Stochastic homogenization of the Landau-Lifshitz-Gilbert equation

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Stochastic homogenization of the Landau-Lifshitz-Gilbert equation

François Alouges, Anne de Bouard, Benoît Merlet, Léa Nicolas

To cite this version:

François Alouges, Anne de Bouard, Benoît Merlet, Léa Nicolas. Stochastic homogenization of the Landau-Lifshitz-Gilbert equation. Stochastics and Partial Differential Equations: Analysis and Com- putations, Springer US, 2021, Stochastic Partial Differential Equations. Analysis and Computations, 9 (4), pp.789-818. �10.1007/s40072-020-00185-4�. �hal-02020241�

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Stochastic homogenization of the Landau-Lifshitz-Gilbert equation

François Alouges^∗, Anne de Bouard^†, Benoît Merlet^‡, Léa Nicolas^§

Abstract

Following the ideas of V. V. Zhikov and A. L. Piatnitski [19], and more precisely the stochastic two-scale convergence, this paper establishes a homogenization theorem in a stochastic setting for two nonlinear equations : the equation of harmonic maps into the sphere and the Landau-Lifschitz equation. These equations have strong nonlinear features, in particular, in general their solutions are not unique.

1 Introduction

Magnetic materials are nowadays at the heart of numerous electric devices (en- gines, air conditioning, transportation, etc.). Very often, the efficiency of the device is directly linked to the magnetic properties of the magnets used [8].

Best magnets are the rare-earth magnets (e.g. Samarium-Cobalt or Neodymium magnets) which have been developed since the 1980’s with no equivalent yet.

Nevertheless, due to the uneven distribution of rare-earth ores [10], the magnet manufacturers aim at developing new types of magnets that do not require rare- earth. Among the best promising candidates are the so-called spring magnets which are made of hard and soft magnets intimately mixed at the nanoscale [12].

Mathematically speaking, the problem of studying such materials is challeng- ing because usual models are highly non-linear and the material dependence of the parameters varies at a very small scale. It is therefore a problem of homogenization. A complete static model, including all relevant classical physical terms, has been derived in [2] in terms ofΓ−convergence of the minimization problems related to the underlying so-called Brown energy of such materials. Neverthe- less, to the knowledge of the authors, there is no such study to characterize the dynamic problem, or, in other words, the evolution of the magnetization inside the composite ferromagnet. This is the first purpose of this paper, while the

∗CMAP, Ecole polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France,francois.alouges@polytechnique.edu

†CMAP, Ecole polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France,anne.debouard@polytechnique.edu

‡LPP, CNRS UMR 8524, Université de Lille, F-59655 Villeneuve d’Ascq Cedex, France and Team RAPSODI, Inria Lille - Nord Europe, 40 av. Halley, F-59650 Villeneuve d’Ascq, France,benoit.merlet@univ-lille.fr

§CMAP, Ecole polytechnique, CNRS, Institut Polytechnique de Paris, 91128 Palaiseau Cedex, France,lea.nicolas@polytechnique.edu

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second is to consider that the different compounds are randomly distributed at the microscopic scale inside the macroscopic composite, whereas a periodic distribution was considered in [2].

To go more precisely into the details, the magnetization inside a homogeneous ferromagnetic materials obeys the Landau-Lifschitz equation (see for instance [14]), a non-linear PDE, which, when only the exchange interactions between magnetic spins are taken into account reads as

∂u

∂t =u×div (a∇u)−λu×(u×div (a∇u)). (1) Here,u(t, x) is a unit vector in R³ that denotes the magnetization at time t ≥ 0 and at x ∈ D, where D is the bounded domain of R³ occupied by the material, and the symbol × stands for the cross product in R³. The so- calledexchange parameter ais a3×3matrix, which depends on the considered material(s). We also assume the different materials to be strongly coupled which amounts to say that the direction of the magnetization u does not jump at the interface between two materials.

Spring magnets being composed of several different materials, the coefficient a is likely to depend on the space variable. Assuming furthermore that the materials are randomly distributed, and on a small scale, we are led to consider the Landau-Lifschitz equation with random coefficients











∂u

∂t =u×div(a(x

, ω)∇u)−λu×

u×(div(a(x

, ω)∇u) , u(x,0) =u0(x),

|u(x, t)|= 1 for a.e. (x, t)∈QT , a(x

, ω)∇u·n= 0on ∂D ×(0, T)

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with T > 0, QT = D ×(0, T), and u0 ∈ H¹(D,R³) satisfying |u₀(x)| = 1a.e.

in D. The exchange parameter a depends on x at scale and on the random parameter ω.

The problem we wish to solve is therefore the stochastic homogenization of the Landau-Lifschitz equation, or in other words, passing to the limit in (2) as goes to 0. Notice that the problem possesses several difficulties that make it not obvious: global solutions of (1) are only known to exist weakly and are not unique [3, 18], the constraint |u(x, t)| = 1 is not convex and the equation is highly non linear.

In order to proceed, we apply the stochastic two-scale convergence method.

Originally defined in a periodic deterministic framework for the first time by G. Nguetseng [15], the theory was further developed by G. Allaire [1] and is by now currently used. Stochastic homogenization of PDEs dates back to [16] for linear equations and [9] for nonlinear problems. Furthermore, a stochastic generalization of two-scale convergence (in the mean) has been first proposed in [6], but turns out to be inadequate for our purposes. Instead, we use in this paper a theory developed in [19]. This latter version allows us to realize the proposed program: we prove that, up to extraction, u converges (weakly inH¹(Q_T)) to

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a (weak) solution of (1) where the effective exchange matrix a = a^eff is fully identified.

The paper is organized as follows. In Section 2, we recall the probability setting and introduce the stochastic two-scale convergence. For the sake of completeness, we present a complete theory, simpler than the one given in [19], but sufficient for our purpose. Most of the arguments are nevertheless borrowed from [19]. Section 3 is devoted to the applications. The classical diffusion equation is first quickly treated and we turn to the stochastic homogenization of the equation of harmonic maps into the sphere. Eventually, the Landau- Lifschitz equation is considered.

2 Stochastic two-scale convergence

A stochastic generalization of the two-scale convergence method, was proposed in [19] in a very general context. In order to have the present paper self- contained, we restrict the approach of [19] to a framework that is sufficient for our needs and we present a complete setting. In our opinion, this method did not have the resonance that it deserves and we hope that the reader will find here a comprehensive introduction to it.

2.1 The stochastic framework

Let(Ω,F,P)be a standard probability space withF a completeσ-algebra and let us consider a d-dimensional random field a(x, ω) ∈ R^d×d, defined for x in R^dand ω inΩ. We consider a group action of R^don the set Ω, forx∈R^d, we noteT_x the action on Ωand we call it the translation of vector x. We assume thataand T satisfy the following classical assumptions:

(H1) Compatibility: the mapping(x, ω)7→T_xωis measurable fromR^d×Ωinto Ω. Moreover, for everyx inR^d andω inΩ,a(·, T_xω) =a(x+·, ω).

(H2) Stationarity: for everyx inR^d, for every kinN, for every BorelianB of (R^d×d)^k, for every y1, . . . , yk inR^d,

P{(a(y₁, Txω), . . . , a(yk, Txω))∈ B}=P{(a(y₁, ω), . . . , a(yk, ω))∈ B}. (H3) Ergodicity: the only measurable sets that are translation invariant (that

is,A∈ F such that (up to a null subset)T_xA=Afor everyx∈R^d) have null or full measure.

(H4) The matrixais symmetric, uniformly bounded and uniformly elliptic:

∃c₁, c₂ >0; ∀ω∈Ω, ∀ξ∈R^d, ∀x∈R^d, c₁|ξ|²≤ξ·a(x, ω)ξ≤c₂|ξ|². (H5) The matrixais stochastically continuous: for every xinR^d,

∀ >0, lim

y→xP(ka(y)−a(x)k_Rd×d ≥) = 0.

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As we shall see, the assumption (H5) allows us to restrict ourselves to the case where Ω is a compact metric space, which is the assumption used in [19]. In fact, all the results given in this paper are applicable if (H5) is replaced by assumptions (H5’-a)–(H5’-c) below.

(H5’-a) Ωis a compact metric space andF is the completion of its Borelσ-algebra.

(H5’-b) The mappingω 7→a(0, ω) is continuous onΩ.

(H5’-c) The group action of R^d on (Ω,F,P) defined by a(·, T_xω) = a(x+·, ω) defines a continuous action ofR^donL¹(Ω). Namely, for everyΦ∈L¹(Ω), Φ◦T_x also belongs to L¹(Ω) and moreover the mapping x 7→ Φ◦T_x is continuous from R^d intoL¹(Ω).

Example 2.1.1. A class of examples satisfying assumptions (H1)–(H5), introduced in [4], can be constructed using a Poisson point process. Let us recall that a Poisson point process on a measurable space(E,E)with intensity measure λ, is a random subsetΠ ofE such that the following properties hold:

• for every measurable setA of E, the number of points inΠ∩A, denoted by N(A), follows a Poisson law with meanλ(A),

• for every pairwise disjoint measurable sets A1, . . . , A_k of E, the random variables N(A₁), . . . , N(A_k) are independent.

We now consider the case where Π is a Poisson point process on R^d with intensity measure λ being the Lebesgue measure, defined on the Borel sets of R^d, anda0,a1 are two matrices in the set

{ea∈R^d×d_sym :∀ξ ∈R^d, c1|ξ|² ≤ξ·eaξ≤c2|ξ|²},

with c₁, c₂ > 0. We define a random field a by setting, for every x ∈R^d, and everyω∈Ω,

a(x, ω) =







a0 if dist(x,Π(ω))≤ 1 2, a₁ otherwise.

Let us choose0< <ka₀−a1k_Rd×d. Then lim sup

y→x P(ka(y,·)−a(x,·)k_Rd×d ≥) =P

dist(x,Π) = 1 2

= 0,

and (H4) is verified. Hypotheses (H1), (H2) and (H3) are obviously satisfied.

In the next three propositions, we prove that (H1)–(H5) imply (H5’) in the case whereais real valued for which the demonstration is simpler. The general case follows from direct modifications that we leave to the reader. (H5’-a) is established in Proposition 2.1.2, (H5’b) in Proposition 2.1.3 and (H5’-c) in Proposition 2.1.5.

Let us start with a technical lemma.

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Figure 1: A sample of the coefficient field defined by the homogeneous Poisson point cloud. The matrix a is equal to a₀ in the black region and to a₁ in the white region [4].

Lemma 2.1.1. Let (Ω,A,P) be a standard probability space with Aa complete σ-algebra and b(x, ω), x in R^d, ω in Ω, be a real bounded random field. Let N be a dense countable subset of R^d and F ⊂ A be the σ-algebra generated by {b(y,·), y∈ N }.

Assume that for everyx in R^d, and for every positive,

y→xlimP(|b(y,·)−b(x,·)| ≥) = 0. Thenb is measurable with respect to F.

Proof. For simplicity, we only show that the set A ={ω ∈ Ω : b(0, ω) > 0} is an element of F. Let (xn)n∈N be a sequence of elements of N that converges to0. By assumption, for every >0,

n→∞lim P(|b(x_n)−b(0)|> ) = 0.

By a diagonal argument and up to extraction, we can assume that for everyn inN,

P |b(x_n)−b(0)|>2⁻ⁿ

≤2⁻ⁿ.

Let us note E = lim sup_n{ω∈Ω :|b(x_n, ω)−b(0, ω)|>2⁻ⁿ}. Borel-Cantelli lemma impliesP(E) = 0. Now, forω∈Ω\E, we haveb(0, ω) = lim infnb(xn, ω), so that

A\E =n

ω∈Ω\E: lim inf

n→∞ b(x_n, ω)>0o

= \

k>0

ω∈Ω : lim inf

n→∞ b(x_n, ω)≥ 1 k

\E . Hence,A is an element of F.

We now build a compact metric space K and a random process b(·, f) indexed byf ∈K with the same law asa(·, ω).

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LetN be a dense countable subset of R^d containing0. Letc₁, c₂ in R be such that for every x inR^d andω inΩ,c1 ≤a(x, ω)≤c2 and let us set

K={{f(x)}_x∈N :∀x∈ N, f(x)∈[c1, c2]}= [c1, c2]^N.

Let {α_x}_x∈N be a summable family of positive numbers, we define a distance on K by

d(f, g) = X

x∈N

α_x|f(x)−g(x)|. (3) According to Tikhonov theorem, K equipped with the distance dis a compact metric space.

Now let us define a probability measure on the Borelians ofK by µ(B) =P{ω∈Ω : (a(x, ω))_x∈N ∈B},

which is well defined according to Lemma 2.1.1. Completing the Borelians with respect toµ, we obtain aσ-algebraE. The space(K,E, µ)is the canonical space for {a(x), x∈ N }.

Finally, let us defineb, for everyf inK, as follows: for everyx inN, b(x, f) =f(x).

Thanks to Lemma 2.1.1, it follows thatb(x)is almost surely uniquely defined for anyxinR^d. We easily check thatbhas the same law asa. We have established (H5’a), namely:

Proposition 2.1.2. The real-valued random field b(x), x ∈R^d over the probability space (K,E, µ) has the same law as a. Moreover (K, d) is a compact metric space and E is the completion of its Borel sets with respect to µ.

We check that (H5’-b) is satisfied.

Proposition 2.1.3. Under the assumptions (H1)–(H5) and with the notation introduced in the previous proposition,b(0,·)is continuous onKfor the distance d defined by (3).

Proof. Letf ∈K. Since0∈ N, we have for every feinK, α0

b(0, f)−b(0,f)e ≤ X

x∈N

αx

b(x, f)−b(x,fe)

=d(f,f)e . Asα0>0, it follows that b(0,·) is continuous in f.

In order to prove (H5’-c) in Proposition 2.1.5 we first establish the following.

Lemma 2.1.4. The spaceCf(K) of continuous functions onK= [c1, c2]^N that only depend on a finite number of variables is dense inC(K). Similarly,C_f(K) is dense in L^p(K, µ) for any 1< p <∞.

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Proof. Let us enumerate N = {x₁, x₂,· · · }. Let Φ ∈ C(K) and N > 0. For f ∈ K, we note fN = ΠNf the element of K defined as fN(xj) = f(xj) for j = 1,· · ·, N and f_N(x_j) = c₁ for j > N. The mapping Π_N is continuous from K into K and only depends on the first N variables. Next, we define ΦN := Φ◦ΠN and by compositionΦN belongs toCf(K). Letmbe the modulus of continuity of Φ. We have

kΦ_N −Φk∞= sup{|Φ(f)−Φ(ΠNf)|:f ∈K}

≤ sup (

m X

x∈N

α_x|f(x)−Π_Nf(x)|

!

: f ∈K )

≤m



|c₂−c₁| X

x∈N \{x₁,···,x_N}

α_x





N↑∞−→ 0.

We conclude thatCf(K) is dense into C(K). Eventually the density of Cf(K) inL^p(K) follows from that of C(K) inL^p(K)for p >1.

Let us define the dynamical system{T_x:K→K}_x∈Rd by b(y, T_xf) =b(y+x, f),

for every f ∈ K and x, y ∈ R^d. Let us notice that for every x in R^d, T_x is uniquely defined thanks to Lemma 2.1.1.

By definition, for everyx, yinR^d,Tx+y =Tx◦Ty andT0 = IdK. Moreover, the stationarity assumption (H2) implies for every eventAof E, for every xinR^d,

µ(T_x⁻¹(A)) =µ(A). (4)

Eventually, (H3) gives thatT is ergodic: if an event AofE isT-invariant, then µ(A) = 0 or µ(A) = 1.

Let us now check thatT complies to (H5’-c).

Proposition 2.1.5. Let Φ∈L¹(K), then Φ◦Tx belongs to L¹(K) for x∈R^d and

y→xlim Z

K

|Φ(T_yf)−Φ(Txf)|dµ(f) = 0, for every x∈R^d, andΦ∈L¹(K). (5) Remark 2.1.1. With the same arguments (see the proof below), we may also prove that for anyΦ∈L^p(K), the mappingx7→Φ◦T_x is continuous from R^d intoL^p(K).

Proof. LetΦ∈L¹(K). By measurability of T_x:K →K, we see that Φ◦T_x is measurable, moreover by the stationarity property (4), we have

µ{Φ◦T_x∈B}=µ{Φ∈B}

for every x ∈ R^d and every Borel set B ⊂ R. In particular, Φ◦Tx ∈ L¹(K) withkΦ◦T_xk_L1(K)=kΦk_L1(K).

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Let us now check the continuity of x ∈ R^d 7→ Φ◦T_x. By density of C(K) in L¹(K), we can assume that Φ is continuous (hence bounded) in K and in fact by Lemma 2.1.4, we can assume that Φonly depends on a finite number of variables. Eventually, by stationarity we only have to check the continuity at x = 0. So, let us assume that for every f in K, Φ(f) = ϕ(f(x1),· · ·, f(xN)) withϕ∈C([c1, c2]^N) andx1,· · · , x_N ∈ N. Let η >0, for y∈R^d we denote

K1(y) ={f ∈K :|(f(x_j+y)−f(xj))j=1,···,N|< η}, K2(y) =K\K1(y).

Decomposing the integration onK over K1 and K2, we have Z

K

|Φ(T_yf)−Φ(f)|dµ(f)≤m(η) + 2kϕk_∞µ(K₂(y)),

wheremis the modulus of continuity ofϕ. Now, by assumption(H5),µ(K2(y)) tends to 0 as y tends to 0 and since η >0 is arbitray, we see that the integral in the left hand side also goes to 0.

From now on, we assume that (Ω,F, µ) is the canonical space associated with a, and that it verifies (H1)-(H5’). As for every x in R^d and ω in Ω, a(x, ω) =a(0, Txω), for simplicity we will denote a(x, ω) =a(Txω). By (H5’), it holds that ais in the space of continuous functions defined on Ω, which will be denoted C(Ω).

Remark 2.1.2. In Section 2.2 below, we introduce the notion of two-scale convergence and use test functions in L²(D ×Ω) where D is a bounded domain of R^d. In the theory developed thereafter, we use the continuous embeding of C_c(D ×Ω) into L²(D ×Ω), which is a consequence of the finiteness of the measureλ⊗µinD ×Ω. The following fact is more crucial: sinceΩis compact, the Banach space (Cc(D ×Ω),k · k_∞) is separable, so there exists a countable subset Γ⊂C_c(D ×Ω) which is dense in(C_c(D ×Ω),k · k_∞) and in L²(D ×Ω).

We end this section by recalling the Birkhov ergodic theorem, introduced in [5], that plays a prominent role in the analysis. We first check that the quantities involved are well defined.

Lemma 2.1.6. Let p≥1andu be a function ofL^p(Ω). Then,µ-almost surely, x7→u(Txω) is inL^p_loc(R^d).

Proof. For every bounded Borel setA ofR^d, Z

Ω

Z

A

|u(T_xω)|^pdxdµ(ω) = Z

A

Z

Ω

|u(T_xω)|^pdµ(ω)dx

= |A|

Z

Ω

|u(ω)|^pdµ(ω) < ∞,

according to Fubini’s theorem and thanks to the stationarity of a. Therefore, R

A|u(T_xω)|^pdx is bounded µ-almost surely.

Theorem 2.1.7. [Birkhov ergodic theorem]. Let f be a function of L¹(Ω).

Then, for µ-almost every ωe in Ω and for every bounded Borel set A of R^d, 1

t^d|A|

Z

tA

f(T_xω)e dx−−−→

t→∞

Z

Ω

f(ω)dµ(ω) =E(f). (6)

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2.2 L² Two-scale convergence

Let us start by noticing that the Birkhov ergodic theorem presented above is not sufficient to obtain results valid almost surely for all functionsf, and thus, cannot be sufficient to obtain an homogenization theorem with almost sure convergence of the solution. One of the reasons is the fact that the set ofeω for which the convergence hold depends onf. Therefore, we introduce the following definition.

Definition 2.2.1. Letωe∈Ω. We say thatωe is a typical trajectory, if,

t→∞lim 1 t^d|A|

Z

tA

g(T_xeω)dx= Z

Ω

g(ω)dµ(ω) =E(g),

for every bounded BorelianA⊂R^d with|A|>0and every g inC(Ω).

Proposition 2.2.1. Let Ωe be the set of typical trajectories. Then µ(Ω) = 1.e Proof. We first notice that the compactness of Ω entails that C(Ω) (endowed with the normkgk_∞= sup_Ω|g|) is separable. We thus consider

Γ ={g_k, k∈N}

a dense countable subset of C(Ω). According to Birkhov ergodic theorem, for everykinN, there existsΩ_kinF such thatµ(Ω_k) = 1, andΩ_k⊂Ωk−1 ⊂ · · · ⊂ Ω₀ and, for everyω⁰∈Ω_k,

t→∞lim 1 t^d|A|

Z

tA

g_k(T_xω⁰)dx= Z

Ω

g_k(ω)dµ(ω) =E(g_k),

for every bounded Borel set A with |A| > 0. Considering Ω⁰ = ∩_k∈NΩ_k, we haveµ(Ω⁰) = 1, and it only remains to show thatΩ⁰ ⊂Ω.e

Let ω⁰ ∈ Ω⁰, g ∈ C(Ω), > 0 and g⁰ ∈ Γ such that kg⁰ −gk∞ < . For every bounded Borel setA⊂R^d andt >0,

1 t^d|A|

Z

tA

g(Txω⁰)dx−E(g)

≤

1 t^d|A|

Z

tA

[g−g⁰](Txω⁰)dx

+

1 t^d|A|

Z

tA

g⁰(Txω⁰)dx−E(g⁰)

+

E(g⁰−g)

≤2kg⁰−gk_∞+

1 t^d|A|

Z

tA

g⁰(T_xω⁰)dx−E(g⁰)

≤3,

fort large enough sinceg⁰ ∈Γ andω⁰ ∈Ω⁰. Therefore,

t→∞lim 1 t^d|A|

Z

tA

g(T_xω⁰)dx= Z

Ω

g(ω)dµ(ω) =E(g),

which givesΩ⁰ ⊂Ω. We deduce thate µ(Ω)e ≥µ(Ω⁰) = 1.

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Proposition 2.2.2. [Mean-value property.] Letg be a function inC(Ω). Then, for every ϕ in Cc(R^d), for everyωe in Ω,e

lim↓0

Z

R^d

ϕ(x)g(T_x/ω)dxe = Z

R^d

ϕ(x)dx Z

Ω

g(ω)dµ(ω) =E(g) Z

R^d

ϕ(x)dx .

Proof. We use Proposition 2.2.1 to get the result for a simple function ϕ = PN

i=1a_i1_A_i where(A_i)1≤i≤N are bounded Borel sets ofR^d, and1_A_i denotes the characteristic function of A_i. The conclusion comes from the approximation of any function ϕ∈Cc(R^d) by simple functions in theL¹(R^d) norm.

We now state the two-scale convergence definition and prove the main compactness theorem.

Definition 2.2.2. Let ωe ∈ Ωe be fixed, let {v}∈I be a family of elements of L²(D) indexed by in a set I ⊂ (0,+∞) with 0 ∈ I and let v ∈ L²(D ×Ω).

Let (_k)k≥0 ⊂ I be a decreasing sequence converging to 0, we say that the subsequence(v^k) weakly two-scale converges to vif, for every ϕinC_c^∞(D) and everyb inC(Ω),

k→∞lim Z

D

v^k(x)ϕ(x)b(T_x/_kω)e dx= Z

D

Z

Ω

v(x, ω)ϕ(x)b(ω)dµ(ω)dx .

We write

v^k ∈L²(D)* v² ∈L²(D ×Ω).

It is worth noticing that this definition of two-scale convergence, and thus the limit, depends on the choice ofωe inΩ. From now on, we assume thate ωe ∈Ωe is fixed.

The main result of this subsection is the following theorem. It is the stochastic equivalent of the two-scale compactness theorem provided in [1] for periodic homogenization.

Theorem 2.2.3. Let {v}_∈I be a bounded family in L²(D), with I as in the above definition. Then, there exist a sequence (_k)k≥0 in I^N that tends to zero, and v⁰ in L²(D ×Ω) such that (v^k)k weakly two-scale converges to v⁰.

Proof. Let K = {ϕ b, ϕ ∈ C_c^∞(D), b ∈ C(Ω)} be the set of test functions and hKi be its linear span, i.e. hKi is the tensor product C_c^∞(D)⊗C(Ω). Using the Cauchy-Schwarz inequality and the boundedness of {v}inL²(D), it holds that, for every >0 and everyΦ inhKi,

Z

D

v(x)Φ(x, T_x/ω)dxe

≤C Z

D

Φ²(x, T_x/ω)dxe 1/2

.

Decomposing Φ as Φ = PN

p=1Φp with Φ1,· · · ,Φ_N ∈ K, and using Proposi- tion 2.2.2, it is easily checked that

→0lim Z

D

Φ²(x, T_x/ω)dxe = Z

D×Ω

Φ²(x, ω)dxdµ(ω).

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Therefore,

lim sup

→0

Z

D

v(x)Φ(x, T_x/ω)dxe

≤CkΦk_L2(D×Ω). (7)

In particular, for every Φ ∈ hKi, the family {R

Dv(x)Φ(x, T_x/eω)dx}_>0 is bounded in R. Recalling Remark 2.1.2, we pick a countable subset Γ ⊂ hKi which is both dense inC(D ×Ω)and in L²(D ×Ω). Using a diagonal process, there exists a sequence(_k)_k that tends to zero, such that for every ΨinΓ,

k→∞lim Z

D

v^k(x)Ψ x, T_x/_keω

dx=`(Ψ), (8)

for some real number`(Ψ). By linearity, this relation extends to the linear span hΓi of Γ, the function ` is a linear form on hΓi and by (7), for every Ψ∈ hΓi, there holds

`(Ψ)≤CkΨk_L2(D×Ω). (9)

By density ofΓ inL²(D ×Ω), we see that` uniquely extends as a continuous linear map` :L²(D ×Ω)→ R satisfying (9). Now, let Φ ∈K and let η >0, there existsΨ∈Γsuch thatkΨ−Φk_∞< η. Using (7), we compute

Z

D

v^k(x)Φ x, T_x/_kωe

dx−`(Φ)

≤ Z

D

v^k(x)(Φ−Ψ) x, T_x/_kωe dx

+

Z

D

v^k(x)Ψ x, T_x/_kωe

dx−`(Ψ)

+|`(Ψ−Φ)|

≤Cp

|D| kΦ−Ψk_∞+ Z

D

v^k(x)Ψ x, T_x/_kωe

dx−`(Ψ)

+CkΨ−Φk_L2

≤2Cp

|D|η+ Z

D

v^k(x)Ψ x, T_x/_keω

dx−`(Ψ) .

Since Ψ∈Γ, the last term tends to 0 as k tends to infinity and since η >0 is arbitrary, we obtain that (8) holds for everyΨ∈K.

Eventually, since`is a continuous linear form on the Hilbert space L²(D ×Ω), by Riesz representation theorem, there exists v⁰ in L²(D ×Ω) such that for everyΦinL²(D ×Ω),

`(Φ) = Z

D×Ω

v⁰(x, ω)Φ(x, ω)dxdµ(ω).

This entails, in particular that for everyϕinC_c^∞(D) and binC(Ω),

k→∞lim Z

D

v^k(x)ϕ(x)b(T_x/_kω)dxe =`(ϕ b) = Z

D×Ω

v⁰(x, ω)ϕ(x)b(ω)dxdµ(ω).

Two-scale convergence is mostly a weak notion. A corresponding strong two- scale convergence exists that allows us to use weak-strong convergence properties as stated in the following.

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Definition 2.2.3. The sequence(v^k)k≥0 of L²(D) is said tostrongly two-scale converge to a function v⁰ inL²(D ×Ω) asgoes to 0 if (v^k) weakly two-scale converges to v⁰ and if

limk→0kv^kk_L2(D)=kv⁰k_L2(D×Ω).

Remark 2.2.1. An important example of strong convergence is the following.

If (v^k) converges towards some function v¯ strongly in L²(D), then, by definition, (v^k) weakly two-scale converges towards v⁰ defined as v⁰(x, ω) := ¯v(x).

Moreover,

kv⁰k_L2(D×Ω) =k¯vk_L2(D)= lim

k↓0kv^kk_L2(D), and (v^k) strongly two-scale converges towardsv⁰.

Strong two-scale convergence allows us to have a two-scale version of the weak-strong convergence principle.

Proposition 2.2.4. Let {v}_>0 and {u}_>0 be two families of functions of L²(D). If {v}_>0 strongly two-scale converges to v⁰ and {u}_>0 weakly two- scale converges to u⁰, for everyϕ in C_c^∞(D) andb in C(Ω),

limk→0

Z

D

u^k(x)v^k(x)ϕ(x)b(T_x/_kω)e dx= Z

D

Z

Ω

u⁰(x, ω)v⁰(x, ω)ϕ(x)b(ω)dµ(ω)dx . Proof. Let {u}_>0, {v}_>0, u₀, v₀ and (_k) satisfying the assumptions of the proposition. For simplicity, we drop the subscript k in the proof below. Let δ >0, there exists Φ∈C_c^∞(D)⊗C(Ω)such that

Φ−v⁰

2

L²(D×Ω) ≤δ².

Sincev * v² ⁰ with strong convergence, and using Proposition 2.2.2, there exists _δ such that for 0< < _δ,

Z

D

Φ(x, T_x/eω)²dx− kΦk²_L2(D×Ω)

≤δ²,

Z

D

v(x)Φ(x, T_x/ω)dxe − Z

D

Z

Ω

v⁰(x, ω)Φ(x, ω)dx dµ(ω)

≤δ²,

Z

D

(v(x))² dx− Z

D×Ω

v⁰(x, ω)2

dx dµ(ω)

≤δ². Combining the preceding estimates leads to

Z

D

v(x)−Φ(x, T_x/ω)e 2

dx≤5δ². (10)

Now, letϕ∈C_c^∞(D) andb∈C(Ω), we write for >0, Z

D

u(x)v(x)ϕ(x)b(T_x/eω)dx= Z

D

u(x)Φ(x, T_x/ω)ϕ(x)b(Te _x/ω)e dx +

Z

D

u(x) v(x)−Φ(x, T_x/ω)e

ϕ(x)b(T_x/ω)e dx .

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Sinceu * u² ⁰, there exists ⁰_δ ∈(0, δ]such that for every 0< < ⁰_δ,

Z

D

u(x)Φ(x, T_x/ω)ϕ(x)b(Te _x/eω)dx− Z

D×Ω

u⁰Φϕ b dx dµ

≤δ . We obtain, for0< < ⁰_δ,

Z

D

u(x)v(x)ϕ(x)b(T_x/ω)e dx− Z

D×Ω

u⁰v⁰ϕ b dx dµ

≤δ+ Z

D×Ω

u⁰ϕb(Φ−v⁰)dxdµ +

Z

D

ϕ(x)b(T_x/ω)e dx

≤δ 1 +ku⁰ϕbk_L2(D×Ω)

+ Z

D

ϕ(x)b(T_x/ω)e dx .

To estimate the last term, we use the Cauchy-Schwarz inequality to get,

Z

D

ϕ(x)b(T_x/ω)dxe

≤Ckuk_L2(D)

Z

D

v(x)−Φ(x, T_x/ω)e 2

dx 1/2

≤C⁰δ, where we have used the boundedness of(u) inL²(D) and (10) to get the last inequality. Consequently, for every0< < ⁰_δ,

Z

D

u(x)v(x)ϕ(x)b(T_x/eω)dx− Z

D×Ω

u⁰v⁰ϕ b dx dµ

≤C⁰⁰δ . This proves the proposition.

The key ingredient in the applications of two-scale convergence to homogenization problems, as introduced for the first time in [1], is the use of a two-scale compactness theorem onH¹(D). This compactness is used to pass to the two- scale limit in the integral formulation of the equations for both the solution and its gradient. In order to extend such property to our setting, we first need to define a spaceH¹(Ω).

2.3 Construction of H¹(Ω)

Definition 2.3.1. Letu be a function ofL²(Ω). We say thatuis differentiable atω inΩif, for everyiin{1, . . . , d}, the limit

δ→0lim

u(T_δe_iω)−u(ω)

δ =: (Diu)(ω) (11)

exists, where(e1, ..., e_d)denotes the canonical basis of R^d. In this case, we note D_ωu= (D₁u,· · · , D_du).

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Lemma 2.3.1. Letu be a function ofL²(Ω), differentiable at every point of Ω.

Then, for every x inR^d andi in {1, . . . , d},

∂

∂x_i[u(T_xω)] = (D_iu)(T_xω). Proof. For everyx inR^d andi in{1, . . . , d},

∂

∂x_i[u(T_xω)] = lim

δ→0

u(T_x+δe_iω)−u(T_xω) δ

= lim

δ→0

u(T_δe_i(T_xω))−u(T_xω)

δ = (D_iu)(T_xω).

Definition 2.3.2. LetC¹(Ω)be the set of functionsuofΩthat are continuous and differentiable at every ω in Ω and such that for every i in {1, . . . , d}, the function D_iuis continuous on Ω.

Lemma 2.3.2. C¹(Ω)is dense in L²(Ω).

Proof. SinceC(Ω)is dense inL²(Ω)(see [17, Theorem 3.14]), it suffices to show thatC¹(Ω)is dense in C(Ω)for the L²-norm.

Let ρ ∈ C_c^∞(R^d,R₊) such that R

R^dρ = 1. We define a standard family of approximations of unity {ρ^δ}_δ>0 as ρ^δ(x) = (1/δ)^dρ(x/δ) for x ∈ R^d, δ > 0.

Now, letϕbe a function inC(Ω), we consider its mollifications{ϕ^δ}defined for δ >0 andω∈Ωas,

ϕ^δ(ω) :=

Z

R^d

ρ^δ(y)ϕ(T_(−y)ω)dy .

We easily check thatϕ^δ∈C¹(Ω)with Diϕ^δ(ω) =

Z

R^d

∂xiρ^δ(y)ϕ(T(−y)ω)dy .

Using assumption (H5’-c) (see also Remark 2.1.1) it is also standard to check that we have

kϕ^δ−ϕk_L2(Ω)

−→δ↓0 0.

This proves the lemma.

Lemma 2.3.3. Let w in C¹(Ω), then E(D_iw) = 0 for i in {1,· · ·, d}. In particular, for u,v in C¹(Ω), there holds

Z

Ω

Diu v dµ+ Z

Ω

u Div dµ = 0. (12)

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Proof. Letw inC¹(Ω)and let χ inC_c^∞(R^d) such that R

χ dx= 1, we have by Proposition 2.2.2,

E(Diw) = lim

↓0

Z

R^d

χ(x)Diw(T_x/ω)dx.e

On the other hand, for > 0, we have, using Lemma 2.3.1 and integrating by parts,

Z

R^d

χ(x)Diw(T_x/ω)e dx= Z

R^d

χ(x) ∂

∂xi

w(T_x/ω)e dx

=−

Z

R^d

∂χ

∂x_i(x)w(T_x/eω)dx.

By Proposition 2.2.2 again, the last expression tends to 0 as goes to0 which proves the first part of the lemma.

Now if u, v are two functions of C¹(Ω), we easily see that w := uv is also in C¹(Ω) with the product rule D_iw = D_iu v+u D_iv. The identity (12) then follows fromE(Diw) = 0.

We now define the weak derivatives of a functionu∈L²(Ω)by duality.

Definition 2.3.3. LetuinL²(Ω),iin{1,· · · , d}andwiinL²(Ω); we say that uadmits w_i as weak derivative in thei^th direction if

− Z

Ω

uD_iv dµ= Z

Ω

w_iv dµ for every v∈C¹(Ω).

We note D_iu = w_i the weak derivative (which is uniquely defined) in L²(Ω), noticing that both definitions ofDicoincide forC¹random variable. Ifuadmits weak derivatives inL²(Ω)in all directions, we say thatubelongs toH¹(Ω)and we noteD_ωu= (D₁u,· · ·, D_du). Obviously,H¹(Ω)is a vector space. We define an inner product onH¹(Ω)as

(u, v)_H1(Ω) := (u, v)_L²_(Ω)+ (Dωu, Dωv)_L²_(Ω)^d.

We easily see that H¹(Ω) shares several properties with the usual Sobolev space H¹(U) when U is a bounded open set ofR^N. First, using the isometric embedding j :u ∈H¹(Ω) 7→ (u, D_ωu) ∈ L²(Ω)×L²(Ω,R^d), the closed graph theorem implies thatj(H¹(Ω))is a closed subspace ofL²(Ω)×L²(Ω,R^d), hence (H¹(Ω),(·,·)_H1(Ω))is a Hilbert space. Moreover, using the mollifying procedure of Lemma 2.3.2, we may prove thatC¹(Ω)is dense inH¹(Ω)and the imbedding

C¹(Ω),→H¹(Ω)

is continuous. Eventually, notice that foru inL²(Ω), if there exists C≥0such that

ku(T_δe_i·)−uk_L2(Ω) ≤Cδ for δ >0and iin{1,· · ·, d}, (13) thenuis inH¹(Ω). This is a classical consequence of (11), of the Lebesgue dom- inated convergence theorem and of the Riesz representation theorem. Gathering theses facts, we have:

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Proposition 2.3.4. H¹(Ω) is a separable Hilbert space; its subspace C¹(Ω) is dense. Moreover, a function u in L²(Ω) belongs to H¹(Ω) if and only if (13) holds true.

Lemma 2.3.5. Let u be a function in H¹(Ω) and let us denote by vω : x 7→

u(T_xω) and z_ω : x 7→ (D_ωu)(T_xω). Then, µ-almost surely, v_ω belongs to H_loc¹ (R^d) andzω=∇_xvω almost everywhere on R^d.

Proof. Let us proceed by density. Let (u_k)_k be a sequence in (C¹(Ω))^N such thatu_k converges to u, and Dωu_k converges toDωu inL²(Ω).

For every bounded Borel setAof R^d, Z

Ω

Z

A

(u_k(T_xω)−v_ω(x))² dx dµ(ω) =|A|

Z

Ω

(u_k(ω)−u(ω))²dµ(ω), because of stationarity. The right-hand side term converges to 0 as k goes to infinity by definition of(u_k)_k. As a consequence, thanks to a diagonal argument, and up to extraction,u_k(T_xω)converges tov_ω inL²_loc(R^d), almost surely onΩ.

We have similarly foriin{1, . . . , d}, Z

Ω

Z

A

∂

∂x_iu_k(T_xω)−z_ω,i(x) 2

dx dµ(ω) =|A|

Z

Ω

(D_iu_k(ω)−D_iu(ω))²dµ(ω), according to Lemma 2.3.1, and the stationarity assumption. The right-hand side converges to 0 asktends to infinity. Then, up to extraction again, _∂x^∂

iu_k(T_xω) converges toz_ω,iinL²_loc(R^d)almost surely inω, and therefore,v_ωis inH_loc¹ (R^d) and _∂x^∂

ivω=zω,i almost everywhere onR^d. Proposition 2.3.6. For every u in H¹(Ω),

D_ωu≡0 =⇒ u is constant µ-a.s.

The measure µ is said to be ergodic with respect to translations.

Proof. Letu∈H¹(Ω)be such thatD_ωu≡0. Then, according to Lemma 2.3.5, µ-a.s., a.e. in x∈Rⁿ,∇_xu(T_xω) = 0 so as a function of L²_loc(R^d),x7→u(T_xω) is constant. As a consequence,µ-a.s., for every tinR^d,

u(ω) = 1 t^d

Z

[0,t]^d

u(T_xω)dx ,

and according to Birkhov theorem, the right-hand side convergesµ-a.s. toE(u) ast goes to infinity, thereforeuis constantµ-a.s.

Now we define spaces that will be used in the definition of the effective matrix for the homogenized problems in the sequel.

Definition 2.3.4. We denote by L²_pot(Ω) the closure of the set {D_ωu : u ∈ C¹(Ω)}inL²(Ω,R^d), and L²_sol(Ω) =L²_pot(Ω)^⊥. We also denote by L²_pot,loc(R^d) the closure of {∇u : u ∈ C_c^∞(R^d)} in L²_loc(R^d,R^d), and L²_sol,loc(R^d) as the closure of {σ ∈C_c^∞(R^d,R^d) : divσ ≡0} in L²_loc(R^d,R^d).

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Definition 2.3.5. Let σ be a random variable inL²(Ω,R^d). We say that σ is inHdiv(Ω)if there exists ginL²(Ω)such that

∀u∈C¹(Ω), Z

Ω

σ·D_ωu dµ(ω) =− Z

Ω

gu dµ(ω).

In that case, we denote bydiv_ωσthe functiong, and call it the divergence ofσ.

Remark 2.3.1. It is clear thatH¹(Ω,R^d)⊂H_div(Ω)and that forσinH¹(Ω,R^d), div_ωσ =

d

X

i=1

D_iσ_i,

where σi, i = 1, ..., d, denote the marginals of σ. Moreover, the definition of L²_sol(Ω)is equivalent to

L²_sol(Ω) ={σ ∈H_div(Ω) : div_ωσ ≡0}.

Theorem 2.3.7. Letσ be a random variable inL²_sol(Ω). Then,µ-almost surely, the functionx7→σ(T_xω) is inL²_sol,loc(R^d).

Proof. Using a mollification as in the proof of Lemma 2.3.2, we see thatC¹(Ω) is dense inH_div(Ω)for the natural norm kσk²_H

div :=kσk²_L₂_(Ω)+kdiv_ωσk²_L₂_(Ω). The proof then proceeds as that of Lemma 2.3.5.

Proposition 2.3.8. The intersection betweenL²_pot(Ω)and the space of constant functions onΩ is null. The measure µ is said to be non-degenerate.

Proof. Let ξ ∈ L²_pot(Ω) be a constant function, and (u_k)_k ∈ (C¹(Ω))^N be a sequence such that Dωu_k converges to ξ in L²(Ω). Lemma 2.3.3 implies E(D_ωu_k) = 0and therefore,

|ξ|=|E(ξ)|=|E(Dωu_k−ξ)| ≤E(|D_ωu_k−ξ|)≤ kD_ωu_k−ξk_L2(Ω) , according to Cauchy-Schwarz inequality. Sending k to infinity, the right-hand side goes to0 which leads toξ≡0.

2.4 Two-scale convergence compactness theorem in H¹(D) Having defined H¹(Ω), the purpose of this subsection is to provide a two-scale compactness theorem inH¹(D)in a manner similar to the periodic case (see [1]).

Lemma 2.4.1. Let {v}_∈I be a family of elements of H¹(D) (with I as in Definition 2.2.2) such that

(i)

kvk_L2(D) is bounded (ii) k∇vk_L2(D)−−→

↓0 0.

Then, up to an extraction, there existsv⁰∈L²(D) such that v(x)* v² ⁰(x) weakly in L²(D ×Ω).

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Proof. According to Theorem 2.2.3, there exists vˆ⁰ ∈L²(D ×Ω) such that, up to the extraction of a subsequence,(v) two-scale converges weakly toˆv⁰(x, ω).

Letσ ∈(C¹(Ω))^dand ϕ∈C_c¹(D), we have for 0< ≤1, Z

D

v(x)ϕ(x) div_ωσ(T_x/ω)e dx=− Z

D

∇_x(vϕ)(x)σ(T_x/ω)e dx .

Passing to the limit asgoes to0, and using Cauchy-Schwarz inequality together with assumption (ii), and Proposition 2.2.2, we infer

Z

D

ϕ(x) Z

Ω

ˆ

v⁰(x, ω) divωσ(ω)dµ(ω)

dx= 0. As a consequence, for almost everyx∈ D and every σ∈C¹(Ω)^d,

Z

Ω

ˆ

v⁰(x, ω) div_ωσ(ω)dµ(ω) = 0.

For suchx, we have by definition that ˆv⁰(x,·) is in H¹(Ω)withD_ωˆv⁰(x,·) = 0 and by Proposition 2.3.6, this means thatˆv⁰(x, ω)does not depend onωup to a µ-negligible set. Settingv⁰(x) :=E(ˆv⁰(x,·)), we havev⁰∈L²(D)andv⁰= ˆv⁰in L²(D ×Ω)so that we still have the weak two-scale convergencev(x)* v² ⁰.

The following compactness theorem is the most important of this section.

Theorem 2.4.2. Let {v}_∈I be a bounded sequence of H¹(D). Then, up to an extraction, there exist v⁰ inH¹(D) and ξ in L²(D, L²_pot(Ω)) such that







v* v² ⁰(x) strongly,

∇v*² ∇v⁰(x) +ξ(x, ω) weakly, in L²(D ×Ω). Moreover, if for every ∈I, v is in H₀¹(D), thenv⁰ is inH₀¹(D).

Proof. We first apply Theorem 2.2.3 to the family {v} so that for some subsequence(_k),(v^k)two-scale converges towards somev₀ ∈L²(Ω× D). According to Lemma 2.4.1, v⁰ does not depend on ω. On the other hand, since {v} is bounded inH¹(D), up to a further extraction, we may assume thatvconverges weakly inH¹(D)and strongly inL²(D)to a limitv¯∈H¹(D). By Remark 2.2.1, we have actually, ¯v=v⁰ and the sequence strongly two-scale converges to v⁰.

Next, we apply Theorem 2.2.3 to the family {∇v}. Extracting again, the sequence (v^k) two-scale converges towards some element w of L²(Ω× D,R^d).

LetϕinC_c^∞(D). Integrating by parts, we compute, Z

D

∇v(x)ϕ(x)dx=− Z

D

v(x)∇ϕ(x)dx , Passing to the two-scale limit, we obtain

Z

D

ϕ(x) Z

Ω

w(x, ω)dµ(ω)

dx =− Z

D

v⁰∇ϕ dx .

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Therefore, the distribution ∇v⁰ is given by the function

∇v⁰= Z

Ω

w(·, ω)dµ(ω),

which belongs toL²(D).

Now, forσ inL²_sol(Ω)∩C¹(Ω,R^d) and ϕinC_c^∞(D), we compute Z

D

∇v(x)·σ(T_x/ω)ϕ(x)e dx= Z

D

∇[v(x)ϕ(x)]·σ(T_x/ω)e dx

− Z

D

v(x)∇ϕ(x)·σ(T_x/ω)e dx

=− Z

D

v(x)∇ϕ(x)·σ(T_x/ω)e dx .

We have used the fact that by Theorem 2.3.7, the mappingx7→σ(T_x/eω) is in L²_sol,loc(R^d). Sendingto 0 and integrating by parts, we get

Z

D

Z

Ω

w(x, ω)·σ(ω)ϕ(x)dµ dx=− Z

D

Z

Ω

v⁰(x)∇ϕ(x)·σ(ω)dµ(ω)dx

= Z

D

Z

Ω

∇v⁰(x)·σ(ω)ϕ(x)dµ(ω)dx . Let us note ξ := w− ∇v⁰. The last identity shows that for almost everyx in D,ξ(x) lies in((C(Ω))^d∩L²_sol(Ω))^⊥. By density ofC(Ω)inL²(Ω)we conclude that

ξ is inL²(D,(L²_sol(Ω))^⊥) =L²(D, L²_pot(Ω)).

This proves the main part of the theorem.

Finally, let us assume moreover, that for every,v is in H₀¹(D). Since the subspaceH₀¹(D) is closed inH¹(D), we see thatv¯=v⁰ is also inH₀¹(D).

3 Applications of stochastic two-scale convergence

With the tools that we just defined, we are now able to provide homogenization theorems for some elliptic and parabolic problems with random coefficients. To introduce the method and set the notations, we start by considering the well- known problem of diffusion equation. We then turn to non-linear problems, namely harmonic maps and the Landau-Lifschitz-Gilbert equation which do not have in general a unique solution. From now on we assumeD to be a bounded domain ofR³.

3.1 Elliptic equations We consider the problem

(−div a(T_x/eω)∇u(x)

=f(x) inD

u∈H₀¹(D), (14)