Omar Anza

Jean-Philippe Mandallena
email: jean-philippe.mandallena@unimes.fr

Hamdi Zorgati
email: hamdi.zorgati@fst.rnu.tn

J.-P Mandallena

J.-P Mandallena

Homogenization of unbounded integrals with quasiconvex growth

Keywords: Homogenization, Quasiconvex growth, Unbounded integrals, Ru-usc Mathematics Subject Classification 49J45, 35B27, 74Q05

We study homogenization by -convergence of periodic nonconvex integrals when the integrand has quasiconvex growth with fixed convex effective domain.

Introduction and main result

Let m, d ≥ 1 be two integers and p ∈[1, ∞[.Let ⊂ R d be a nonempty bounded open set with Lipschitz boundary. We consider the periodic homogenization problem of integral functionals by -convergence. More precisely, for each ε>0, we define I ε : W 1, p ( ; R m ) → [0, ∞] by

I ε (u) := ˆ W x ε , ∇u(x) dx,
where the integrand W : R d × M m×d → [0, ∞] is Borel measurable and 1-periodic with respect to the first variable. The homogenization of {I ε } ε>0 consists to show theconvergence with respect to the strong topology of L p as ε → 0 to an homogenized functional integral and to give a representation of the homogenized integrand.

Nonconvex homogenization by -convergence of the family {I ε } ε>0 was mainly studied in the framework of p-polynomial growth conditions on W. Unfortunately, this framework is not compatible with two basic conditions of hyperelasticity: the non-interpenetration of the matter, i.e., W (x, ξ) = ∞ if and only if det(I +ξ) ≤ 0, and the necessity of an infinite amount of energy to compress a finite volume into zero volume, i.e., for every x ∈ R d , W (x, ξ) → ∞ as det(I + ξ) → 0. At present, it seems difficult to take these conditions into account in homogenization problems. Generally, the attempts to go beyond the p-polynomial growth are not easy due to the lack of available techniques. However, in the scalar case, we refer to the book [START_REF] Carbone | Unbounded functionals in the calculus of variations[END_REF] where relaxation and homogenization of unbounded functionals were studied (see also [START_REF] Carbone | Homogenization of unbounded functionals and nonlinear elastomers. The general case[END_REF][START_REF] Carbone | Homogenization of unbounded functionals and nonlinear elastomers. The case of the fixed constraints set[END_REF][START_REF] Cardone | Homogenization of Dirichlet and Neumann problems with gradient constraints[END_REF][START_REF] De Arcangelis | On the relaxation of some classes of pointwise gradient constrained energies[END_REF]). In the vectorial case, i.e., when min{d, m} > 1, the homogenization in W 1,∞ without growth conditions but with W having fixed bounded convex domain is studied in [START_REF] Hafsa | Homogenization of unbounded singular integrals in W 1,∞[END_REF] , and in [START_REF] Hafsa | Homogenization of nonconvex integrals with convex growth[END_REF] the homogenization with convex growth (independent of x) on W is carried out (for the relaxation problems see [START_REF] Hafsa | On the integral representation of relaxed functionals with convex bounded constraints[END_REF][START_REF] Hafsa | On the relaxation of unbounded multiple integrals[END_REF][START_REF] Sychev | First general lower semicontinuity and relaxation results for strong materials[END_REF]).

To go beyond the p-polynomial growth we consider G-growth and p-coercivity conditions on W as follows: (H 1 ) G-growth conditions, i.e., there exist α, β > 0 such that for every x ∈ R d and every

ξ ∈ M m×d αG(ξ ) ≤ W (x, ξ) ≤ β(1 + G(ξ ));
(H 2 ) W is p-coercive, i.e., there exists c > 0 such that for every

(x, ξ) ∈ R d × M m×d c|ξ | p ≤ W (x, ξ),
where G : M m×d → [0, ∞] is a Borel measurable function. Denote by G the effective domain of G, i.e., G = {ξ ∈ M m×d : G(ξ ) < ∞}. We can remark that (H 1 ) implies that the effective domain of W is independent of x and domW (x, •) = G for all x ∈ R d . In [START_REF] Hafsa | Homogenization of nonconvex integrals with convex growth[END_REF] the case G convex with 0 ∈ int(G) and p > d was studied.

In this paper we consider the following conditions on G: (C 1 ) 0 ∈ int(G); (C 2 ) there exists C > 0 such that for every ξ, ζ ∈ M m×d and every t ∈]0, 1[

G(tξ + (1 -t)ζ ) ≤ C(1 + G(ξ ) + G(ζ )); (C 3 ) G is W 1, p -quasiconvex, i.e., for every ξ ∈ M m×d G(ξ ) = inf ˆY G(ξ + ∇ϕ(x))dx : ϕ ∈ W 1, p 0 (Y ; R m ) where Y =]0, 1[ d .
Note that (C 2 ) implies that G is convex, but G is not necessarily convex (see Sect. 9 for an example). The condition (C 2 ) prevents the possible "strong bumps" of G.

We say that W is periodically radially uniformly upper semicontinuous (periodically ruusc) if there exists a ∈ L 1 loc (R d ; ]0, ∞[) 1-periodic such that lim Here is the main result of our paper.

(kY ; R m ) if ξ ∈ G ∞ otherwise.
Theorem 1.1 is an extension of the homogenization result in [START_REF] Hafsa | Homogenization of nonconvex integrals with convex growth[END_REF], to the case where W has quasiconvex growth conditions.

The assumption that W is periodically ru-usc (already in [START_REF] Hafsa | Homogenization of nonconvex integrals with convex growth[END_REF]), allows us to consider a suitable extension (in a radial way) of the homogenized integrand to the boundary ∂G of G. The reason is that the weak limits of the sequences of gradients can be located at ∂G during the homogenization process by -convergence. In fact, we will see that the homogenized integrand HW is nothing but the lower semicontinuous envelope of the Braides-Müller homogenization formula HW (see formula (2.1) and Remark 2.2).

The assumption that p > d (already in [START_REF] Hafsa | Homogenization of nonconvex integrals with convex growth[END_REF]) allows, by using the p-coercivity condition (H 2 ) and Sobolev compact imbedding, to work with the convergence in L ∞ ( ; R m ) instead of L p ( ; R m ). Moreover, the functions of W 1, p ( ; R m ) are almost everywhere differentiable in since Sobolev imbedding. When it is combined with the fact that G has a local upper bound property (see Lemma 4.1) this allows, in both proofs of the lower and upper bound of the -limit (see Step 2 of Sect. 6 and Step 3 of Sect. 7), to obtain suitable bounds in cut-off techniques.

The main difficulty of proving Theorem 1.1 comes from the proof of the upper bound of the -limit. Indeed, in the setting of convex growth conditions on W we can use mollifier techniques to construct approximations of Sobolev functions by smooth ones. However, we need to develop other techniques when we deal with quasiconvex growth. We will consider a set function which is a pointwise limit of local Dirichlet minimization problems associated to the family {I ε } ε>0 and localization arguments introduced by [START_REF] Bouchitté | A global method for relaxation[END_REF] which reduce the proof of the upper bound to cut-off techniques, avoiding then any approximation arguments.

Outline of the paper

In Sect. 2 we present definitions and notations needed in this paper. The proof of Theorem 1.1 is based on two propositions which are stated in this section. Proposition 2.1 is concerned with, first the lower bound of the -limit, and second, with the upper bound of the -limit in the restrictive case where the gradients belong to the interior of the effective domain. Next, we need to extend the homogenized integrand to the boundary of the effective domain, this is the purpose of Proposition 2.2.

In Sect. 3 we show how to recover the classical homogenization theorem with ppolynomial growth in the case p > d from Theorem 1.1.

In Sect. 4 we present some preliminary results needed in the proof of the main result. We first give an analogue property of convex functions for nonconvex integrands satisfying (C 1 ) and (C 2 ). Then, we give the definition and some properties of radially uniformly upper semicontinuous integrands. In Sect. 4.3, we recall some basic facts about subadditive invariant set functions which allow easily to characterize the homogenized formula. Section 4.4 is devoted to the introduction of the pointwise limit of local Dirichlet minimization problems associated to a family of variational functionals.

In Sect. 5 we prove Proposition 2.2.

In Sect. 6 we prove the lower bound for the -limit by the method of localization and cut-off techniques.

In Sect. 7 we prove the upper bound for the -limit for gradients in the interior of the effective domain in three steps. The first step consists in proving that the -limsup is lower than a suitable envelope (similar to a Carathéodory type envelope in measure theory) of a set function given by the pointwise limit of local Dirichlet minimization problems associated to the family {I ε } ε>0 . This envelope turns out to be a nonnegative finite Radon measure by a domination condition coming from the G-growth conditions. Then, the second step is devoted to prove the local equivalence of the envelope with the set function through Radon-Nikodym derivative. In the last step, we use cut-off functions techniques allowing to substitute the Sobolev functions with their affine tangents maps and we conclude by a subadditive argument which gives the homogenized formula.

In Sect. 8 we prove Theorem 1.1 using Propositions 2.1 and 2.2. In Sect. 9 we give an example, when d = m = 2, of W and G satisfying the requirements of the homogenization result. Moreover, we show that the two basic conditions of hyperelasticity can be considered.

Preliminaries

If L : × M m×d → [0, ∞] is a Borel measurable integrand which is 1-periodic with respect to the first variable, i.e., for every x ∈ , z ∈ Z d and ξ ∈ M m×d we have

L(x + z, ξ) = L(x, ξ), then HL : M m×d → [0, ∞] defined by HL(ξ ) := inf k∈N * inf kY L(x, ξ + ∇ϕ(x))dx : ϕ ∈ W 1, p 0 (kY ; R m ) (2.1)
is usually called the Braides-Müller homogenization formula (see [START_REF] Braides | Homogenization of some almost periodic coercive functional[END_REF][START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF]). When L has p-polynomial growth, the corresponding homogenized functional by -convergence with respect to the strong topology of L p ( ; R m ) is given by ˆ

HL(∇u(x))dx for all u ∈ W 1, p ( ; R m ) (see Sect. 3).
Remark 2.1 Some comments concerning the effective domain of HW and G are in order.

(i) If (H 1 ) and (C 3 ) hold then domHW = G. Indeed, by a change of variables and periodicity arguments we see that

inf ϕ∈W 1, p 0 (nY ;R m ) nY G(ξ + ∇ϕ(x))dx = G(ξ )
for any ξ ∈ M m×d and n ∈ N. So, using the G-growth conditions (H 1 ) we obtain

αG(ξ ) ≤ HW (ξ ) ≤ β(1 + G(ξ ))
which implies domHW = G.

(ii) Note that the effective domain G is convex when (C 2 ) holds. Moreover, if both (C 1 ) and (C 2 ) hold, then we have the well-known property

tξ ∈ int(G)
for all t ∈ [0, 1[ and all ξ ∈ G, where G denotes the closure of G.

The homogenization of the family {I ε } ε>0 is achieved by -convergence. We refer to the book of G. Dal Maso [START_REF] Dal Maso | An introduction to -Convergence[END_REF] for a good introduction to the -convergence theory. We give a brief description and specify some notation. We denote by O( ) the set of all open subsets of . Define I -,

I + : W 1, p ( ; R m ) × O( ) → [0, ∞] by I -(u; O) := inf lim ε→0 I ε (u ε ; O) : u ε → u in L p ( ; R m ) ; I + (u; O) := inf lim ε→0 I ε (u ε ; O) : u ε → u in L p ( ; R m ) .
I 0 (•; O) = -lim ε→0 I ε (•; O).
When O = we simply write

I 0 (•) = -lim ε→0 I ε (•).
Note that the functionals I -(•; O) and I + (•; O) are lower semicontinuous with respect to the strong topology of L p ( ; R m ).

We define the radial extension of HW by

HW (ξ ) := lim t→1 - HW (tξ) for all ξ ∈ M m×d . Define G, HI, HI : W 1, p ( ; R m ) × O( ) → [0, ∞] by ♦ G(u; O) := ˆO G(

Application to homogenization with p-polynomial growth

We want to show how to recover, from Theorem 1.1, the classical homogenization result with p-polynomial growth on the integrand for p > d (see [START_REF] Braides | Homogenization of some almost periodic coercive functional[END_REF][START_REF] Müller | Homogenization of nonconvex integral functionals and cellular elastic materials[END_REF]).

Let L : R d × M m×d → [0, ∞] be a Borel measurable, 1-periodic with respect the first variable, and satisfying p-polynomial growth: there exist α, β > 0 such that for every

ξ ∈ M m×d α|ξ | p ≤ L(x, ξ) ≤ β(1 + |ξ | p ). ( 3.1)
For each ε >0 we define

J ε : W 1, p ( ; R m ) → [0, ∞] by J ε (u) := ˆ L x ε , ∇u(x) dx. Theorem 3.1 Let p > d. The family {J ε } ε>0 -converges with respect to the strong topology of L p ( ; R m ) to J 0 : W 1, p ( ; R m ) → [0, ∞] given by J 0 (u) := ˆ HL(∇u(x))dx. (3.2)
Proof Since (3.1) we consider the quasiconvexification of L, denoted by W, given by the Dacorogna formula

W (x, ξ) = QL(x, ξ) := inf ˆY L(x, ξ + ∇ϕ(y))dy : ϕ ∈ W 1,∞ 0 (Y ; R m ) for all (x, ξ) ∈ R d × M m×d . We set G(•) := |•| p .
It is easy to see that G satisfies (C 1 ), (C 2 ), (C 3 ), and W satisfies (H 1 ) and (H 2 ). We show now that W (= QL) is periodically ru-usc. Fix any t ∈ [0, 1], any x ∈ R d and any ξ ∈ G. As W is quasiconvex and satisfies (3.1), there exists K > 0 such that

|W (x, ζ ) -W (x, ζ )| ≤ K |ζ -ζ |(1 + |ζ | p-1 + |ζ | p-1 ) (3.
W (x, tξ) -W (x, ξ) ≤ K (1 -t)(1 + W (x, ξ)) (3.4)
with K := 3K max{1, 1 α }. Dividing by 1 + W (x, ξ) and passing to the supremum in x ∈ R d and ξ ∈ M m×d , we have 1

W (t) ≤ K (1 -t).
Passing to the limit t → 1, we have that W is periodically ru-usc.

Applying Theorem 1.1 to the family {I ε } ε>0 defined by

W 1, p ( ; R m ) u → I ε (u) := ˆ W x ε , ∇u(x) dx.
We obtain that for every

u ∈ W 1, p ( ; R m ) ( -lim ε→0 I ε )(u) = I 0 (u) = ˆ H(Q L)(∇u(x))dx. ( 3.5)
By Remark 2.2, we have

H(Q L) = H(Q L).
Using Dacorogna-Acerbi-Fusco relaxation result [START_REF] Acerbi | Semicontinuity problems in the calculus of variations[END_REF][START_REF] Dacorogna | Quasiconvexity and relaxation of nonconvex problems in the calculus of variations[END_REF] (see "Dacorogna-Acerbi-Fusco" of Appendix section) we have H(Q L) = HL since the p-growth (3.1). Assume for the moment that H(Q L) is lower semicontinuous, i.e.,

H(Q L) = H(Q L) (3.6)
then we have

HL = H(Q L). (3.7)
For each ε >0 we apply the Dacorogna-Acerbi-Fusco relaxation theorem to J ε and we have

J ε (u) = ˆ QL x ε , ∇u(x) dx = I ε (u) for all u ∈ W 1, p ( ; R m ),
where the bar over J ε denotes the lower semicontinuous envelope with respect to the strong topology of L p ( ; R m ). On the other hand, it is well known that

J 0 (u) := ( -lim ε→0 J ε )(u) = ( -lim ε→0 J ε )(u) = I 0 (u) (3.8)
for all u ∈ W 1, p ( ; R m ). Collecting (3.5), (3.7) and (3.8), we finally obtain (3.2).

The only thing which remains to prove is (3.6). Let {ξ ε } ε>0 , ξ 0 ∈ M m×d be such that ξ ε → ξ 0 as ε → 0. Without loss of generality, we can assume that

lim ε→0 H(Q L)(ξ ε ) = lim ε→0 H(Q L)(ξ ε )<∞, and M := sup ε>0 H(Q L)(ξ ε )<∞. Fix ε ∈]0, 1[. We choose k ε ∈ N and ϕ ε ∈ W 1, p (k ε Y ; R m ) such that M + 1 ≥ ε + H(Q L)(ξ ε ) ≥ k ε Y QL(x, ξ ε + ∇ϕ ε (x))dx = ˆY QL(k ε y, ξ ε + ∇ϕ ε (k ε y))dy, where a change of variable is used. Set φ ε (•) := 1 k ε ϕ(k ε •).
It is easy to see that QL satisfies (3.1) with the same constants. So, it follows that

sup ε>0 ˆY |ξ ε + ∇φ ε (x)| p dx ≤ M + 1 c . (3.9)
By Hölder inequality and (3.9) we have for every ε >0

ξ ε + ∇φ ε L p-1 (Y ;R m ) ≤ ξ ε + ∇φ ε L p (Y ;R m ) ≤ M + 1 c 1 p , ( 3.10)
and also by norm inequality

ξ 0 + ∇φ ε L p-1 (Y ;R m ) ≤ |ξ 0 -ξ ε | + ξ ε + ∇φ ε L p-1 (Y ;R m ) (3.11) ≤ |ξ 0 -ξ ε | + M + 1 c 1 p
.

By the definition of

H(Q L)(ξ 0 ), (3.
3), (3.10) and (3.11), we have for every ε >0

ε + H(Q L)(ξ ε ) ≥ ˆY QL(k ε y, ξ ε + ∇φ ε (y)) -QL(k ε y, ξ 0 + ∇φ ε (y))dy + H(Q L)(ξ 0 ) ≥ -K |ξ ε -ξ 0 | 1 + ˆY |ξ ε + ∇φ ε (x)| p-1 + |ξ 0 + ∇φ ε (x)| p-1 dx + H(Q L)(ξ 0 ) ≥ -K |ξ ε -ξ 0 | ⎛ ⎝ 1 + M + 1 c p-1 p + |ξ 0 -ξ ε | + M + 1 c 1 p p-1 ⎞ ⎠ + H(Q L)(ξ 0 ),
letting ε → 0 we obtain the lower semicontinuity of H(Q L).

Auxiliary results

Consequence of assumptions (C 1 ) and (C 2 )

The following lemma is an extension, for nonconvex functions satisfying (C 1 ) and (C 2 ), of the classical local upper bound property for convex functions.

Lemma 4.1 Let L : M m×d → [0, ∞] be a Borel measurable integrand. If L satisfies (C 1 )
and (C 2 ) then there exists ρ 0 > 0 such that

r 0 := sup ζ ∈B ρ 0 (0) L(ζ ) < ∞.
Proof From (C 1 ) there exists ρ 0 > 0 such that L(ξ ) < ∞ for all ξ ∈ B ρ 0 (0). Each matrix ξ ∈ B ρ 0 (0) is identified to the vector ξ = (ξ 11 , . . . , ξ 1d , . . . , ξ i1 , . . . , ξ id , . . . , ξ m1 , . . . , ξ md ) .

Consider the finite subset

S := (ξ 11 , . . . , ξ md ) ∈ M m×d : ξ i j ∈ {-ρ 0 , 0, ρ 0 } ⊂ B ρ 0 (0)
and we define

L * := max ξ ∈S L(ξ ) < ∞. Let ζ = (ζ 11 , . . . , ζ 1d , . . . , ζ i1 , . . . , ζ id , . . . , ζ m1 , . . . , ζ md ) ∈ S and ξ ∈ B ρ 0 (0) with ξ i j = ζ i j for all i = 1 and j = 1. If ξ 11 = 0 then by (C 2 ) we have L(ξ ) =L |ξ 11 | ρ 0 sgn (ξ 11 )ρ 0 + 1 - |ξ 11 | ρ 0 0, . . . , ξ 1d , . . . , ξ m1 , . . . , ξ md (4.1) ≤C (1 + L(ρ 0 , . . . , ξ md ) + L(0, . . . , ξ md )) ≤2C 1 + L *
where sgn (ξ i j ) denotes the sign of ξ i j . The same upper bound in (4.1) holds for L(ξ ) when ξ 11 = 0. Assume now that ξ i j = ζ i j for all i = 1 and j / ∈ {1, 2}. Then using (4.1) and (C 2 ), we have

L(ξ ) =L ξ 11 , |ξ 12 | ρ 0 sgn (ξ 12 )ρ 0 + 1 - |ξ 12 | ρ 0 0, . . . , ξ 1d , . . . , ξ m1 , . . . , ξ md ≤C 1 + 2C(1 + L * ) + L * ≤C(1 + 2C) 1 + L * .
Recursively, we obtain C * > 0 which depends on C only, such that

L(ξ ) ≤ C * (1 + L * )
for all ξ ∈ B ρ 0 (0).

Ru-usc functions

Let U ⊂ R d be an open set and let L : U × M m×d → [0, ∞] be a Borel measurable function. For each x ∈ U , we denote the effective domain of L(x, •) by L x and, for each

a ∈ L 1 loc (U ; ]0, ∞[), we define a L : [0, 1] →] -∞, ∞] by a L (t) := sup x∈U sup ξ ∈L x L(x, tξ) -L(x, ξ) a(x) + L(x, ξ) . Definition 4.1 We say that L is radially uniformly upper semicontinuous (ru-usc) if there exists a ∈ L 1 loc (U ; ]0, ∞[) such that lim t→1 a L (t) ≤ 0.
If moreover a is 1-periodic then we say that L is periodically ru-usc.

For a detailed study of ru-usc functions see [START_REF] Hafsa | Radial representation of lower semicontinuous envelope[END_REF].

Remark 4.1 If L is ru-usc then lim t→1 - L(x, tξ) ≤ L(x, ξ) (4.2)
for all x ∈ U and all ξ ∈ L x . Indeed, given x ∈ U and ξ ∈ L x , we have

L(x, tξ) ≤ a L (t) (a(x) + L(x, ξ)) + L(x, ξ) for all t ∈ [0, 1], which gives (4.2) since a(x) + L(x, ξ) > 0 and lim t→1 a L (t) ≤ 0. Define L : U × M m×d → [0, ∞] by L(x, ξ) := lim t→1 - L(x, tξ).
The following lemma gives some properties of L when L is ru-usc (for the proof see also [START_REF] Hafsa | Homogenization of nonconvex integrals with convex growth[END_REF]Lemma 3.4 and Theorem 3.5 (ii)]).

Lemma 4.2 If L is ru-usc and if for every

x ∈ U , tL x ⊂ L x for all t ∈]0, 1[ (4.3) then (i) for every ξ ∈ L x it holds L(x, ξ) = lim t→1 - L(x, tξ) for all x ∈ U ; (ii) L is ru-usc.
Proof First we prove (i). Fix x ∈ U . We have to prove that for every

ξ ∈ L x lim t→1 - L(x, tξ) = lim t→1 - L(x, tξ). Fix ξ ∈ L x .
(t) = lim n→∞ (t n ); lim t→1 (t) = lim n→∞ (s n ).
It is possible because, once the sequences

{t n } n , {s n } n ⊂]0, 1[ satisfying t n → 1, s n → 1 choosen, we can extract a subsequence {s σ (n) } n such that t n s σ (n) < 1 for all n ∈ N.
Indeed, it suffices to consider the increasing map σ : N → N defined by σ (0) := min{ν ∈ N : s ν > t 0 } and σ (n + 1) := min{ν ∈ N : ν > σ(n) and s ν > t n+1 } for all n ∈ N.

Since (4.3) we have t n ξ ∈ L x for all n ∈ N, so we can assert that for every n ∈ N

(t n ) = t n s n s n = L x, t n s n (s n ξ) (4.5) ≤ a L t n s n (a(x) + (s n )) + (s n ).
Letting n → ∞ we deduce (4.4) from (4.5) since L is ru-usc. It remains to prove (ii), i.e., L is ru-usc. Fix t ∈ [0, 1[ and ξ ∈ L x . By (i) we can assert that

L(x, ξ) = lim s→1 L(x, sξ) L(x, tξ) = lim s→1 L(x, s(tξ )).
So, we have

L(x, tξ) -L(x, ξ) a(x) + L(x, ξ) = lim s→1 L(x, t (sξ)) -L(x, sξ) a(x) + L(x, sξ) ≤ a L (t).
It follows that a L (t) ≤ a L (t). Letting t → 1, we finish the proof.

Assume that U = R d and consider HL given by (2.1). The following result shows that the ru-usc property is stable by homogenization.

♦ φ n ∈ W 1, p 0 (k n Y ; R m ) for all n ≥ 1; ♦ HL(ξ ) = lim n→∞ k n Y L(x, ξ + ∇φ n (x))dx; ♦ ξ + ∇φ n (x) ∈ L x for all n ≥ 1 and a.a. x ∈ k n Y .
Moreover, for every n ≥ 1,

HL(tξ ) ≤ k n Y L(x, t (ξ + ∇φ n (x)))dx since tφ n ∈ W 1, p 0 (k n Y ; R m
), and so

HL(tξ ) -HL(ξ ) ≤ lim n→∞ k n Y L(x, t (ξ + ∇φ n (x))) -L(x, ξ + ∇φ n (x)) dx.
As L is periodically ru-usc it follows that

HL(tξ ) -HL(ξ ) ≤ a L (t) a + HL(ξ )
with a := ´Y a(y)dy, which implies that

a HL (t) ≤ a L (t) for all t ∈ [0, 1]
, and the proof is complete. (i) We say that S is subadditive if

S( A) ≤ S(B) + S(C) for all A, B, C ∈ O b (R d ) with B, C ⊂ A, B ∩ C = ∅ and |A \ B ∪ C| = 0. (ii) We say that S is Z d -invariant if S( A + z) = S( A) for all A ∈ O b (R d ) and all z ∈ Z d . Let Cub(R d ) be the class of all open cubes in R d and let Y :=]0, 1[ d .
The following theorem is due to Akcoglu and Krengel (see [START_REF] Akcoglu | Ergodic theorems for superadditive processes[END_REF], see also [START_REF] Licht | Global-local subadditive ergodic theorems and application to homogenization in elasticity[END_REF], and [START_REF] Hafsa | Homogenization of nonconvex integrals with convex growth[END_REF]Theorem 3.11]).

for all A ∈ O b (R d ). Then, for every Q ∈ Cub(R d ), lim ε→0 S 1 ε Q 1 ε Q = inf k≥1 S(kY ) k d .
Given a Borel measurable function W :

R d × M m×d → [0, ∞], we define for each ξ ∈ M m×d , S ξ : O b (R d ) → [0, ∞] by S ξ (A) := inf ˆA W (x, ξ + ∇φ(x))dx : φ ∈ W 1, p 0 (A; R m ) . ( 4.7)
It is easy to see that the set function S ξ is subadditive. If we assume that W is 1-periodic with respect to the first variable, then

S ξ is Z d -invariant. Moreover, if W is such that there exist a Borel measurable function G : M m×d → [0, ∞] and β > 0 satisfying W (x, ξ) ≤ β(1 + G(ξ )) (4.8)
for all ξ ∈ M m×d , then

S ξ (A) ≤ β(1 + G(ξ ))|A| for all A ∈ O b (R d ).
From the above, we see that the following result is a direct consequence of Theorem 4.1.

Corollary 4.1 Assume that W is 1-periodic with respect to the first variable and satisfies (4.8). Then, for every ξ

∈ G lim ε→0 S ξ 1 ε Q 1 ε Q = inf k≥1 S ξ (kY ) k d .

Local Dirichlet problems associated to a family of functionals

For any family of (variational) functionals

{H δ } δ>0 , H δ : W 1, p ( ; R m ) × O( ) → [0, ∞] we set ♦ m δ (u; O) := inf H δ (v; O) : v ∈ u + W 1, p 0 (O; R m ) ; ♦ M (u; O) := lim δ→0 m δ (u; O), where v ∈ u + W 0 1, p (O; R m ) means that v ∈ W 1, p ( ; R m ) and v -u = 0in \ O (this
i := Q ρ i (x i )} i∈I of disjointed (pairwise disjoint) closed balls of O with x i ∈ O and ρ i ∈]0, ε[ such that |O \ ∪ i∈I Q i | = 0. Let u ∈ W 1, p ( ; R m ) and ε >0. Consider M ε (u; •) : O( ) → [0, ∞] defined by M ε (u; O) := inf i∈I M (u; Q i ) : {Q i } i∈I ∈ V ε (O) ,
and define M * (u; •) : O( ) → [0, ∞] by M * (u; O) := sup ε>0 M ε (u; O) = lim ε→0 M ε (u; O). Lemma 4.3 Assume that for each δ > 0 and each v ∈ W 1, p ( ; R m ) the set function H δ (v; •) is countably subadditive. Let (u, O) ∈ W 1, p ( ; R m ) × O( ) satisfying i∈I sup δ>0 H δ (u; Q i )<∞ (4.9)
for all countable disjointed closed balls

{Q i } i∈I of O satisfying |O \ ∪ i∈I Q i | = 0. Then we have M (u; O) ≤ M * (u; O). (4.10) Proof Fix (u, O) ∈ W 1, p ( ; R m ) × O( ) satisfying (4.9). Fix ε > 0. Choose {Q i } i≥1 ∈ V ε (O) such that i≥1 M (u; Q i ) ≤ M ε (u; O) + ε 2 ≤ M * (u; O) + ε 2 . (4.11)
Fix δ >0. For each i ≥ 1 there exists

ϕ i ∈ W 1, p 0 (Q i ; R m ) such that H δ (u + ϕ i ; Q i ) ≤ δ 2 i+1 + m δ (u; Q i ). (4.12) Set ϕ δ,ε := i≥1 ϕ i I Q i ∈ W 1, p 0 (O; R m ).
By the countable subadditivity of H δ (u + ϕ δ,ε ; •) and (4.12) we have

M (u; O) ≤ lim δ→0 H δ (u + ϕ δ,ε ; O) ≤ lim δ→0 i≥1 H δ (u + ϕ i ; Q i ) ≤ lim δ→0 i≥1 m δ (u; Q i ). (4.13)
But, for every δ >0 and every i ≥ 1 it holds

sup η∈]0,δ[ m η (u; Q i ) ≤ sup η>0 H η (u; Q i ). (4.14)
Applying the dominated convergence theorem and using (4.9) together with (4.14), we have

lim δ→0 i≥1 m δ (u; Q i ) ≤ i≥1 lim δ→0 m δ (u; Q i ) = i≥1 M (u; Q i ). ( 4
G : W 1, p ( ; R m )×O( ) → [0, ∞] be such that G(u, O) < ∞ and G(v, •) is a measure for all v ∈ W 1, p ( ; R m ).
If there exists β >0 such that for every δ >0 it holds

H δ (u; U ) ≤ β (|U | + G(u; U ))
for all U ∈ O(O), then (4.9) is satisfied. Indeed, we have

i∈I sup δ>0 H δ (u; Q i ) ≤ i∈I β (|Q i | + G(u; Q i )) = β (|O| + G(u; O)) < ∞.
The following result is needed for the proof of Lemma 7.

M (u; Q) ≤ μ u (Q)
then M * (u; •) can be extended to a Radon measure λ u on satisfying 0 ≤ λ u ≤ μ u .

Proof of Proposition 2.2

The function HW is ru-usc since Proposition 4.1, and so HW is ru-usc by Lemma 4.2 (ii). Since Remark 2.1 (i) we have domHW = G. It is easy to deduce that dom HW = G. From Lemma 4.2 (i) it holds that

HW (ξ ) = lim t→1 - HW (tξ) for all ξ ∈ G.
The proof is complete.

Proof of Proposition 2.1 (i)

Let O ∈ O( ) and let u ∈ W

Consider {u ε } ε>0 ⊂ W 1, p ( ; R m ) satisfying u ε -u L p ( ;R m ) → 0.
u ε -u L ∞ ( ;R m ) → 0. (6.5)
Step 1: Localization

For each ε > 0, we define the nonnegative Radon measure μ ε on O by

μ ε := W • ε , ∇u ε (•) dx O .
From (6.2) we see that sup ε μ ε (O) < ∞, and so there exists a Radon measure μ on O such that (up to a subsequence) μ ε * μ. By Lebesgue's decomposition theorem, we have μ = μ a + μ s where μ a and μ s are nonnegative Radon measures such that μ a dx O and μ s ⊥ dx O , and from Radon-Nikodym's theorem we deduce that there exists f ∈ L 1 (O; [0, ∞[), given by

f (x) = lim ρ→0 μ a (Q ρ (x)) ρ d = lim ρ→0 μ(Q ρ (x)) ρ d a.
ρ > 0 such that Q ρ (x) ⊂ O \ spt (μ s ) with Q ρ (x) := x + ρY . Thus, for a.a. x ∈ , μ(Q ρ (x)) = μ a (Q ρ (x)
) for all ρ > 0 sufficiently small.

To prove (6.1) it suffices to show that f (x) ≥ HW (∇u(x)) a.e. in O. (6.7) Indeed, by Alexandrov theorem (see "Alexandrov theorem" of Appendix section) we see that

lim ε→0 I ε (u ε ; O) = lim ε→0 μ ε (O) ≥ μ(O) = μ a (O) + μ s (O) ≥ μ a (O) = ˆ f (x)dx.
But, by (6.

G(t * ∇u(x 0 )) < ∞; (6.11) lim ρ→0 1 ρ u -u x 0 L ∞ (Q ρ (x 0 );R m ) = 0. (6.12)
Note that G(t * ∇u(•)) < ∞ a.e. in O since Remark 2.1 (ii) and ∇u(•) ∈ G a.e. in O. Note also that u is almost everywhere differentiable, i.e., lim

ρ→0 1 ρ u -u x L ∞ (Q ρ (x);R m ) = 0 a.e. in O since p > d (where u x (•) := u(x) + ∇u(x)(• -x) is the affine tangent map of u at x ∈ O).
We have to prove that f (x 0 ) ≥ HW (∇u(x 0 )). As μ(O) < ∞ we have μ(∂ Q ρ (x 0 )) = 0 for all ρ ∈]0, 1] \ D where D is a countable set. From (6.6) and Alexandrov theorem (see "Alexandrov theorem" of Appendix section) we deduce that

f (x 0 ) = lim ρ→0 μ(Q ρ (x 0 )) ρ d = lim ρ→0 lim ε→0 μ ε (Q ρ (x 0 )) ρ d ,
and so we are reduced to show that lim

ρ→0 lim ε→0 Q ρ (x 0 ) W x ε , ∇u ε (x) dx ≥ HW (∇u(x 0 )). (6.13)
Using ru-usc property of W we can see that

lim ρ→0 lim ε→0 Q ρ (x 0 ) W x ε , ∇u ε (x) dx ≥ lim t→1 - lim ρ→0 lim ε→0 Q ρ (x 0 ) W x ε , t∇u ε (x) dx.
So to prove (6.13), it is enough to show that lim

t→1 - lim ρ→0 lim ε→0 Q ρ (x 0 ) W x ε , t∇u ε (x) dx ≥ HW (∇u(x 0 )). (6.14)
Step 2: Cut-off technique to substitute tu ε with tv ε ∈ tu x 0 +W (6.15) and if τ := t t * ∈]0, 1[ then

1, p 0 (Q ρ (x 0 );R m ) Fix any ε >0 and any s ∈]0, 1[. Let φ ∈ W 1,∞ 0 (Q ρ (x 0 ); [0, 1]) be a cut-off function between Q sρ (x 0 ) and Q ρ (x 0 ) such that ∇φ L ∞ (Qρ(x 0 )) ≤ 4 ρ(1-s) . Setting v ε := φu ε + (1 -φ)u x 0 where u x 0 (•) := u(x 0 ) + ∇u(x 0 )(• -x 0 ), we have tv ε ∈ tu x 0 + W 1, p 0 (Q ρ (x 0 ); R m ),
t∇v ε := t∇u ε on Q sρ (x 0 ) τ (φt * ∇u ε + (1 -φ)t * ∇u(x 0 )) + (1 -τ ) ε,ρ on S ρ (6.16) with S ρ := Q ρ (x 0 ) \ Q sρ (x 0 ) and ε,ρ := t 1-τ ∇φ ⊗ u ε -u x 0 . Using the G-growth conditions (H 1 ) we have Q ρ (x 0 ) W x ε , t∇v ε dx ≤ Q sρ (x 0 ) W x ε , t∇u ε dx + 1 ρ d ˆSρ W x ε , t∇v ε dx ≤ Q ρ (x 0 ) W x ε , t∇u ε dx + β(1 -s d ) + β ρ d ˆSρ G(t∇v ε )dx.
On the other hand, taking (6.16) into account and using (C 2 ), we have

G(t∇v ε ) ≤2C 1 1 + G(t * ∇u ε ) + G(t * ∇u(x 0 )) + G( ε,ρ ) ≤2C 1 1 + 1 α W x ε , t * ∇u ε + G(t * ∇u(x 0 )) + G ε,ρ a.e. in S ρ with C 1 := C 2 + C. Moreover, it is easy to see that ε,ρ L ∞ (Q ρ (x 0 );M m×d ) ≤ 4t (1 -τ )(1 -s) 1 ρ u -u x 0 L ∞ (Q ρ (x 0 );R m ) + 4t ρ(1 -τ )(1 -s) u ε -u L ∞ ( ;R m )
where lim

ρ→0 4t (1 -t)(1 -s) 1 ρ u -u x 0 L ∞ (Q ρ (x 0 );R m ) = 0 (6.17)
since (6.12), i.e., lim ρ→0

1 ρ u -u x 0 L ∞ (Q ρ (x 0 );R m ) = 0, and lim ε→0 4t ρ(1 -t)(1 -s) u ε -u L ∞ ( ;R m ) = 0 for all ρ > 0 (6.18)
since (6.5), i.e., lim

ε→0 u ε -u L ∞ ( ;R m ) = 0
. By Lemma 4.1 we have for some ρ 0 > 0

r 0 := sup ξ ∈B ρ 0 (0) G(ξ ) < ∞.
By (6.17) there exists ρ > 0 such that

4t (1-t)(1-s) 1 ρ u -u x 0 L ∞ (Q ρ (x 0 );R m ) < ρ 0 2 for all ρ ∈]0, ρ[.
Fix any ρ ∈]0, ρ[. Taking (6.18) into account we can assert that there exists ε ρ > 0 such that for every

ε ∈]0, ε ρ [ G ε,ρ ≤ r 0 a.e. in Q ρ (x 0 ).
Thus, for every ε ∈]0, ε ρ [, we have

Q ρ (x 0 ) W x ε , t∇v ε dx ≤ Q ρ (x 0 ) W x ε , t∇u ε dx + 1 -s d β + C 1 + r 0 +G(t * ∇u(x 0 )) + C α 1 ρ d ˆSρ W x ε , t * ∇u ε dx (6.19)
where 2βC 1 := C. Since W is periodically ru-usc, for every ε ∈]0, ε ρ [ we have the estimate for the last term of (6.19) shown as follows

1 ρ d ˆSρ W x ε , t * ∇u ε dx ≤ a W (t * ) 1 ρ d ˆSρ a x ε dx + 1 + a W (t * ) 1 ρ d μ ε (S ρ ). (6.20)
Step 3: End of the proof Taking (6.15) into account we see that for every ε ∈]0, ε ρ [

Q ρ (x 0 ) W x ε , t∇v ε dx ≥ 1 | 1 ε Q ρ (x 0 )| S t∇u(x 0 ) 1 ε Q ρ (x 0 ) ,
where S ξ (A) is given by (4.7) for all ξ ∈ M m×d and all open set A ⊂ R d . By (6.9) we have ∇u(x 0 ) ∈ G, and so t∇u(x 0 ) ∈ G because G is convex and 0 ∈ int(G) since (C 1 ) and (C 2 ).

From Corollary 4.1 we deduce that lim

ε→0 Q ρ (x 0 ) W x ε , t∇v ε dx ≥ HW (t∇u(x 0 )) (6.21)
for all ρ ∈]0, ρ[. On the other hand, as μ ε (S ρ ) ≤ μ ε (S ρ ) for all ε ∈]0, ε ρ [, S ρ is compact and μ ε * μ, we have lim ε→0 μ ε (S ρ ) ≤ μ(S ρ ) by Alexandrov theorem. But μ(S ρ ) = μ a (S ρ )

since S ρ ⊂ Q ρ (x 0 ) ⊂ \ spt (μ s ) (see Remark 6.1). Hence, for every ρ ∈]0, ρ[,

lim ε→0 1 ρ d μ ε (S ρ ) ≤ 1 ρ d ˆSρ f (x)dx = Q ρ (x 0 ) f (x)dx -s d Q sρ (x 0 ) f (x)dx,
and consequently

lim ρ→0 lim ε→0 1 ρ d μ ε (S ρ ) ≤ -s d ) f 0 ). (6.22)
Taking (6. [START_REF] Dal Maso | An introduction to -Convergence[END_REF]) and ( 6.20) into account, from (6.21) and ( 6.22) we deduce that

HW (t∇u(x 0 )) ≤ lim ε→0 Q ρ (x 0 ) W x ε , t∇u ε dx + 1 -s d β + C 1 + r 0 +G(t * ∇u(x 0 ) + 1 α a W (t * ) a + f (x 0 ) .
Taking (6.10), (6.11) and (6.8) into account and passing to the limits ρ → 0 and s → 1, we obtain

HW (t∇u(x 0 )) ≤ lim ρ→0 lim ε→0 Q ρ (x 0 ) W x ε , t∇u ε dx,
and (6.14) follows when t → 1.

Proof of Proposition 2.1 (ii)

For each

(u, O) ∈ W 1, p ( ; R m ) × O( ) we recall that m ε (u; O) := inf I ε (v; O) : v ∈ W 1, p 0 (O; R m ) and M (u; O) := lim ε→0 m ε (u; O).
We give a sketch of the proof which is divided into three steps. The first step consists in proving that

I + (u; O) ≤ M * (u; O) for all (u, O) ∈ W 1, p ( ; R m ) × O( ).
When we assume that G(u; O) < ∞, Lemma 4.4 and the G-growth conditions imply that M * (u; •) is a Radon measure which is absolutely continuous with respect to the Lebesgue measure on O. Thus, we can write

I + (u; O) ≤ M * (u; O) = ˆO lim ρ→0 M * (u; Q ρ (x)) ρ d dx. ( 7.1)
The second step consists in showing that M * (u; •) is locally equivalent to M (u; •), i.e., for a.a.

x ∈ O lim ρ→0 M * (u; Q ρ (x)) ρ d = lim ρ→0 M (u; Q ρ (x)) ρ d . ( 7.2)
This is carrying out by measure theoretic arguments (see Step 2).

In the third and last step we replace u by tu with t ∈]0, 1[ and we show, using cut-off techniques, that for a.a.

x ∈ O lim ρ→0 M (tu; Q ρ (x)) ρ d ≤ lim s→1 lim ρ→0 M (tu x ; Q sρ (x)) (sρ) d (7.3)
where u x (•) := u(x) + ∇u(x)(•x). The right hand term of (7.3) is equal to HW (t∇u(x)). Indeed, it is easy to see that for any ε, ρ > 0 and any x ∈ O we can write

m ε (tu x ; Q sρ (x)) (sρ) d = 1 1 ε Q sρ (x) inf ˆ1 ε Q sρ (x) W (y, ∇v(εy))dy : v ∈ tu x + W 1, p 0 (Q sρ (x); R m ) = 1 1 ε Q sρ (x) inf ˆ1 ε Q sρ (x) W (y, t∇u(x 0 ) + ∇φ(εy))dy : φ ∈ W 1, p 0 (Q sρ (x); R m ) = 1 1 ε Q sρ (x) S t∇u(x) 1 ε Q sρ (x)
which give

M (tu x ; Q sρ (x)) (sρ) d = HW (t∇u(x))
since a subadditive argument (see Corollary 4.1). The proof is achieved by taking (7.2) and (7.3) into account in the inequality (7.1)

I + (tu; O) ≤ ˆO lim ρ→0 M * (tu; Q ρ (x)) ρ d dx = ˆO lim ρ→0 M (tu; Q ρ (x)) ρ d dx ≤ ˆO lim s→1 lim ρ→0 M (tu x ; Q sρ (x)) (sρ) d dx = ˆO HW (t∇u(x))dx.
Step 1:

Prove that I + (u; O) ≤ M * (u; O) when G(u; O)<∞ Fix (u, O) ∈ W 1, p ( ; R m ) × O( ) such that G(u; O) < ∞.
Without loss of generality we assume that

M * (u; O)<∞. (7.4) Fix ε ∈]0, 1[. Choose {Q i } i∈I ∈ V ε (O) such that i∈I M (u; Q i ) ≤ M ε (u; O) + ε 2 ≤ M * (u; O) + ε 2 . (7.5) Fix δ ∈]0, 1[. Given any i ∈ I there exists v i ∈ u + W 1, p 0 (Q i ; R m ) such that I δ (v i ; Q i ) ≤ m δ (u; Q i ) + δ 2 |Q i | |O| (7.6) by definition of m δ (u; Q i ). Define u δ,ε ∈ u + W 1, p 0 (O; R m ) by u δ,ε := i∈I v i I Q i + uI \ ∪ i∈I Q i .
From (7.6) we have that

I δ (u δ,ε ; O) = i∈I I δ (v i ; Q i ) ≤ i∈I m δ (u; Q i ) + δ 2 .
Letting δ → 0 we obtain

lim δ→0 I δ (u δ,ε ; O) ≤ lim δ→0 i∈I m δ (u; Q i ). ( 7.7)
By the G-growth conditions (H 1 ) we have sup

η∈]0,δ[ m η (u; Q i ) ≤ β(|Q i | + G(u; Q i )) for all δ >0 and all i ∈ I with i∈I β(|Q i | + G(u; i )) = β(|O| + G(u; O)) < ∞,
then applying the dominated convergence theorem we have lim

δ→0 i∈I m δ (u; Q i ) ≤ i∈I M (u; Q i ). ( 7.8) Therefore collecting (7.5), (7.7), (7.8
) and passing to the limit ε → 0, we have

lim ε→0 lim δ→0 I δ (u δ,ε ; O) ≤ M * (u; O). ( 7.9)
From the p-coercivity of W (H 2 ), (7.9) and (7.4), we deduce

lim ε→0 lim δ→0 ˆO |∇u δ,ε | p dx < ∞. (7.10)
By Poincaré inequality there exists K > 0 depending only on p and d such that for each

v i ∈ u + W 1, p 0 (Q i ; R m ) ˆQi |v i -u| p dx ≤ K ε p ˆQi |∇v i -∇u| p dx since diam(Q i )<ε. Summing on i ∈ I we obtain ˆO |u δ,ε -u| p dx ≤ 2 p-1 K ε p ˆO |∇u δ,ε | p dx + ˆO |∇u| p dx which shows that lim ε→0 lim δ→0 ˆ |u δ,ε -u| p dx = 0 (7.11)
since (7.10). A simultaneous diagonalization of (7.9) and (7.11) gives a sequence

{u δ := u δ,ε(δ) } δ>0 ⊂ u + W 1, p 0 (O; R m ) such that u δ → u in L p ( ; R m ) and I + (u; O) ≤ lim δ→0 I δ (u δ ; O) ≤ M * (u; O)
since the definition of I + (u; O). The proof is complete.

Step 2: Prove that M * (u; •) is locally equivalent to M (u; •)

In this step we use the following result from [START_REF] Bouchitté | Regularization of a set function: application to integral representation[END_REF][START_REF] Bouchitté | A global method for relaxation[END_REF]. For a sake of completeness we give a proof.

Lemma 7.1 If G(u; O)<∞ then we have

lim ρ→0 M * (u; Q ρ (x 0 )) ρ d = lim ρ→0 M (u; Q ρ (x 0 )) ρ d x 0 -a.e. in O. Proof Let u ∈ W 1, p ( R m ) be such that G(u; O)<∞. Then for each U ∈ O(O) M (u; U ) ≤ lim ε→0 ˆU W x ε , ∇u(x) dx ≤ β |O| + ˆO G(∇u)dx < ∞, so, using Lemma 4.4 with u := β (| • | + G(∇u(•))dx)) O , we have M (u; •) is the trace of a Radon measure λ u on O satisfying 0 ≤ λ u ≤ μ u .
M * (u; Q ρ (x 0 )) ρ d ≥ lim ρ→0 M (u; Q ρ (x 0 )) ρ d x 0 -a.e. in O.
It remains to prove that

lim ρ→0 M * (u; Q ρ (x 0 )) ρ d ≤ lim ρ→0 M (u; Q ρ (x 0 )) ρ d x 0 -a.e. in O. ( 7.12)
Fix any θ >0. Consider the following sets

G θ := Q ρ (x) : x ∈ O, ρ > 0 and M * (u; Q ρ (x)) > M (u; Q ρ (x)) + θ Q ρ (x) , N θ := x ∈ O : ∀δ >0 ∃ρ ∈]0, δ[ Q ρ (x) ∈ G θ .
It is sufficient to prove that N θ is a negligible set for the Lebesgue measure on O. Indeed, given x 0 ∈ O \ N θ there exists δ 0 > 0 such that

M * (u; Q ρ (x 0 )) ≤ M (u; Q ρ (x 0 )) + θ Q ρ (x 0 ) for all ρ ∈]0, δ 0 [. Hence lim ρ→0 M * (u; Q ρ (x 0 )) Q ρ (x 0 ) ≤ lim ρ→0 M (u; Q ρ (x 0 )) Q ρ (x 0 ) + θ,
then we obtain (7.12) when θ → 0. Fix δ >0. Consider the set

F δ := Q ρ (x) : x ∈ N θ , ρ ∈]0, δ[ and Q ρ (x) ∈ G θ .
Using the definition of N θ we can see that inf

Q∈F δ diam (Q) = 0.
By the Vitali covering theorem there exists a (pairwise) disjointed countable subfamily {Q i } i≥1 of F δ such that

N θ \ ∪ i≥1 Q i = 0. (7.13) We have N θ ⊂ ∪ i≥1 Q i ∪ N θ \ ∪ i≥1 Q i .
To prove that N θ is a negligible set is equivalent to prove that |V j | = 0 for all j ≥ 1 where

V j := j ∪ i=1 Q i . Fix j ≥ 1. Let {Q i } i≥1 ∈ V δ O\ ∪ j i=1 Q i satisfying i≥1 M (u; Q i ) ≤ M * u; \ j ∪ i=1 Q i + δ. (7.14)
Recalling that M * (u; •) is the trace on O(O) of a nonnegative finite Radon measure, we see that

M * (u; O) ≥M * u; O \ j ∪ i=1 Q i + M * u; V j =M * u; O \ j ∪ i=1 Q i + 1≤i≤ j M * (u; Q i ).
Since each Q i ∈ G θ , by (7.14) we have

M * (u; O) ≥ i≥1 M (u; Q i ) -δ + j i=1 M (u; Q i ) + θ |V j |.
It is easy to see that the countable family

{Q i : i ≥ 1} ∪ {Q i : 1 ≤ i ≤ j} belongs to V δ (O), thus M * (u; O) ≥ M δ (u; O) + θ|V j | -δ.
Letting δ → 0, we have M δ (u; O) → M * (u; O), and so |V j | = 0 since θ >0.

Step 3: Cut-off technique to locally substitute tu with tu x 0 in

M (•;Q ρ (x 0 )) ρ d Fix (u, O) ∈ W 1, p ( ; R m ) × O( ). Let t ∈]0,
; Q sρ (x 0 )) (sρ) d = lim s→1 lim ρ→0 lim ε→0 1 1 ε Q sρ (x 0 ) S t∇u(x 0 ) 1 ε Q sρ (x 0 ) = HW (t∇u(x 0 )) x 0 -a.e. in O.
since t∇u(•) ∈ G a.e. in O by (7.15). We are reduced to prove (7.17). Consider

x 0 ∈ O satisfying lim ρ→0 Q ρ (x 0 ) G(t∇u(x))dx = G(t∇u(x 0 )) < ∞; (7.18) lim ρ→0 Q ρ (x 0 ) G(t * ∇u(x))dx = G(t * ∇u(x 0 )) < ∞. (7.19) Fix s ∈]0, 1[, ρ >0 and ε >0. Choose v ∈ u x 0 + W 1, p 0 (Q sρ (x 0 ); R m satisfying I ε (tv; Q sρ (x 0 )) ≤ m ε (tu x 0 ; Q sρ (x 0 )) + ρ d+1 . Consider a cut-off function φ ∈ W 1,∞ 0 (Q ρ (x 0 ); [0, 1]) between Q sρ (x 0 ) and Q ρ (x 0 ) such that ∇φ L ∞ (Q ρ (x 0 )) ≤ 4 (1-s)ρ . Define w := φv + (1 -φ)u which belongs to u + W 1, p 0 (Q ρ (x 0 ); R m ). We have m ε (tu; Q ρ (x 0 )) ≤ I ε (tv; Q sρ (x 0 )) + I ε (tw; Q ρ (x 0 ) \ Q sρ (x 0 )) (7.20) ≤ m ε (tu x 0 ; Q sρ (x 0 )) + ρ d+1 + I ε (tw; Q ρ (x 0 ) \ Q sρ (x 0 )).
Let us estimate the last term of (7.20) divided by

ρ d . Set τ := tt * -1 , ρ := t 1-τ ∇φ ⊗ (u x 0 -u) and S ρ := Q ρ (x 0 ) \ Q sρ (x 0 ).
I ε (tw; S ρ ) ρ d ≤ β lim ρ→0 (1 -s d ) + 1 ρ d ˆSρ G τ (φt * ∇u(x 0 ) + (1 -φ)t * ∇u) + (1 -τ ) ρ dx ≤ C 1 lim ρ→0 (1 -s d ) + 1 ρ d ˆSρ G(φt * ∇u(x 0 ) + (1 -φ)t * ∇u)dx + 1 ρ d ˆSρ G( ρ )dx ≤ C 2 lim ρ→0 (1 -s d )(1 + G(t * ∇u(x 0 ))) + 1 ρ d ˆSρ G(t * ∇u)dx + 1 ρ d ˆSρ G( ρ )dx ≤ 2C 2 (1 -s d )(1 + G(t * ∇u(x 0 ))) + lim ρ→0 1 ρ d ˆSρ G( ρ )dx . (7.21)
where

C 1 := β(1 + C) and C 2 := C 1 (1 + C).
Since (6.12), we choose ρ >0 such that for every

ρ ∈]0, ρ[ it holds 4t (1 -τ )(1 -s) 1 ρ u x 0 -u L ∞ (Q ρ (x 0 );R m ) ≤ ρ 0 2 . It follows that ρ L ∞ (Q ρ (x 0 );M m×d ) ≤ ρ 0 .
Using Lemma 4.

(x 0 )) ρ d ≤ s d lim ρ→0 M (tu x 0 ; Q sρ (x 0 )) (sρ) d +2C 2 (1 -s d )(1+G(t * ∇u(x 0 ))+r 0 ).
Letting s → 1, we finally find (7.17 Let u ∈ W 1, p ( ; R m ). The proof of the lower bound is already done in Proposition 2.1 (i), so it remains to prove the upper bound I + (•) ≤ HI (•).

Assume without loss of generality that HI (u)<∞. First, we assume that ∇u(•) ∈ int(G) a.e. in . In this case, from Proposition 2.2, coercivity conditions (H 1 ) and (C 2 ) we have

HI (u) = ˆ lim t→1 - HW (t∇u(x))dx ≥ α ˆ lim t→1 - G(t∇u(x))dx.
But G is W 1, p -quasiconvex, so it is continuous in int(G) (see for instance [START_REF] Fonseca | The lower quasiconvex envelope of the stored energy function for an elastic crystal[END_REF] or [START_REF] Dacorogna | Direct Methods in the Calculus of Variations[END_REF]).

HW (t∇u(x))dx = HI (u).

Using the lower semicontinuity of I + with respect to the strong topology of L p ( ; R m ), we then have

I + (u) ≤ lim t→1 - I + (tu) ≤ lim t→1 - HI (tu) ≤ HI (u).
Now, we assume the general case ∇u(•) ∈ G a.e. in . We have for every t ∈]0, 1[ that t∇u(•) ∈ int(G) a.e. in since G is convex with 0 ∈ int(G) (see Remark 2.1 (ii)). We can apply the first part of the proof to get I + (tu) ≤ HI (tu) for all t ∈]0, 1[. But HW is ru-usc since Proposition 2.2, so, for every t ∈]0, 1[

HI (tu) ≤ a HW (t) a | | + HI (u) + HI (u).
Letting t → 1 and using the lower semicontinuity of I + , we obtain the desired result.

A two-dimensional example

We show, when m = d = 2, how to construct an example of W with G-growth conditions satisfying the assumptions of Theorem 1.1. We want to point out that the following discussed example is not covered by [START_REF] Hafsa | Homogenization of nonconvex integrals with convex growth[END_REF], indeed the growth considered here are not convex see Remark 9.3.

Consider a set G ⊂ M 2×2 with the following properties:

(A 1 ) 0 ∈ int(G); (A 2 ) G is convex; (A 3 ) det(I + ξ)>0 for all ξ ∈ G; (A 4 ) tr (cof(I + ξ) (I + ζ )) > 0 for all ξ, ζ ∈ G,
where I is the identity matrix and cof(F) is the matrix of cofactors of F ∈ M 2×2 .

Remark 9.1 The set I + G can be interpreted as internal constraints of an elastic material. However, the properties of I + G do not fit with the requirements of the theory of internal constraints as developed by [START_REF] Gurtin | The thermodynamics of constrained materials[END_REF]. Indeed, due to the frame indifference principle, we should have

SO(2)(I + G) ⊂ I + G, (9.1
cof(I + ξ) (I + ζ ) = tr (I + ζ ) ≤ 0, so (A 4 ) cannot be satisfied. Let g : M 2×2 → [0, ∞] be defined by g(ξ ) := h(det(I + ξ)) if ξ ∈ G ∞ otherwise where h :]0, ∞[→ [0, ∞] is a nonincreasing convex function satisfying for every λ ∈]0, 1[ and every x ∈]0, ∞[ h(λx) ≤ 1 λ r h(x) (9.2)
where r ≤ 1. Note that the function h can be chosen to satisfy lim x→0 h(x) = ∞. The following proposition shows that such a W is consistent with the assumptions of Theorem 1.1 as well as with the two basic conditions of hyperelasticity, i.e., the noninterpenetration of the matter and the necessity of an infinite amount of energy to compress a finite volume of matter into zero volume. As is quasiconvex and satisfies (9.6), then repeating the same arguments as in the proof of Theorem 3.1 (Sect. 3) we see that is periodically ru-usc with a ≡ 1.

On the other hand, as g is ru-usc by Proposition 9.1 (iii) we have g(tξ)g(ξ ) ≤ We can take g : M 2×2 → [0, ∞] defined by

g(ξ ) := h(det(I + ξ)) if ξ ∈ G ∞ otherwise
where h is the convex and nonincreasing function defined by h(x) := 1 x for all x > 0 satisfying (9.2) with r = 1.

Remark 9.3 It is easy to see that g(•) := g(• -I ) is polyconvex but not convex. Indeed, consider F ∈ I + G defined by

F := 1 1 2 -1 2 1 then g 1 2 F + 1 2 F = 1 > 4 5 = 1 2 ( g(F) + g(F )).
Remark 9.4 A necessary condition for g to be frame indifferent is that P(I + G) ⊆ I + G for all P ∈ SO(2), which in particular means that SO(2) ⊆ I + G since (A 1 ). But, this is not true because the rotation of angle π 2 does not belong to I + G.

W

  (t) := sup x∈U sup ξ ∈domW (x,•) W (x, tξ) -W (x, ξ) a(x) + W (x, ξ)where domW (x, •) is the effective domain of W (x, •) (see Sect. 4.2).

  For any O ∈ O( ), the functional I -(•; O) (resp. I + (•; O)) is called the -liminf (resp. the -limsup) with respect to the strong topology of L p ( ; R m ) of the family {I ε (•; O)} ε>0 . Note that we always have I + (•; O) ≥ I -(•; O). When I + (•; O) = I -(•; O) we say that the family {I ε (•; O)} ε>0 -converges with the -limit given by the common value and we write

3 )

 3 for all x ∈ R d and all ζ, ζ ∈ M d×d . Using (3.3) with ζ = tξ and ζ = ξ and taking the left inequality in (3.1) into account, we obtain

4. 3

 3 Subadditive theorem Let O b (R d ) be the class of all bounded open subsets of R d . Definition 4.2 Let S : O b (R d ) → [0, ∞] be a set function.

Theorem 4 . 1

 41 Let S : O b (R d ) → [0, ∞] be a subadditive and Z d -invariant set function for which there exists c > 0 such that S( A) ≤ c|A| (4.6)

). 8 Proof of Theorem 1. 1

 81 To shorten notation we set G(•; ) := G(•), HI (•; ) := HI (•), and HI (•; ) := HI (•).

Proposition 9 . 1

 91 We have (i) g is polyconvex where g(•) := g(• -I ); (ii) for every ξ, ζ ∈ G and everyλ ∈]0, 1[ it holds g(λξ + (1 -λ)ζ ) ≤ g(ξ ) + g(ζ ); (iii) g is ru-usc. Proof We have (i) because we can write g(F) = ϕ(F, det(F)) with ϕ : M 2×2 ×R → [0, ∞] is the convex function defined by ϕ(F, s) := h(s) if F ∈ I + G ∞ otherwise.Now, we show (ii). Fix ξ, ζ ∈ G and λ ∈]0, 1[. Using (A 3 ), (A 4 ) and properties of h, we haveg(λξ + (1 -λ)ζ )) = h(det(λ(I + ξ) + (1 -λ)(I + ζ ))) = h(λ 2 det(I + ξ) + (1 -λ) 2 det(I + ζ ) + λ(1 -λ)tr cof(I + ξ) (I + ζ ) ) ≤ h(λ 2 det(I + ξ) + (1 -λ) 2 det(I + ζ )) ≤ λh(λ det(I + ξ)) + (1 -λ)h((1 -λ) det(I + ζ ))≤ λ 1-r h(det(I + ξ)) + (1 -λ)

Proposition 9 . 2

 92 We have (i) W is 1-periodic with respect to the first variable; (ii) W satisfies (H 1 ) and (H 2 ) with G given by (9.4);(iii) for every (x, ξ) ∈ R d × G, W(x, ξ) < ∞ if and only if det(I + ξ) > 0; (iv) if lim x→0 h(x) = ∞ then for every x ∈ R d , W(x, ξ) → ∞ as det(I + ξ) → 0; (v) W is periodically ru-usc.Proof The only not direct property is (v). Fix any t ∈ [0, 1], any x ∈ R d and any ξ ∈ G.

  Assume that p > d. Assume that (C 1 ), (C 2 ), (C 3 ), (H 1 ) and (H 2 ) hold. If W is periodically ru-usc then {I ε } ε>0 -converges with respect to the strong topology of L p ( ; R m ) to I 0 : W 1, p ( ; R m ) → [0, ∞] given by

			ˆ
			I 0 (u) =	HW (∇u(x))dx
	with HW (ξ ) =	⎧ ⎨ ⎩ t→1 -lim	inf k∈N * inf	1, p 0

Theorem 1.1 kY W (x, tξ + ∇ϕ(x))dx : ϕ ∈ W

  The following proposition gives some properties of HW and HW when W is periodically ru-usc. HW where the bar denotes the lower semicontinuous envelope of HW . Indeed, first it is easy to see that HW ≥ HW . On the other hand, HW is ru-usc by Proposition 2.2. By Remark 4.1 we have HW ≤ HW on domHW and domHW = G by Remark 2.1 (i). So it follows that HW ≤ HW . But Theorem 1.1 implies that HW is lower semicontinuous, i.e., HW = HW , and then HW ≤ HW .

	(ii) Let t ∈]0, 1[. If G(tu; O) < ∞ and if there exists t * ∈]t, 1[ such that G(t * u; O) < ∞
	then
		I + (tu; O) ≤ HI (tu; O).
	Proposition 2.2 Assume that (C 1 ), (C 2 ), (C 3 ) and (H 1 ) hold. If W is periodically ru-usc
	then HW and HW are ru-usc. Moreover, we have
		HW (ξ ) = lim t→1 -	HW (tξ)
	for all ξ ∈ M m×d .
	Remark 2.2 Under the assumptions of Theorem 1.1 it holds that
		∇u(x))dx;
	♦ HI (u; O) := ˆO HW (∇u(x))dx;
	ˆO
	♦ HI (u; O) :=	HW (∇u(x))dx.
	The proof of Theorem 1.1 is based on the following result.
	(i) If W is periodically ru-usc then
		I -(u; O) ≥ HI (u; O).

Proposition 2.1 Assume that p > d. Assume that (C 1 ), (C 2 ), (H 1 ) and (H 2 ) hold. Let u ∈ W 1, p ( ; R m ) and O ∈ O( ). HW =

  AH96, Chap. 9, p. 233]). It is easy to see that we may also write m δ (u; O) = inf H δ (u + ϕ; O) : ϕ ∈ W R m ) for all δ >0 and all u ∈ W 1, p ( ; R m ).For each ε > 0 and each O ∈ O( ), denote by V ε (O) the class of all countable family {Q

	1, p 0 (O;

definition is equivalent to the classical definition of u + W 0 1, p (O; R m ),seefor instance [

  1. Let u ∈ W 1, p ( ; R m ). If there exists a finite Radon measure μ u on such that for every cube Q ∈ O( )

	Lemma 4.4 ([10] and [9, Prop. 2.1.])

  Since μ u is absolutely continuous with respect to dx O the Lebesgue measure on O, the limit lim ρ→0 ∈ O as the Radon-Nikodym derivative of λ u with respect to dx O . Moreover, using Lemma 4.3, the G-growth conditions together with Remark 4.2 we have lim ρ→0

	λ u (Q ρ (x 0 )) ρ d	exists for
	a.a. x 0

  Using the G-growth conditions, (C 2 ) and (7.19) we have

	lim ρ→0	lim ε→0

  ≤ 1-t 2r t 2r . Letting t → 1 we obtain (iii). We may think that the function g could be convex but it is not the case in general, see Remark 9.3.We define the functionG : M 2×2 → [0, ∞] by Let W : R 2 × M 2×2 → [0, ∞] be defined by W (x, ξ) := (x, ξ) + g(ξ ) if ξ ∈ G ∞ otherwise,(9.5)where : R 2 × M 2×2 → [0, ∞] is a quasiconvex function, 1-periodic with respect to its first variable and of p-polynomial growth, i.e., there exist c, C > 0 such thatc|ξ | p ≤ (x, ξ) ≤ C(1 + |ξ | p ) (9.6)for all (x, ξ) ∈ R 2 × M 2×2 .

	≤	1 t 2r h(det(I + ξ))
	=	1 t 2r h(det(I + ξ)) -h(det(I + ξ)) + g(ξ )
	≤	1 -t 2r t 2r (1 + g(ξ )) + g(ξ )
	which implies 1 g (t) Remark 9.2

1-r

h(det(I + ζ )) ≤ g(ξ ) + g(ζ ). (

9.3)

From (9.3) and properties of h we have for every ξ ∈ G and every t ∈]0, 1[

g(tξ) = h(t 2 det(I + ξ) + (1t) 2 + t (1t)tr (I + ξ )) ≤ h(t 2 det(I + ξ)) G(ξ ) := |ξ | p + g(ξ ) if ξ ∈ G ∞ otherwise. (

9

.4) Using (A 1 ), Proposition 9.1 (i) and (ii) it is easy to see Lemma 9.1 The function G defined in (9.4) satisfies (C 1 ), (C 2 ) and (C 3 ).

  + ξ 22 ) -ξ 12 ξ 21 > |ξ 12 ξ 21 | -ξ 12 ξ 21 ≥ 0. To verify (A 4 ), we note that for every ξ, ζ ∈ G it holds tr cof(I + ξ) (I + ζ ) = (1 + ξ 11 ) (1 + ζ 11 ) + (1 + ξ 22 ) (1 + ζ 11 ) -ξ 12 ζ 21 -ξ 21 ζ 12 > |ξ 12 ζ 21 | + |ξ 21 ζ 12 | -ξ 12 ζ 21 -ξ 21 ζ 12 ≥ 0.

				1 g (t)(1 + g(ξ )).	(9.7)
	From (3.4) and (9.7) we deduce that
	Passing to the supremum on x and ξ , we obtain
	sup x∈R d	sup ξ ∈G	W (x, tξ) -W (x, ξ) 2 + W (x, ξ)	≤ max	2 (t), 1 g (t) ,
	and (v) follows when t → 1.
	Concrete example
	Every ξ ∈ M 2×2 is denoted by
			ξ :=	ξ 11 ξ 12 ξ 21 ξ 22
	Define the set

W (x, tξ) -W (x, ξ) ≤ max 2 (t), 1 g (t) (2 + W (x, ξ)). G := ξ ∈ M 2×2 : min {1 + ξ 11 , 1 + ξ 22 } > max {|ξ 12 |, |ξ 21 |} , Property (A 1 ) is evident. The subset G is open and convex as intersection of open convex sets, so (A 2 ) holds. The assertion (A 3 ) is satisfied because for every ξ ∈ G we have det(I + ξ) = (1 + ξ 11 ) (1

Acknowledgments

The first author would like to thank the University of Tunis El Manar for the invitation to their Department of Mathematics in May 2012 where this work was initiated. The authors would also like to thank the anonymous referees for their valuable comments and suggestions to improve the manuscript.

Appendix

Let ⊂ R d be an open bounded set with Lipschitz boundary.

be a Borel measurable integrand with p-polynomial growth, i.e., there exist c, C > 0 such that for every (x, ξ) ∈ × M m×d it holds

Then for every u ∈ W 1, p ( ; R m )

where Q f : × M m×d → [0, ∞[ is the quasiconvexification of f given by the Dacorogna formula

As a consequence we have inf

(9.9)

The same equalities (9.8) and (9.9) hold when replacing W 1, p ( ; R m ) by W