Homogenization of periodic nonconvex integral functionals in terms of Young measures

January 2006, Vol. 12, 35–51 DOI: 10.1051/cocv:2005031

HOMOGENIZATION OF PERIODIC NONCONVEX INTEGRAL FUNCTIONALS IN TERMS OF YOUNG MEASURES

Omar Anza Hafsa

, Jean-Philippe Mandallena

and G´ erard Michaille

Abstract. Homogenization of periodic functionals, whose integrands possess possibly multi-well struc- ture, is treated in terms of Young measures. More precisely, we characterize the Γ-limit of sequences of such functionals in the set of Young measures, extending the relaxation theorem of Kinderlherer and Pedregal. We also make precise the relationship between our homogenized density and the classical one.

Mathematics Subject Classification. 35B27, 49J45, 74N15.

Received June 4, 2004. Revised February 16, 2005.

1. Introduction and main results

Letm, N≥1 be two integer,p >1, Ω⊂R^N a bounded open set with Lipschitz boundary∂Ω, andY =]0,1[^N. We denote the space of all realm×N matrices byM. Consider the integral

F_ε u_ε

:=

Ω

fx

ε,∇u_ε(x)

dx, (1)

where ε >0, u_ε : Ω →R^m andf :R^N ×M →[0,+∞[ is aCarath´eodory integrand, possibly with multi-well structure, satisfying the following two conditions:

(C₁) for everyξ∈M,f(·, ξ) isY-periodic,i.e.,f(x+z, ξ) =f(x, ξ) for allx∈R^N and allz∈Z^N. (C₂) α|ξ|^p≤f(x, ξ)≤β(1 +|ξ|^p) for allx∈R^N, allξ∈Mand someα, β >0.

In pseudo¹-nonlinear elasticity, whenm=N = 3,F_εin (1) is the free-energy functional at a microscopic scaleε of an elastic material which occupies the bounded open set Ω ⊂R³ in a reference configuration, the body is assumed to have a periodic structure with period εY at any scale ε. Roughly, following the idea of Ball and

Keywords and phrases. Young measures, homogenization.

1 Institut f¨ur Mathematik, Universit¨at Z¨urich, Winterthurerstrasse 190, CH-8057 Z¨urich, Switzerland;anza@math.unizh.ch

2 EMIAN (´Equipe de Math´ematiques, d’Informatiques et Applications de Nˆımes), Centre Universitaire de Formation et de Recherche de Nˆımes, Site des Carmes, Place Gabriel P´eri, Cedex 01, 30021 Nˆımes, France;

jean-philippe.mandallena@unimes.fr

3 I3M (Institut de Math´ematiques et Mod´elisation de Montpellier) UMR-CNRS 5149, Universit´e Montpellier II, Place Eug`ene Bataillon, 34090 Montpellier, France;micha@math.univ-montp2.fr

1We use the term “pseudo” to mean thatf(x,·) does not satisfy the basic condition that infinite energy is needed to compress the material to zero volume,i.e.,f(x, ξ)→+∞as det(ξ)→0⁺.

c EDP Sciences, SMAI 2006

Article published by EDP Sciences and available at http://www.edpsciences.org/cocvor http://dx.doi.org/10.1051/cocv:2005031

James [4], the fine microstructure of the material can be thought of as an element of anε-minimizing sequence {u_ε}εforF_εin U with

U :=

u∈W^1,p(Ω;R^m) :u= 0 on Γ ,

where Γ is a subset of∂Ω with positive (N−1)-dimensional Hausdorff measure. The homogenization theorem², firstly established by Braides in [6] and then completed by M¨uller in [19], states that iff satisfies (C₁) and (C₂) then the homogenized free-energy functional of the material in terms of Sobolev functions

F_hom(u) :=

Ω

f_hom

∇u(x)

dx, (2)

wheref_hom:M→[0,+∞[ (the homogenized free-energy density of the material) is defined by f_hom(ξ) := inf

k∈N^∗inf

−

kY

f(x, ξ+∇φ(x))dx:φ∈W₀^1,p(kY;R^m) ,

characterizes the W^1,p-weak limits of ε-minimizing sequences for F_ε in U. More precisely: lim_ε→0inf_UF_ε = min_UF_hom; theW^1,p-weak limituof anyε-minimizing sequence{u_ε}εforF_εinU is a minimizer forF_hom inU; conversely, any minimizerufor F_hom in U is the W^1,p-weak limit of someε-minimizing sequence for F_ε in U. Such aucan be thought of as a “macroscopic representation” of the fine microstructure of the material.

In the homogeneous case, when f does not depend onx(so that F_ε=F), another characterization can be obtained by using the notion of gradient Young measure due to Kinderlehrer and Pedregal [15,16]: aW-gradient Young measure, with W ⊂W^1,p(Ω;R^m), is a Young measure µ on Ω×M for which there exists a bounded sequence{u_ε}εin W such that µ is the narrow limit ofδ_∇uε(x)⊗dxas ε→0 (cf. Sect 2.1)³. The relaxation theorem of Kinderlehrer and Pedregal states that under (C₂), the relaxed free-energy functional of the material in terms of Young measures

F(µ) :=

Ω

M

f(ξ)dµ_x(ξ)

dx, (3)

where the variable µ = µ_x⊗dx is a W^1,p(Ω;R^m)-gradient Young measure, characterizes the weak limits of minimizing sequences for F in U as follows: inf_UF = min_UF, where U is the set of all U-gradient Young measures; the narrow limitµof anyδ_∇uε(x)⊗dxasε→0, where{u_ε}εis minimizing forF inU, is a minimizer for F in U; conversely, any minimizer µfor F in U is the narrow limit of someδ_∇uε(x)⊗dxas ε→0, where {u_ε}εis minimizing for F in U. Moreover, min_UQF = min_UF, where

QF(u) :=

Ω

Qf

∇u(x) dx

(the relaxed free-energy functional of the material in terms of Sobolev functions) with Qf :M→[0,+∞[ the quasiconvexification off (the relaxed free-energy density of the material) given by⁴

Qf(ξ) = inf

Y

f(ξ+∇φ(x))dx:φ∈W₀^1,p(Y;R^m) .

Finally, uis a minimizer forQF in U if and only if there exists µ=µ_x⊗dxminimizer for F in Usuch that

∇u(x) =

Mζdµ_x(ζ) for a.e. x∈Ω. Such aµ can be thought of as a “microscopic representation” of the fine microstructure of the material.

2In the convex case, the homogenization theorem was proved by Marcellini in [18].

3To simplify the presentation of the paper, we will denote by dxthe Lebesgue measure restricted to any bounded open subset ofR^N.

4The quasiconvexification formula was established by Dacorogna in [10].

In our paper we extend the relaxation theorem of Kinderlherer and Pedregal to the periodic homogenization by means of a Γ-convergence procedure (for an other approach about Γ-convergence through Young measures, we refer to [21]). In the classical homogenization process, gradient solutions of min_UF_hom capture the oscillations due to the periodic structure. Unfortunately, as the density f_hom is quasiconvex, we loose the information about the oscillations developed by the gradient minimizing sequences because of the multi-well structure. By considering our process, every probability solution of the new limit problem captures two kinds of oscillations:

those due to theε-periodicity (by its barycenter) and those due to the multi-well structure (see Cor. 1.2(iv) and Rem. 3.1). However, the homogenized density ¯gin (4) is given by a complicated formula, and our paper can be seen as a first attempt in the scope of homogenization with gradient oscillations analysis.

Denote the set of all Young measures on Ω×MbyY(Ω;M). For each ε >0, letFε:Y(Ω;M)→[0,+∞] be defined by

Fε(µ) :=

⎧⎨

⎩

Ω

M

fx ε, ξ

dµ_x(ξ)

dx ifµ=µ_x⊗dx∈∆(U)

+∞ otherwise,

where

∆(U) :=

µ_x⊗dx∈ Y(Ω;M) :µ_x=δ_∇u(x) withu∈ U .

LetP(M) be the set of all probability measures onM, and, for everyξ∈M, letHξ(M) be the set ofλ∈ P(M) fulfilling the following three conditions (cf. also Rem. 2.2):

–

Mζdλ(ζ) =ξ;

– h(ξ)≤

Mh(ζ)dλ(ζ) for every quasiconvex function h: M→R bounded below and satisfyingh(ζ)≤ c

1 +|ζ|^p

for allζ∈M and somec >0;

–

M|ζ|^pdλ(ζ)<+∞.

For each bounded open setA⊂R^N, we define S_A(ξ,·) :H_ξ(M)→[0,+∞[ by S_A(ξ, λ) := inf

A

M

f(x, ζ)dσ_x(ζ)

dx:σ_x⊗dx∈ ∇Y_ξ(A), −

A

σ_xdx=λ ,

where∇Yξ(A) is the set of alll_ξ+W₀^1,p(A;R^m)-gradient Young measures,l_ξ denoting the affine function with constant gradient ξ. The equality −

Aσ_xdx = λ means that −

A Mϕ(ζ)dσ_x(ζ)

dx = λ, ϕ for all ϕ in the space C_c(M) of all real-valued and continuous functions with compact support on M. Let g : M× P(M) → [0,+∞] be the measurable function defined by

g(ξ, λ) :=

⎧⎨

⎩ inf

k∈N^∗

SkY(ξ, λ)

k^N ifλ∈ Hξ(M)

+∞ otherwise.

Then, for each fixed ξ in M, we consider ¯g(ξ,·) : P(M) → [0,+∞], the weak lower semicontinuous envelope ofg(ξ,·),i.e., the function defined by

¯g(ξ, λ) := inf

n→+∞lim g(ξ, λ_n) :P(M)λ_n λ^∗ , (4)

where λ_n λ^∗ means that for every ϕ ∈ C_c(M), lim_n→+∞ λ_n, ϕ = λ, ϕ. It is worth noticing that, if f does not depend on x, then ¯g(ξ, λ) =

Mf(ζ)dλ(ζ) for all ξ ∈ M and all λ ∈ Hξ(M) (cf. Prop. 2.7). Let

F :Y(Ω;M)→[0,+∞] be defined by

F(µ) :=

⎧⎨

⎩

Ω

¯g

bar(µ_x), µ_x

dx ifµ=µ_x⊗dx∈U

+∞ otherwise

(5) with bar(µ_x) :=

Mζdµ_x(ζ). Here are the main results of the paper.

Theorem 1.1. Under (C₁) and(C₂), we haveF = Γ(nar)-lim_ε→0F_ε.

The Γ-convergence process stated in Theorem 1.1 is taken with respect to the narrow convergence of Young measures because of its compactness property (cf. Prokhorov’s compactness theorem in Sect. 2 and Cor. 1.2(iii)).

According to the previous discussion, F (resp. ¯g) in (5) (resp. (4)) can be called the homogenized free- energy functional (resp. homogenized free-energy density) of the material in terms of Young measures. Since F_ε(u) =Fε

δ_∇u(x)⊗dx

for allu∈ U, as a direct consequence of Theorem 1.1, Proposition 2.3 and Remark 2.1, we obtain

Corollary 1.2. Under the hypotheses of Theorem1.1, we have:

(i) lim_ε→0inf_UF_ε= min_UF= min_UF_hom;

(ii) if δ_∇uε(x)⊗dx^nar µ, with{u_ε}ε anε-minimizing sequence for F_ε inU, thenµis a minimizer for F in U;

(iii) ifµis a minimizer forF inU, then, there exists anε-minimizing sequence{u_ε}εforF_εinU, such that δ_∇uε(x)⊗dx^nar µ;

(iv) uis a minimizer for F_hom in U if and only if, there exists a minimizer µ=µ_x⊗dx for F inU, such that ∇u(x) = bar(µ_x)for a.e. x∈Ω.

Remark 1.3. In contrast to the relaxed functional in (3), the homogenized functional in (5) is “quasi-local”:

it is local with respect to dx but, in general, non-local with respect to µ_x, i.e., ¯g has, a priori, no integral representation with respect toµ_x.

Finally, the following result makes clear the link betweenf_hom and ¯g.

Theorem 1.4. Assume that conditions (C₁) and(C₂)hold. Then, f_hom(ξ) = inf

Y ¯g(ξ, µ_x)dx:µ_x⊗dx∈ ∇Y(Y), bar(µ_x) =ξ (6) for allξ∈M, where∇Y(Y)denotes the set of all W^1,p(Y;R^m)-gradient Young measures.

A concrete example of our nonlinear homogenization process can be given when considering the free-energy functional of a polycristal with shape Ω. In this particular case

f(x, ξ) =h_t

R(x)ξR(x) ,

whereRis a spatially periodic (or random) piecewise constant rotation-valued function, andhhas a finite num- ber of wells (in the setting of linear elasticity these wells are called the stress-free strains of martensite variants, see Bhattacharya and Kohn [5] for more details). Minimizers of the homogenized free-energy functionalF_hom in (2) characterize the mixtures of martensite variants in the austenite/martensite phase transformation below the transition temperature. According to the point of view of Bhattacharya and Kohn, and taking formula (6) into account, we can say that ¯gis the microscopic free-energy density corresponding to the macroscopic onef_hom. The plan of the paper is as follows. Section 2 presents some preliminaries. In Section 2.1 we review some of the standard facts on Young (and gradient Young) measures. In Section 2.2, we briefly recall the notion of Γ-convergence. In Section 2.3 we point out a subadditive result (cf. Prop. 2.4) for the set functionA→SA(ξ, λ)

with ξ ∈ M and λ∈ Hξ(M) (Prop. 2.4 is used in the proof of the upper bound in Th. 1.1). Properties of ¯g (cf. Props. 2.5 and 2.7) are established in Section 2.4. Section 3 (resp Sect. 4) is devoted to the proof of Theorem 1.1 (resp. Th. 1.4). Finally, in Section 5 we discuss on some open questions and possible extensions.

2. Preliminaries 2.1. Young measures

LetN, m≥1 be two integers letMdenote the space of all realm×N matrices, and consider a bounded open set Ω⊂R^N. AYoung measureon Ω×Mis a positive Radon measureµon Ω×M such thatµ(B×M) =|B| for all Borel setB⊂Ω, where|B|denotes the Lebesgue measure ofB. The set of all Young measures on Ω×M is denoted byY(Ω;M). ByCth^b(Ω;M) we denote the space of all bounded Carath´eodory integrands on Ω×M.

Given µ∈ Y(Ω;M) and{µ_n}n≥1 ⊂ Y(Ω;M), we say thatµ_n narrow converges to µ, and we writeµ_n ^nar µ, if for everyϕ∈Cth^b(Ω;M),

n→+∞lim

Ω×M

ϕ(x, ξ)dµ_n(x, ξ) =

Ω×M

ϕ(x, ξ)dµ(x, ξ).

We denote the set of all probability measures onM byP(M). For a proof of the following theorem we refer to [13], Theorem 10, p. 14 (see also [22], Th. A4 and Cor. A5).

Slicing theorem. Given µ∈ Y(Ω;M), there exists a unique (up to the equality a.e.) family {µ_x}_x∈Ω⊂ P(M) such that:

(i) the function x→

Ω×Mϕ(x, ξ)dµ_x is measurable;

(ii)

Ω×Mϕ(x, ξ)dµ(x, ξ) =

Ω Mϕ(x, ξ)dµ_x(ξ) dx,

for every ϕ∈L¹_µ(Ω×M). To summarize it, we will writeµ=µ_x⊗dx.

The slicing theorem leads to the following version of the narrow convergence, (see [22] for more details).

Givenµ∈ Y(Ω;M) and{µ_n=µⁿ_x⊗dx}_n≥1⊂ Y(Ω;M), we have: µ_n^nar µ if and only if for every ψ∈C_c(M),

M

ψ(ξ)dµⁿ_x(ξ)

M

ψ(ξ)dµ_x(ξ) inL^∞(Ω) weak^∗,

i.e., for every ψ∈C_c(M)and every ϕ∈L¹(Ω),

n→+∞lim

Ω

ϕ(x)

M

ψ(ξ)dµⁿ_x(ξ)

dx=

Ω

ϕ(x)

M

ψ(ξ)dµ_x(ξ)

dx.

We say that{µ_n} ⊂ Y(Ω;M) istightif for everyδ >0, there exists a compact setK⊂Msuch that sup µ_n

Ω×

(M\K)

:n≥1

< δ. A proof of the following compactness result can be found in [22], Theorem 11 (see also [23], Th. 7 and Comments 1), 2) and 3)).

Prokhorov’s compactness theorem. If {µ_n} ⊂ Y(Ω;M)is tight, then there exists µ∈ Y(Ω;M) such that, up to a subsequence,µ_n ^nar µ.

Remark 2.1. A straightforward consequence of Prokhorov’s compactness theorem is the following: if{ξ_n}_n≥1 is a bounded sequence inL¹(Ω;M), then the sequence{δ_ξn(x)⊗dx}n≥1 is narrow relatively compact. Indeed, by Markov’s inequality,i.e., |{x∈Ω :|ξ_n(x)| ≥c}| ≤(1/c)

Ω|ξ_n(x)|dxfor anyc >0 and anyn≥1, it is obvious that {δ_ξn(x)⊗dx}n≥1is tight.

The result below is usually referred as the continuity theorem. For a proof we refer to [23], Theorem 6 and Comments 1), 2), 3) and 4).

Continuity theorem. Let {ξ_n}n≥1 be a sequence of measurable functions fromΩtoM, let ϕ: Ω×M→Rbe a Carath´eodory integrand, and letµ∈ Y(Ω;M). Ifδ_ξn(x)⊗dx^nar µand the sequence

ϕ(·, ξ_n)

n≥1 is uniformly integrable, then

n→+∞lim

Ω

ϕ

x, ξ_n(x) dx=

Ω×M

ϕ(x, ξ)dµ(x, ξ).

Finally, we say that µ∈ Y(Ω;M) is a U-gradient Young measure if there exists a bounded sequence{u_n}n≥1

inU such thatδ_∇un(x)⊗dx^nar µ. The set of allU-gradient Young measures is denoted byU. A characterization ofUwas established by Kinderlehrer and Pedregal in [16], Theorem 1.1 (see also [20], Th. 8.14 p. 150).

Kinderlehrer-Pedregal’s characterization theorem. µ ∈ U if and only if the following three conditions hold:

(i) there existsu∈ U such that ∇u(x) =

Mζdµ_x(ζ) =: bar(µ_x)for a.e. x∈Ω;

(ii) h _Mζdµ_x(ζ)

≤

Mh(ζ)dµ_x(ζ)for a.e. x∈Ωand for every quasiconvex function h:M→Rbounded below, and satisfying h(ζ)≤c

1 +|ζ|^p

for allζ∈Mand some c >0;

(iii)

Ω M|ζ|^pdµ_x(ζ)

dx <+∞.

Remark 2.2. Given ξ∈Mand a bounded open setA⊂R^N, the set of alll_ξ+W₀^1,p(A;R^m)-gradient Young measures is denoted by ∇Y_ξ(A). Similarly, we have: µ ∈ ∇Y_ξ(A) if and only if (i), (ii) and (iii) are satisfied with Ω replaced byAandU byl_ξ+W₀^1,p(A;R^m). Thus, the elements ofHξ(M) are the homogeneous elements of∇Yξ(A) whose barycenter is equal to ξ,i.e.,

Hξ(M)≡

µ_x⊗dx∈ ∇Yξ(A) :∀x∈Ω, µ_x=λwithλ∈ P(M) and bar(λ) =ξ .

2.2. Γ-convergence

Let {F_ε}_ε be a sequence of functionals fromY(Ω;M) to [0,+∞] and let F : Y(Ω;M) →[0,+∞]. We say that Fε Γ(nar)-converges to F as ε→0, and we write F = Γ(nar)- lim_ε→0Fε,if the following two assertions hold:

Lower bound: for everyµ∈ Y(Ω;M), and everyµ_ε^nar µ, F(µ)≤lim

ε→0Fε(µ_ε);

Upper bound: for everyµ∈ Y(Ω;M), there existsµ_ε^nar µ such that F(µ)≥lim

ε→0Fε(µ_ε).

The following proposition is a well-known result that makes precise the variational nature of Γ-convergence.

Proposition 2.3. IfF = Γ(nar)-lim_ε→0Fεand if{µ_ε}εis anε-minimizing sequence for{Fε}εwhich is narrow relatively compact, then any cluster point µof {µ_ε}_ε is a minimizer forF, andlim_ε→0infF_ε(µ_ε) =F(µ).

For a proof and a deeper discussion of the Γ-convergence theory we refer the reader to the books [3, 7, 11].

2.3. A subadditive result

Denote the class of all bounded open subsets ofR^N byOb. A set function S:Ob →[0,+∞[, A→SA, is called subadditive ifSA ≤SA+SA for allA, A, A ∈ Ob such that A ⊂ A, A ⊂ Aand |A∩A|= 0,

|A\A∪A|= 0. The following well-known result is substantially the subadditive ergodic theorem of Akcoglu and Krengel (see [1]) in the deterministic case. For a proof we refer to [17], Theorem 2.1 (see also [2], Lem.

B.1).

Akcoglu-Krengel’s subadditive theorem. LetCub(R^N)be the class of all open cubes in R^N, and consider a subadditive set function S:Ob →[0,+∞[satisfying the following two conditions:

(S₁) S_A≤c|A|for all A∈ O_b and somec >0;

(S₂) S isZ^N-invariant, i.e., S_A+z=S_A for all z∈Z^N and all A∈ O_b.

Then, for every Q∈Cub(R^N)and every real sequence {r_k}_k≥1 withr_k →+∞as k→+∞,

k→+∞lim SrkQ

|r_kQ| = inf

k∈N^∗

SkY

k^N ·

In our framework, we are led to consider, for eachξ∈Mand eachλ∈ P(M), the set functionObA→SA(ξ, λ) given by

S_A(ξ, λ) := inf

A

M

f(x, ζ)dσ_x(ζ)

dx:σ_x⊗dx∈Γ_A(ξ, λ) , where Γ_A:M× P(M)^−→−→Y(A;M) is the multifunction defined by

Γ_A(ξ, λ) :=

σ_x⊗dx∈ ∇Y_ξ(A) :−

A

σ_xdx=λ . (7)

According to Remark 2.2, it is clear that for everyA∈ Ob,

Γ_A(ξ, λ) =∅ if and only ifλ∈ Hξ(M). (8)

For λ ∈ Hξ(M) with ξ ∈ M, we see that λ⊗dx∈ Γ_A(ξ, λ). From the second inequality in (C₂), it follows that S_A(ξ, λ)≤ β

1 +

M|ζ|^pdλ(ζ)

|A| for all A ∈ O_b. Thus S_(·)(ξ, λ) satisfies (S₁). Condition (C₁) makes it is obvious that (S₂) holds, and we let the reader to verify thatS_(·)(ξ, λ) is subadditive. Applying Akcoglu- Krengel’s subadditive theorem, we obtain the following proposition used in the proof of the upper bound in Theorem 1.1 (cf. Sect. 3.2).

Proposition 2.4. If (C₁)and the second inequality in (C₂)hold, then for everyξ∈Mand every λ∈ H_ξ(M),

k→+∞lim

S_kY(ξ, λ) k^N = inf

k∈N^∗

S_kY(ξ, λ) k^N ·

2.4. Properties of ¯ g

We begin with the following proposition.

Proposition 2.5. For every µ_x⊗dx∈U, we have:

(i) the functionx→¯g

bar(µ_x), µ_x

is measurable;

(ii) if (C₂)holds then α

M|ζ|^pdµ_x(ζ)≤¯g

bar(µ_x), µ_x

≤β 1 +

M|ζ|^pdµ_x(ζ)

for a.e. x∈Ω.

Remark 2.6. As a consequence of Kinderlehrer-Pedregal’s characterization theorem(iii) and the second in- equality in Proposition 2.5(ii), we have dom

F

=U.

Proof of Proposition 2.5. (i) For everyA∈ O_b, the mesurability of the function M× P(M)(ξ, λ)→S_A(ξ, λ) = inf

F

σ_x⊗dx

:σ_x⊗dx∈Γ_A(ξ, λ) , whereF:Y(A;M)→[0,+∞] is defined byF

σ_x⊗dx

=

A Mf(x, ζ)dσ_x(ζ)

dxand Γ_A:M×P(M)−→^−→Y(A;M) is given by (7), comes from [9], Lemma III.39. Taking (8) into account, we see thatg(ξ, λ) = inf_k∈N∗SkY(ξ, λ)/k^N for allξ∈Mand allλ∈ P(M), henceg is measurable and (i) follows.

(ii) Fixx∈Ω. From the second inequality in (C₂), we have

Y

M

f(y, ζ)dµ_x(ζ)

dy≤β

1 +

M|ζ|^pdµ_x(ζ)

,

and the second inequality in (ii) follows since ¯g(bar(µ_x), µ_x)≤SY(bar(µ_x), µ_x) andµ_x⊗dy∈Γ_Y(bar(µ_x), µ_x).

On the other hand, considering{λ_n}n≥1⊂ P(M) such thatλ_n µ^∗ _x and ¯g(bar(µ_x), µ_x) = lim_n→+∞g(bar(µ_x), λ_n), and using the first inequality in (C₂), we see that

g¯

bar(µ_x), µ_x

≥ lim

n→+∞α

M|ζ|^pdλ_n(ζ)≥α

M|ζ|^pdµ_x(ζ),

which completes the proof.

The next proposition shows that the relaxation theorem of Kinderlehrer and Pedregal is a particular case of Corollary 1.2.

Proposition 2.7. If the second inequality in (C₂) holds and if f does not depend on x, then ¯g(ξ, λ) =

Mf(ζ)dλ(ζ)for allξ∈Mand all λ∈ H_ξ(M).

Proof. Taking the second inequality in (C₂) into account, it is clear that for every k ≥ 1, everyξ ∈ M and everyλ∈ Hξ(M),−

kY Mf(ζ)dσ_x(ζ) dx=

Mf(ζ)dλ(ζ) wheneverσ_x⊗dx∈ ∇Yξ(kY) with−

kyσ_xdx=λ. It follows thatg(ξ, λ) =

Mf(ζ)dλ(ζ) for allξ∈Mand allλ∈ Hξ(M), which gives the desired conclusion because the mappingλ→

Mf(ζ)dλ(ζ) is weakly lower semicontinuous onP(M).

3. Proof of Theorem 1.1 3.1. Proof of the lower bound

Letµ=µ_x⊗dx∈ Y(Ω;R^m), andµ_ε^nar µ. We have to prove that F(µ)≤lim

ε→0Fε(µ_ε). (9)

Without loss of generality we can assume that

ε→0limF_ε(µ_ε)<+∞. (10)

Thus,µ_ε∈∆(U),i.e., there existsu_ε∈ U such thatµ_ε=δ_∇uε(x)⊗dx, and so F_ε(µ_ε) =

Ω

fx

ε,∇u_ε(x) dx.

From the first inequality in (C₂), we see that{µ_ε}_εis tight. Using Prokorov’s compactness theorem, we deduce that there exists µ∈Usuch that (up to a subsequence)µ_ε ^nar µ (cf. Rem. 2.1). Thenµ=µ, and soµ∈U.

According to Kinderlehrer-Pedregal’s characterization theorem, there existsu∈ U such that bar(µ_x) =∇u(x) fora.e. x∈Ω.

In order to obtain (9) we proceed in three steps. Firstly, using a standard blow-up technique near x₀, we show that is sufficient to prove

¯g

∇u(x₀), µ_x0

≤lim

ρ→0lim

ε→0−

Qρ(x0)

fx

ε,∇u_ε(x)

dx. (11)

The two last steps consist in establishing (11) by means of De Giorgi’s slicing method together with a lower semicontinuous regularization.

Step 1(localization and blow-up). Denote the space of all Radon measures on Ω byM(Ω), and setM⁺(Ω) :=

{Θ∈ M(Ω) : Θ≥0}. Let{Θε}ε⊂ M⁺(Ω) be defined by Θ_ε:=f·

ε,∇u_ε dx.

By (10),{Θ_ε}_εis bounded inM⁺(Ω), hence there exists Θ∈ M⁺(Ω) such that (up to a subsequence) Θ_ε^∗ Θ.

As Θ(Ω)≤lim_ε→0Θ_ε(Ω), if we prove that

Ω

g¯

∇u(x), µ_x

dx≤Θ(Ω),

then (9) will follow. Consider the Lebesgue decomposition of Θ = Θ^a + Θ^s, where Θ^a,Θ^s ∈ M⁺(Ω) are respectively the absolutely continuous and the singular part with respect to dx. Radon-Nikodym’s theorem asserts that there exists θ∈L¹(Ω;R⁺) such that Θ^a=θdx, and by Lebesgue’s differentiation theorem,

θ(x₀) = lim

ρ→0

Θ^a

Qρ(x₀) Qρ(x₀) = lim

ρ→0

Θ

Qρ(x₀)

Qρ(x₀) (12) for a.e. x₀ ∈ Ω, where Q_ρ(x₀) is the open cube centered at x₀ and of side ρ. From now on, one fix any x₀ outside a negligible set of Ω, such that the following four assertions hold:

– bar(µ_x0) =∇u(x₀);

– (12) holds;

– for everyψ∈ D,

ρ→0lim−

Qρ(x0)

M

ψ(ξ)dµ_x(ξ)

dx=

M

ψ(ξ)dµ_x0(ξ), (13)

whereDis a countable subset of Lipschitz function fromMtoRwhich is dense inC_c(M);

– for ¯u:R^N →R^m denoting the affine function defined by ¯u(x) :=u(x₀) +∇u(x₀)·(x−x₀), we have (see [24], Th. 3.4.2)

ρ→0lim 1 ρ−

Qρ(x0)

u−u¯dx= 0. (14) As Θ_ε ^∗ Θ, one has Θ

Q_ρ(x₀)

= lim_ε→0Θ_ε

Q_ρ(x₀)

whenever Θ

∂Q_ρ(x₀)

= 0. Since Θ is finite, Θ

∂Qρ(x₀)

= 0 for all but countably many ρ >0. In the sequel, we will take ρsuch that Θ

∂Qρ(x₀)

= 0.

Consequently, it is sufficient to prove (11).

Step 2 (decreasing the energy by slicing De Giorgi’s method). Fix any t ∈]0,1[ and any ∈ N^∗. For each i∈ {0,· · · , }, defineQi:=Qtρ+i(1−t)ρ/(x₀) and consider a cut-off functionφ_ibetweenQi−1andQi(i≥1) such that ∇φ_i∞≤_(1−t)ρ² . Settinguⁱ_ε(x) := ¯u(x) +φ_i(x)(u_ε(x)−u(x)),¯ we haveuⁱ_ε∈l_∇u(x0)+W^1,p(Qρ(x₀);R^m) and

∇uⁱ_ε=

⎧⎨

⎩

∇u_ε onQ_i−1

∇u(x₀) + (u_ε−u)¯ ⊗ ∇φ_i+φ_i(∇u_ε− ∇u(x₀)) onQ_i\ Q_i−1

∇u(x₀) onQ_ρ(x₀)\ Q_i. (15) Using the second inequality in (C₂), we obtain

1

i=1

−

Qρ(x0)

fx

ε,∇uⁱ_ε(x) dx≤ −

Qρ(x0)

fx

ε,∇u_ε(x)

dx+ ˆA(ε, ρ) (16)

with ˆA(ε, ρ) :=¹

i=1A_i(ε, ρ) and A_i(ε, ρ) := 1

ρ^N

Qi\Qi−1

fx

ε,∇uⁱ_ε(x)

dx+β

1 +|∇u(x₀)|^p 1−t^N

.

Let k_ε ∈ N^∗ be the smallest integer such that ¹_εQρ(x₀) ⊂ k_εY +z_ε for an appropriate z_ε ∈ Z^N. Consider

¯ı(ε, ρ, t, ) =: ¯ı∈ {1,· · ·, }such that

−

Qρ(x0)

fx

ε,∇u^¯ı_ε(x) dx≤1

i=1

−

Qρ(x0)

fx

ε,∇uⁱ_ε(x)

dx, (17)

and definew^¯ı_ε∈l_∇u(x0)+W₀^1,p

k_εY;R^m by w^¯ı_ε(x) :=

₁

εv^¯ı_n

ε(x+z_ε)

ifx∈ ¹_εQ_ρ(x₀)−z_ε l_∇u(x0)(x) ifx∈k_εY \₁

εQ_ρ(x₀)−z_ε withv^¯_ε^ı ∈l_∇u(x0)+W₀^1,p

Qρ(x₀);R^m

given byv^¯ı_ε(x) :=u^¯ı_ε(x)−u(x₀) +∇u(x₀)·x₀. By (C₁), we thus have

−

1εQρ(x0)

f

x,∇w^¯ı_ε(x−z_ε)

dx=−

Qρ(x0)

fx

ε,∇u^¯ı_ε(x)

dx. (18)

Setting γ:=β

1 +|∇u(x₀)|

and ∆_ε:=k^−N_ε

k_ε^N−(k_ε−2)^N

, it is easily seen that

−

kεY

f

x,∇w^¯ı_ε(x) dx≤ −

1εQρ(x0)

f

x,∇w^¯_ε^ı(x−z_ε)

dx+γ∆_ε.

Letλ^¯ı_ε∈ H_∇u(x0)(M) be defined by

λ^¯ı_ε:=−

kεYδ_∇w¯ı ε(x)dx.

By definition, lim_ε→0k_ε= +∞, hence limε→0∆_ε= 0, and consequently

ε→0lim inf

k∈N^∗

S_kY(ξ, λ^¯ı_ε) k^N = lim

ε→0g

∇u(x₀), λ^¯ı_ε

(19)

≤ lim

n→+∞−

1εQρ(x0)

f

x,∇w^¯_n^ı(x−z_ε) dx.

Using (14), we see that lim_ρ→0lim_ε→0A(ε, ρ)ˆ ≤c 1−t^N

+¹

, wherec >0 is a constant independent ofε, ρ,tand . Taking (16), (17) and (18) into account, from (19) we deduce that

→+∞lim lim

t→1lim

ρ→0lim

ε→0g

∇u(x₀), λ^¯ı_ε

≤ lim

ρ→0lim

ε→0−

Qρ(x0)

fx

ε,∇u_ε(x)

dx. (20)

Step 3(end of the proof). There is no loss of generality in assuming that there existλ_ρ(t, ), λ_t(), λ, λ∈ P(M) such that λ^¯ı_ε λ^∗ _ρ(t, ) asε→0,λ_ρ(t, ) λ^∗ _t() asρ→0,λ_t() λ^∗ ast→1 andλ λ^∗ as →+∞. We claim that

λ=µ_x0.

Indeed, given anyϕ∈C_c(M) and anyη >0, considerψ∈ Da C-Lipschitz function such thatϕ−ψ∞≤η.

Then,

λ^¯ı_ε−µ_x0, ϕ≤ λ^¯ı_ε−µ_x0, ψ+ 2η.

Moreover,

λ^¯ı_ε−µ_x0, ψ≤ −

Qρ(x0)

ψ

∇u_ε(x)

dx− µ_x0, ψ +ψ

∇u(x₀)∆_ε+ B_¯ı(ε, ρ) with

B_¯ı(ε, ρ) :=−

Qρ(x0)

ψ

∇u^¯ı_ε(x)

−ψ

∇u_ε(x)dx.

Sinceµ_ε^nar µ, from (13) we have

ρ→0limlim

ε→0

−

Qρ(x0)

ψ

∇u_ε(x)

dx− µ_x0, ψ = 0.

Asψis C-Lipschitz, using (C₂) and (15), we obtain the following estimate:

B_¯ı(ε, ρ)≤Cˆ

1−t^N_p−1/p 1 +

Θ_ε(Q_ρ(x₀))

|Qρ(x₀)|

_1/p

+ 2C

(1−t)ρ−

Qρ(x0)

u−u¯dx with ˆC:= max

C∇u(x₀),(1/α)^1/p

. Taking (14) into account, we see that

ρ→0limlim

ε→0B_¯ı(ε, ρ)≤Cˆ

1−t^N_p−1/p∇u(x₀)+θ(x₀)^1/p

withθ(x₀) given by (12). Thus lim_→+∞lim_t→1lim_ρ→0lim_ε→0 λ^¯ı_ε−µ_x0, ψ= 0.Asηis arbitrary, we deduce that for everyϕ∈C_c(M),

→+∞lim lim

t→1lim

ρ→0lim

ε→0 λ^¯ı_ε, ϕ= µ_x0, ϕ, and the claim follows. We thus have:

– ¯g

∇u(x₀), λ_ρ(t, )

≤lim

ε→0g

∇u(x₀), λ^¯ı_ε

; – ¯g

∇u(x₀), λ_t())

≤lim

ρ→0g¯

∇u(x₀), λ_ρ(t, )

; – ¯g

∇u(x₀), λ

≤lim

t→1g¯

∇u(x₀), λ_t()

; – ¯g

∇u(x₀), µ_x0

≤ lim

→+∞¯g

∇u(x₀), λ . Hence, ¯g

µ_x0,∇u(x₀)

≤lim_→+∞lim_t→1lim_ρ→0lim_ε→0g

λ^¯ı_ε,∇u(x₀)

,and (11) follows from (20).

3.2. Proof of the upper bound

Letµ∈ Y(Ω;M). We have to prove that there exists{µ_ε}ε⊂ Y(Ω;M) such thatµ_ε^nar µand F(µ)≥lim

ε→0Fε(µ_ε).

Without loss of generality we can assume thatF(µ)<+∞.Thusµ∈U(cf. Rem. 2.6), and F(µ) =

Ω

g¯

bar(µ_x), µ_x dx.

We proceed in three steps. For a comprehensive reading we refer to Remark 3.1 before Step 3.