m 0 (x))dydx = Ω×Q ϕ(x/ε, m(x)) -ϕ(y, m(x)) Q dydx + Ω×Q ϕ(y, m(x)) -ϕ(y, m 0 (x))dydx. (89)
Next, we observe that since ϕ(x/ε, m(x)) ϕ(y, m(x)) Q weakly * in L ∞ (Ω) (cfr. Lemma 2.5), there exists a sufficiently small ε 0 such that for every ε < ε 0

Ω×Q ϕ(x/ε, m(x)) -ϕ(y, m(x))dydx < η 2 . ( 90
)
On the other hand, by the global lipschitz continuity in the second variable of ϕ an , we have

Ω×Q |ϕ(y, m(x)) -ϕ(y, m 0 (x))|dydx c L Ω |m(x) -m 0 (x)|dx (91) c L |Ω| 1/2 m -m 0 Ω . (92)
Substituting estimates (90) and ( 92) into (89) we get

|A ε (m) -A hom (m 0 )| η 2 + c m -m 0 Ω with c := c L |Ω| 1/2 . (93)
Therefore for every m ∈ L 2 (Ω, S 2 ) such that mm 0 Ω < η/(2c ) we get |A ε (m) -A hom (m 0 )| η, and this concludes the proof.

Corollary 6.3 (Uniaxial anisotropy energy density). If ϕ(y, m) = κ(y)|m(x) ∧ u(y)| 2 then A hom (m) = Ω κ Q -κu ⊗ u Q : m ⊗ mdτ.

The continuous limit of interaction energy functionals Z ε

The convergence of (Z ε ) ε∈R + to Z ε is straightforward. Indeed this energy term is expressed by the product, with respect to the L 2 (Ω) scalar product, of the constant function h a and the weakly converging sequence (M ε m) ε∈R + M s Q weakly * in L ∞ (Ω) (cfr. Lemma 2.5). Therefore repeating the same argument given in the previous subsection:

Z ε Γcont ---→ ε→0 Z hom with Z hom (m) := -M s Q Ω h a • mdτ.

Proof of Theorem 1.1 completed

It is now easy to complete the proof of Theorem 1.1. Indeed the equicoercivity of the family of Gibbs-Landau free energy functionals (G ε L ) ε∈R + expressed by [START_REF] Babadjian | Homogenization of variational problems in manifold valued Sobolev spaces[END_REF] has been proved in Section 3. It is therefore sufficient to recall the stability properties of the Γ-limit under the sum of a continuously convergent family of functionals. In fact, what has been proved in the previous subsections, can be summarized by the following convergence scheme

E ε Γ -→ ε→0 E hom , W ε Γcont ---→ ε→0 W hom , A ε Γcont ---→ ε→0 A hom , Z ε Γcont ---→ ε→0 Z hom . (94)
Thus, Proposition 2.4 completes the proof.

Conclusions and acknowledgment

We have given in this paper a complete theory for periodic microstructured magnetic materials. Obtained through a process of Γ-convergence the model derives rigorously the energy terms from the parameters of each constituent of the sample and the mixing geometry of the different materials in the unit periodic cell. We believe that the result applies to most of magnetic composites that are nowadays considered, e.g. those obtained from a mixing of hard and soft phases [START_REF] Hadjipanayis | The incredible pull of nanocomposite magnets[END_REF][START_REF] Skomski | Giant energy product in nanostructured two-phase magnets[END_REF] or the multilayer magnetic materials [START_REF] Hu | Remanence characteristic of nanostructure of Hard/Soft magnetic multilayered systems[END_REF][START_REF] Renard | Magnetic multilayers[END_REF]. In this latter case, the formula obtained furthermore simplifies since the exchange coefficient can be analytically computed. We leave the exploration of potential applications to a forthcoming work.

A Restatement of some well-known result

In this appendix we state, in a bit general form, two results that we mentioned in the previous sections. We start with a result whose proof can be extracted from [START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF], which permits to pass, in the characterization of the homogenized energy density, from homogeneous boundary conditions to periodic ones. The proof, although well-known in the homogenization community, is difficult to find in the classical monographs on the subject:

Proposition A.1 Let V be a vector subspace of R 3 and let g : R 3 × R 3×3 → R + be a Carathéodory function satisfying the hypotheses of (thm) with M := V (see Proposition 4.1). Then, for every (s, ξ) ∈ V × V 3 one has Tg hom (s, ξ) = T # g hom (ξ) with Proof A.2 Since T s (V) ≡ V it is obvious that Tg hom does not depend on the s-variable. Moreover, since H 1 0 (Q t , V) ⊆ H 1 # (Q t , V) one trivially has T # g hom (ξ) Tg hom (s, ξ). Let us therefore focus on the converse inequality. For any ϕ ∈ H 1 0 (Q t , V) and any ξ ∈ V 3 we consider the family (m ε (x) := ξx+εϕ(x/ε)) ∈ H 1 (Ω, V) which converges to ξx in the metric space (H 1 (Ω, V), d L 2 (Ω,V) ). By the characterizing properties of the Γ-limit we have:

Tg hom (s, ξ) = 1 |Ω| E hom (ξx) 1 |Ω| lim inf ε→0 F ε (m ε ) = lim ε→0 1 |Ω| Ω g x ε , ξ + ∇ϕ x ε dx. ( 96
)
Since the integrand in the last member of the previous equation is Q t -periodic and satisfies the hypotheses of the generalized Riemann-Lebesgue Lemma (cfr. Proposition 2.5) we finish with the estimate Tg hom (s, ξ)

1 |Q t | Qt g(y, ξ + ∇ϕ(y))dy ∀ξ ∈ V 3 , ∀ϕ ∈ H 1 0 (Q t , V). ( 97
)
Passing to the infimum we get the stated result.

Finally, in the hope to achieve our goal of making the material contained in this paper accessible to as large audience as possible, we prefer to state -again in a bit general form -and give a more commented proof of the following well-known result (cfr. [START_REF] Braides | Homogenization of Multiple Integrals[END_REF]) which permits, in the convex case, to pass from variations that are periodic over an ensemble of t 3 1-cells to variations that are periodic in the unit cell Q.

Proposition A.3 Let g : R 3 × R 3×3 → R + be a Carathéodory integrand satisfying the hypotheses of (thm) (Proposition 4.1). If for any y ∈ R 3 the function g(y, •) is convex, then for any convex subset C of R 3 and every t ∈ N the following equality holds: Proof A.4 One of the main ingredients of the argument, is in observing that for every t ∈ N the following decomposition of the t-cube in 1-cubes holds:

Q t = (i1,i2,i3)∈N 3 t Q (i1,i2,i3) 1
with Q (i1,i2,i3) 1

:= Q 1 + e (i1,i2,i3) (99)
and e (i1,i2,i3) := (i 1 -1)e 1 + (i 2 -1)e 2 + (i 3 -1)e 3 . Thus, from the 1-periodicity of g in the first variable, we arrive to the intuitive and well-known fundamental equality:

Qt g(y, ξ + ∇ϕ(y))dy = (i1,i2,i3)∈N g(y, ξ + ∇ϕ(y))dy.

To complete the proof we now prove that the previous inequality is actually an equality, and therefore (in particular) that the left hand side of (98) does not depend on t ∈ N. To this end it is sufficient to show that for every t ∈ N and every ψ ∈ H

  [m] generated by m. From the mathematical point of view, assuming Ω to be open, bounded and with a Lipschitz boundary, a given magnetization m ∈ L 2 Ω, R 3 generates the stray field h d [m] = ∇u m where the potential u m solves:

  y) does not depend on y. Then, passing to the average over Q we get m 1 (x, y) ∈ T m0(x) (M). The rigorous formulation of the previous idea is the object of the next Proposition: Proposition 4.2 Let M be a connected smooth submanifold of R 3 , and let (m ε ) be a sequence in H 1 (Ω, M) that converges weakly to a limit m ∈ H 1 Ω, R 3 . Then m ∈ H 1 (Ω, M) and m ε m. Moreover there exists a null average function v ∈ L 2 [Ω; H 1 # (Q)/R] such that, up to the extraction of a a subsequence: ∇m ε ∇m + ∇ y v and v(x, y) ∈ T m0(x) (M) for a.e. (x, y) ∈ Ω × Q. Proof 4.3 In view of Proposition 2.8, we only need to prove that v(x, y) ∈ T m0(x) (M) for a.e. (x, y) ∈ Ω×Q.

  Now, we observe that for every x ∈ Ω the scalar function v m (x, •) is the unique solution of the variational cell problem (71), therefore setting ψ(•) := v m (x, •) in (71) we get m(x) • Q M s (y)∇ y v m (x, y)dy = Q |∇ y v m (x, y)| 2 dy (83)and therefore lim ε→0

T # g hom (ξ) := lim t→∞ inf ϕ∈H 1 #

 1 (Qt,V)1 |Q t | Qt g(y, ξ + ∇ϕ(y))dy .(95)

inf ϕ∈H 1 #

 1 (Qt,C) 1 |Q t | Qt g(y, ξ + ∇ϕ(y))dy = inf ϕ∈H 1 # (Q,C) Q g(y, ξ + ∇ϕ(y))dy.(98)

  Next, we observe thatH 1 # (Q, C) ⊆ H 1 # (Q t , C) for every t ∈ N, and therefore, from the previous equality we get:

				3 t	Q	1 (i 1 ,i 2 ,i 3 )	g(y, ξ + ∇ϕ(y))dy
			=			g(y, ξ + ∇ϕ(y))dy
			t (i1,i2,i3)∈N 3	Q1
			= |Q t |	g(y, ξ + ∇ϕ(y))dy.	(100)
			Q1
	inf ϕ∈H 1 # (Qt,C)	1 |Q t | Qt	g(y, ξ + ∇ϕ(y))dy	inf ϕ∈H 1 # (Q,C)	1 |Q t | Qt	g(y, ξ + ∇ϕ(y))dy
			=	inf ϕ∈H 1 # (Q,C) Q

  Moreover, by the convexity and 1-periodicity of g with respect to the first variable, from equality (100), we get: Qt g(y + e (i1,i2,i3) , ξ + ∇ψ(y + e (i1,i2,i3) ))dy

			ϕ(y) :=	1 |Q t |	t (i1,i2,i3)∈N 3	ψ(y + e (i1,i2,i3) ).	(102)
	Clearly ϕ belongs to H 1
	Q	g(y, ξ + ∇ϕ(y))dy =	1 |Q t | Qt	g(y, ξ + ∇ϕ(y))dy
			1 |Q t |	t (i1,i2,i3)∈N 3	1 |Q t | Qt	g(y, ξ + ∇ψ(y + e (i1,i2,i3) ))dy
		1 |Q t | |Q t | = 1 = (i1,i2,i3)∈N 3 t 1 |Q t | t (i1,i2,i3)∈N 3 1 |Q t | Qt	g(y, ξ + ∇ψ(y))dy
		=	1 |Q t | Qt	g(y, ξ + ∇ψ(y))dy,
	and this completes the proof.

1 # (Q t , C) there exists a ϕ ∈ H 1 # (Q, C) such that Q g(y, ξ + ∇ϕ(y))dy 1 |Q t | Qt g(y, ξ + ∇ψ(y))dy.

(101)

With this goal in mind, for every ψ ∈ H 1 # (Q t , C), we define the convex combination # (Q, C).

The two-scale limit of the demagnetizing field

In this subsection we make use of the notion of two-scale convergence, to characterize the behavior of the demagnetizing field operator under two-scale convergence. More precisely, we suppose to have a bounded sequence (m ε ) ∈ L 2 (R 3 ) which two scale converges to some m ∞ (x, y) ∈ L 2 and we want to understand if the two-scale limit of the sequence h d [m ε ] exists, and in the affirmative case to characterize in some analytic sense such a limit. This problem has already been treated in [START_REF] Santugini-Repiquet | Homogenization of the demagnetization field operator in periodically perforated domains[END_REF] although without justifying the use of two-scale compactness results in weighted space. That is why in this subsection we start by proving two compactness results concerning two-scale convergence in the weighted spaces L 2 ω R 3 and BL 1 (R 3 ). The first one is a weighted variant of the compactness result stated in Proposition 2.7, and shows that a notion of two-scale convergence in L 2 ω (R 3 ) makes sense.

) and, up to the extraction of a subsequence,

In this case we say that the sequence (u ε ) L 2 ω -two-scale converges to u.

This implies that for every bounded domain Ω ⊆ R 3 , one has u ε(n) u ∞ weakly in L 2 (Ω). We now consider a sequence of bounded domain (Ω i ) i∈N covering R 3 . Let us start with the index i = 1, i.e. with Ω 1 . According to the two-scale compactness result (see Proposition 2.7), there exists a subsequence u ε(n k 1 ) and an element

Now we consider i = 2, i.e. Ω 2 . Since

Moreover, due to the unicity of the two-scale limit, one has

whenever Ω 1 ∩ Ω 2 = ∅. Proceeding in this way, we find for every i ∈ N a subsequence u ε(n k i ) such that

for some

We then define the diagonal sequence of indices defined by n k∞(1) := n k1(1) , n k∞(2) := n k2(2) , . . . , n k∞(i) = n ki(i) , . . . .

From (56) we get that for every i ∈ N, up to the first i -1 terms, the sequence of indices n k∞ is included in n ki , and this means that for every i ∈ N

By observing again that u i|(Ωi∩Ωj )×Q ≡ u j|(Ωi∩Ωj )×Q if Ω i ∩ Ω j = ∅, from the principe du recollement des morceaux (cfr. [START_REF] Schwartz | Théorie des distributions[END_REF]) there exists a unique distribution u ∈ L 2 loc (R 3 × Q) such that u |Ωi×Q ≡ u i , and therefore lim k∞→∞ R 3 u ε(n k∞ ) (x)ϕ(x, x/ε)dx = we get also u(x, y) Q ≡ u ∞ ∈ L 2 ω (R 3 ). This completes the proof.