PERIODIC HOMOGENIZATION OF INTEGRAL ENERGIES UNDER SPACE-DEPENDENT DIFFERENTIAL CONSTRAINTS

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1 PERIODIC HOMOGENIZATION OF INTEGRAL ENERGIES UNDER SPACE-DEPENDENT DIFFERENTIAL CONSTRAINTS ELISA DAVOLI AND IRENE FONSECA Abstract. A homogenization result for a family of oscillating integral energies u ε fx, x, uεx dx, ε 0+ ε is presented, where the fields u ε are subjected to first order linear differential constraints depending on the space variable x. The work is based on the theory of A -quasiconvexity with variable coefficients and on two-scale convergence techniques, and generalizes the previously obtained results in the case in which the differential constraints are imposed by means of a linear first order differential operator with constant coefficients. The identification of the relaxed energy in the framework of A -quasiconvexity with variable coefficients is also recovered as a corollary of the homogenization result. 1. Introduction In this paper we continue the study of the problem of finding an integral representation for limits of oscillating integral energies u ε f x, x ε α, u εx dx, where f : R N R d [0, + has standard p-growth, R N is a bounded open set, ε 0, and the fields u ε : R d are subjected to x dependent differential constraints of the type A i x uε x ε β 0 strongly in W 1,p ; R l, 1 < p < +, 1.1 or in divergence form A i x ε β u ε x 0 strongly in W 1,p ; R l, 1 < p < +, 1.2 with A i x LinR d ; R l for every x R N, i = 1,, N, d, l 1, and where α, β are two nonnegative parameters. Different regimes are expected to arise, depending on the relation between α and β. We recently analyzed in [10] the limit case in which α = 0, β > 0, the energy density is independent of the first two variables, and the fields {u ε are subjected to 1.2. We will consider here the case in which α > 0, β = 0 and 1.1, i.e., the 2010 Mathematics Subject Classification. 49J45; 35D99; 49K20. Key words and phrases. Homogenization, two-scale convergence, A quasiconvexity. 1

2 2 E. DAVOLI AND I. FONSECA energy density is oscillating but the differential constraint is fixed and in nondivergence form. The situation in which there is an interplay between α and β will be the subject of a forthcoming paper. The key tool for our study is the notion of A quasiconvexity with variable coefficients, characterized in [20]. A quasiconvexity was first investigated by Dacorogna in [8] and then studied by Fonseca and Müller in [12] in the case of constant coefficients see also [9]. More recently, in [20] Santos extended the analysis of [12] to the case in which the coefficients of the differential operator A depend on the space variable. In order to illustrate the main ideas of A -quasiconvexity, we need to introduce some notation. For i = 1, N, consider matrix-valued maps A i C R N ; M l d, where for l, d N, M l d stands for the linear space of matrices with l rows and d columns, and for every x R N define A as the differential operator such that A u := A i x ux, x 1.3 for u L 1 loc ; Rd, where u is to be interpreted in the sense of distributions. We require that the operator A satisfies a uniform constant-rank assumption see [18], i.e., there exists r N such that rank A i xw i = r for every w S n 1, 1.4 uniformly with respect to x, where S N 1 is the unit sphere in R N. The definitions of A -quasiconvex function and A -quasiconvex envelope in the case of variable coefficients read as follows: Definition 1.1. Let f : R d R be a Carathéodory function, let be the unit cube in R N centered at the origin, = 1 2 2, 1 N, and denote by CperR N ; R d the set of smooth maps which are -periodic in R N. For every x consider the set { C x := w CperR N ; R d : wy dy = 0, A i x wy = y i For a.e. x and ξ R d, the A quasiconvex envelope of f in x is defined as { A fx, ξ := inf fx, ξ + wy dy : w C x. f is said to be A -quasiconvex if fx, ξ = A fx, ξ for a.e. x and ξ R d. Denote by A c a generic differential operator, defined as in 1.3 and with constant coefficients, i.e. such that A i x A i c for every x R N, with A i c M l d, i = 1,, N. We remark that when A = A c = curl, i.e., when v = φ for some φ W 1,1 loc ; Rm, and if is connected, then d = m N, and A -quasiconvexity reduces to Morrey s notion of quasiconvexity see [1, 4, 15, 17].

3 HOMOGENIZATION FOR A x UASICONVEXITY 3 The first identification of the effective energy associated to periodic integrands evaluated along A c -free fields was provided in [5], by Braides, Fonseca and Leoni. Their homogenization results were later generalized in [11], where Fonseca and Krömer worked under weaker assumptions on the energy density f. Recently, Matias, Morandotti, and Santos extended the previous results to the case p = 1 [16], whereas Kreisbeck and Krömer performed in [13] simultaneous homogenization and dimension reduction in the framework of A c -quasiconvexity. This paper is devoted to extending the results in [11] to the framework of A quasiconvexity with variable coefficients. To be precise, in [11] the authors studied the homogenized energy associated to a family of functionals of the type F ε u ε := f x, x ε, u εx dx, where is a bounded, open subset of R N, u ε u weakly in L p ; R d and the sequence {u ε satisfies a differential constraint of the form A c u ε = 0 for every ε. We analyze the analogous problem in the case in which A depends on the space variable and the differential constraint is replaced by the condition A u ε 0 strongly in W 1,p ; R l. Our analysis leads to a limit homogenized energy of the form: { E hom u := f homx, ux dx if A u = 0, + otherwise in L p ; R d, where f hom : R d [0, + is defined as f hom x, ξ := lim inf inf fx, ny, ξ + vy dy. n + v C x Our main result is the following. Theorem 1.2. Let 1 < p < +. Let A i C perr N ; M l d, i = 1,, N, and assume that A satisfies the constant rank condition 1.4. Let f : R N R d R be a function satisfying fx,, ξ is measurable, 1.6 f, y, is continuous, 1.7 fx,, ξ is periodic, fx, y, ξ C1 + ξ p for all x, ξ R d, and for a.e. y R N. 1.9 Then for every u L p ; R d there holds { inf lim inf f x, x ε 0 ε, u εx dx : u ε u weakly in L p ; R d and A u ε 0 strongly in W 1,p ; R l { = inf lim sup f x, x ε 0 ε, u εx dx : u ε u weakly in L p ; R d and A u ε 0 strongly in W 1,p ; R l = E hom u.

4 4 E. DAVOLI AND I. FONSECA As in [10] and [11], the proof of this result is based on the unfolding operator, introduced in [6, 7] see also [21, 22]. In contrast with [11, Theorem 1.1] i.e. the case in which A = A c, here we are unable to work with exact solutions of the system A u ε = 0, but instead we consider sequences of asymptotically A vanishing fields. This is due to the fact that for A -quasiconvexity with variable coefficients we do not project directly on the kernel of the differential constraint, but construct an approximate projection operator P such that for every field v L p, the W 1,p norm of A P v is controlled by the W 1,p norm of v itself for a detailed explanation we refer to [20, Subsection 2.1]. In [10] the issue of defining a projection operator was tackled by imposing an additional invertibility assumption on A and by exploiting the divergence form of the differential constraint. We do not add this invertibility requirement here, instead we use the fact that in our framework the differential operator depends on the macro variable x but acts on the micro variable y see 1.5. Hence it is possible to define a pointwise projection operator Πx along the argument of [12, Lemma 2.14] see Lemma 4.1. As a corollary of our main result we recover an alternative proof of the relaxation theorem [5, Theorem 1.1] in the framework of A quasiconvexity with variable coefficients, that is we obtain the identification see Corollary 4.10 A fx, ux dx = Iu, D D for every open subset D of, and for every u L p ; R d satisfying A u = 0, where the functional I is defined as { Iu, D := inf lim inf fx, u ε x : 1.10 ε 0 D u ε u weakly in L p ; R m and A u ε 0 strongly in W 1,p ; R l. We point out here that a proof of this relaxation theorem follows directly combining [5, Proof of Theorem 1.1] with the arguments in [20]. The interest in Corollary 4.10 lies in the fact that it is obtained as a by-product of our homogenization result, and thus by adopting a completely different proof strategy. In analogy to [10] one might expect to be able to apply an approximation argument and extend the results in Theorem 4.3 to the situation in which A i W 1, R N ; M l d, i = 1, N, which is the least regularity assumption in order for A to be well defined as a differential operator from L p to W 1,p. We were unable to achieve this generalization, mainly because the projection operator here plays a key role in the proof of both the liminf and the limsup inequalities. In order to work with approximant operators A k having smooth coefficients, we would need to finally construct an approximate projection operator P associated to A, whereas the projection argument provided in [20] applies only to the case of smooth differential constraints. The article is organized as follows. In Section 2 we establish the main assumptions on the differential operator A and we recall some preliminary results on two-scale convergence. In Section 3 we recall the definition of A -quasiconvex envelope and we construct some examples of A quasiconvex functions. Section 4 is

5 HOMOGENIZATION FOR A x UASICONVEXITY 5 devoted to the proof of our main result. Notation Throughout this paper, R N is a bounded open set, O is the set of open subsets of, denotes the unit cube in R N centered at the origin and with normals to its faces parallel to the vectors in the standard orthonormal basis of R N, {e 1,, e N, i.e., = 1 2 2, 1 N. Given 1 < p < +, we denote by p its conjugate exponent, that is 1 p + 1 p = 1. Whenever a map u L p, C,, is periodic, that is ux + e i = ux i = 1,, N for a.e. x R N, we write u L p per, C per,, respectively. We will implicitly identify the spaces L p and L p perr N. We will designate by, the duality product between W 1,p and W 1,p 0. We adopt the convention that C will denote a generic constant, whose value may change from expression to expression in the same formula. 2. Preliminary results In this section we introduce the main assumptions on the differential operator A and we recall some preliminary results about A quasiconvexity and two-scale convergence Preliminaries. For i = 1,, N, consider the matrix-valued functions A i C R N ; M l d. For 1 < p < + and u L p ; R d, we set A u := A i x ux W 1,p ; R l. For every x 0 and u L p ; R d we define A x 0 u := A i x 0 ux W 1,p ; R l. We will also consider the operators and A x w := A y w := A i wx, y x A i wx, y x y i for every w L p ; R d. Finally, for every x 0 and for w L p ; R d, we set A x x 0 w := A i wx, y x 0

6 6 E. DAVOLI AND I. FONSECA and A y x 0 w := A i wx, y x 0. y i For every x R N, λ R N \ {0, let Ax, λ be the linear operator Ax, λ := A i xλ i M l d. We assume that A satisfies the following constant rank condition: N rank A i xλ i = r for some r N and for all x R N, λ R N \ { For every x R N, λ R N \ {0, let Px, λ : R d R d be the linear projection on Ker Ax, λ, and let x, λ : R l R d be the linear operator given by x, λax, λξ := ξ Px, λξ for all ξ R d, x, λξ = 0 if ξ / Range Ax, λ. The main properties of P, and, are stated in the following proposition see [20, Subsection 2.1]. Proposition 2.1. Under the constant rank condition 2.1, for every x R N the operators Px, and x, are, respectively, 0 homogeneous and 1 homogeneous. Moreover, P C R N R N \ {0; M d d and C R N R N \ {0; M d l Two-scale convergence. We recall here the definition and some properties of two-scale convergence. For a detailed treatment of the topic we refer to, e.g., [2, 14, 19]. Throughout this subsection 1 < p < +. Definition 2.2. If v L p ; R d and {u ε L p ; R d, we say that {u ε weakly two-scale converge to v in L p ; R d 2 s, u ε v, if u ε x ϕ x, x dx vx, y ϕx, y dy dx ε for every ϕ L p ; C perr N ; R d. We say that {u ε strongly two-scale converge to v in L p ; R d, u ε 2 s v, if u ε 2 s v and lim u ε Lp ;R ε 0 d = v Lp ;R. d Bounded sequences in L p ; R d are pre-compact with respect to weak two-scale convergence. To be precise see [2, Theorem 1.2], Proposition 2.3. Let {u ε L p ; R d be bounded. Then, there exists v L p ; R d 2 s such that, up to the extraction of a non relabeled subsequence, u ε v weakly two-scale in L p ; R d, and, in particular u ε vx, y dy weakly in L p ; R d. The following result will play a key role in the proof of the limsup inequality see [11, Proposition 2.4, Lemma 2.5 and Remark 2.6].

7 HOMOGENIZATION FOR A x UASICONVEXITY 7 Proposition 2.4. Let v L p ; C per R N ; R d or v L p perr N ; C; R d. Then, the sequence {v ε defined as v ε x := v x, x ε is p equiintegrable, and v ε 2 s v strongly two-scale in L p ; R d The unfolding operator. We collect here the definition and some properties of the unfolding operator see e.g. [7, 6, 21, 22]. Definition 2.5. Let u L p ; R d. For every ε > 0, the unfolding operator T ε : L p ; R d L p R N ; L p perr N ; R d is defined componentwise as x T ε ux, y := u ε + εy y for a.e. x and y R N, 2.2 ε where u is extended by zero outside and denotes the least integer part. The next proposition and the subsequent theorem allow to express the notion of two-scale convergence in terms of L p convergence of the unfolding operator. Proposition 2.6. see [7, 22] T ε is a nonsurjective linear isometry from L p ; R d to L p R N ; R d. The following theorem provides an equivalent characterization of two-scale convergence in our framework see [22, Proposition 2.5 and Proposition 2.7],[14, Theorem 10]. Theorem 2.7. Let be an open bounded domain and let v L p ; R d. Assume that v is extended to be 0 outside. Then the following conditions are equivalent: 2 s i u ε v weakly two scale in L p ; R d, ii T ε u ε v weakly in L p R N ; R d. Moreover, u ε 2 s v strongly two scale in L p ; R d if and only if T ε u ε v strongly in L p R N ; R d. The following proposition is proved in [11, Proposition A.1]. Proposition 2.8. For every u L p ; R d extended by 0 outside, as ε 0. u T ε u L p R N ;R d 0 3. A -quasiconvex functions In this section we recall the notion of A -quasiconvexity and A -quasiconvex envelope, and we provide some examples of A -quasiconvex functions in the case in which A has variable coefficients. We start by recalling the main definitions when A = A c, where A c is a first order differential operator with constant coefficients, that is, for every u L p ; R d, A c ux := A i c ux W 1,p ; R l,

8 8 E. DAVOLI AND I. FONSECA with A i c M l d for i = 1,, N. Definition 3.1. Let f : R d [0, + be a Carathéodory function, let A c be a first order differential operator with constant coefficients, and consider the set { C const := w CperR N ; R d : wy dy = 0 and A i wy c = 0. y i The A c -quasiconvex envelope of f is the function A c f : R d [0, +, given by { A c fx, ξ := inf fx, ξ + wy dy : w C const, 3.1 for a.e. x and for all ξ R d. We say that f is A c -quasiconvex if fx, ξ = A c fx, ξ for a.e. x and for all ξ R d. Similarly, in the case in which A depends on the space variable, the definitions of A -quasiconvex envelope and A -quasiconvex function read as in Definition 1.1. We stress that A -quasiconvexity and pointwise A x-quasiconvexity are related by the following fixed point relation: A fx, ξ = A x fx, ξ for a.e. x and for all ξ R d. 3.2 The remaining part of this section is devoted to illustrating these concepts with some explicit examples of A -quasiconvex functions. We first exhibit an example where A -quasiconvexity reduces to A c -quasiconvexity for a suitable operator A c with constant coefficients. Example 3.2. Let 1 < p < + and define fx, ξ := axbξ for a.e x and every ξ R d, with a L p and b CR d L p R d. In order for f to be A -quasiconvex, the function b must satisfy { bξ = inf bξ + wy dy : w x C x. Consider the case in which C x is the same for every x, for example when the differential constraint is provided by the operator: A wx := MxA i wx c, where M : M l l and det Mx > 0 for every x. In this case, { C x = w CperR N ; R d : wy dy = 0 and A c w = 0 for every x, where A c wx := A i wx c. Hence, f is A -quasiconvex if and only if b is A c -quasiconvex.

9 HOMOGENIZATION FOR A x UASICONVEXITY 9 In the previous example, A -quasiconvexity could be reduced to A c -quasiconvexity owing to the fact that the class C x was constant in x. We provide now an example where an analogous phenomenon occurs, despite the fact that C x varies with respect to x. To be precise, we consider the case in which A is a smooth perturbation of the divergence operator. In this situation, the A -quasiconvex envelope of f coincides with its convex envelope. Example 3.3. We consider a smooth perturbation of the divergence operator in a set R 2, that is ax 0 A u := u for every u L p ; R 2, 1 < p < +, 0 1 with a C and We notice that and therefore ker Ax, λ = 1 2 ax < 1 for every x. { ξ R 2 : axλ 1 ξ 1 + λ 2 ξ 2 = 0, rank Ax, λ = 1 for every x and λ R 2 \ {0. In this situation, the class C x depends on x, since we have { C x = w CperR 2 ; R 2 : wy dy = 0 and ax w 1y y 1 + w 2y = 0, y 2 although λ S 1 ker Ax, λ = R 2. Let now f : R d [0, + be a continuous map. By [12, Proposition 3.4], it follows that A fξ coincides with the convex envelope of f evaluated at ξ, exactly as in the case of the divergence operator see [12, Remark 3.5 iv]. We conclude this section with an example in which the notion of A -quasiconvexity can not be reduced to A c -quasiconvexity with respect to a constant operator A c. Example 3.4. Here we consider a smooth perturbation of the curl operator in a set R 2. Let A : L p ; R 2 W 1,p ; R 4 be given by A u = a 1 x u 2 x 1 + u 1 x 2, where a 1 C is not constant, and satisfies We first notice that ker Ax, λ := = 1 2 a 1x 1 for every x. {ξ R 2 : λ 1 a 1 xξ 2 = λ 2 ξ 1 { ξ R 2 : ξ 2 = αλ 2 and ξ 1 = αa 1 xλ 1, α R, hence rank Ax, λ = 3 for every x, λ S 1. The class C x depends on x and there holds { C x = w CperR 2 ; R 2 : wy dy = 0 and a 1 x w 2y y 1 = w 1y y 2 3.3

10 10 E. DAVOLI AND I. FONSECA { = w CperR 2 ; R 2 : wy dy = 0, w 1 y = a 1 x ϕy, y 1 and w 2 y = ϕy, where ϕ C y perr 2. 2 Let now g : R 2 [0, + be a quasiconvex function and let f : R 2 [0, + be defined as ξ1 fx, ξ := g x, a 1 x, ξ 2 for a.e. x and for every ξ R 2. We claim that A fx, ξ = fx, ξ for a.e. x and for every ξ R 2. Indeed, by 3.3 there holds fx, ξ + wy dy inf w C x { = inf { = inf = g x, f x, ξ1 g x, ξ1 a 1 x, ξ 2 ξ 1 + a 1 x ϕy y 1 a 1 x + ϕy y 1, ξ 2 + ϕy dy : ϕ C y perr 2 2, ξ 2 + ϕy y 2 dy : ϕ C perr 2 for a.e. x and for every ξ R 2, where g denotes the quasiconvex envelope of the function g. The claim follows by the definition of f and the quasiconvexity of g. 4. A homogenization result for A -free fields In this section we prove a homogenization result for oscillating integral energies under weak L p convergence of A vanishing maps. Fix 1 < p < + and consider a function f : R N R d [0, + satisfying In analogy with the case of constant coefficients see [11, Definition 2.9], we define the class of A -free fields as the set F := {v L p ; R d : A y v = 0 and A x vx, y dy = 0, 4.1 where both the previous differential conditions are in the sense of W 1,p. We aim at obtaining a characterization of the homogenized energy { inf lim inf f x, x ε 0 ε, u εx dx : u ε u weakly in L p ; R d 4.2 and A u ε 0 strongly in W 1,p ; R l. We start with a preliminary lemma, which will allow us to define a pointwise projection operator. We will be using the notation introduced in Sections 2 and 3. Lemma 4.1. Let 1 < p < +. Let A i C perr N ; M l d, i = 1, N, and assume that the associated first order differential operator A satisfies 2.1. Then, for every x there exists a projection operator Πx : L p ; R d L p ; R d

11 HOMOGENIZATION FOR A x UASICONVEXITY 11 such that P1 Πx is linear and bounded, and vanishes on constant maps, P2 Πx Πxψy = Πxψy and A y xπxψy = 0 in W 1,p ; R l for a.e. x, for every ψ L p ; R d, P3 there exists a constant C = Cp > 0, independent of x, such that ψy Πxψy Lp ;R d C A y xψy W 1,p ;R l for a.e. x, for every ψ L p ; R d with ψy dy = 0, P4 if {ψ n is a bounded p-equiintegrable sequence in L p ; R d, then {Πxψ n y is a p-equiintegrable sequence in L p ; R d, P5 if ϕ C 1 ; CperR N ; R d then the map ϕ Π, defined by ϕ Π x, y := Πxϕx, y for every x and y R N, satisfies ϕ Π C 1 ; C perr N ; R d. Proof. For every x, let Πx be the projection operator provided by [12, Lemma 2.14]. Properties P1 and P2 follow from [12, Lemma 2.14]. In order to prove P3, fix x and ψ CperR N ; R d. Let A, P and be the operators defined in Subsection 2.1. Writing the operator Πx explicitly, we have Πxψy := Px, λ ψλe 2πiy λ, where ψλ := λ Z N \{0 ψye 2πiy λ dy, for every λ Z N \ {0 are the Fourier coefficients associated to ψ. By the -1-homogeneity of the operator see Proposition 2.1 we deduce ψy Πxψy p = x, λax, λ ψλe 2πiy λ p 4.3 = λ Z N \{0 1 λ x, λ Z N \{0 λ λ Ax, λ ψλe 2πiy λ p. For 1 < p < 2, by the smoothness of and by applying first Hölder s inequality and then Hausdorff-Young inequality, we obtain the estimate ψy Πxψy p 4.4 p 1 max x, λ λ Z N, λ =1 λ p Ax, λ ψλ p p p C λ Z N \{0 λ Z N \{0 λ Z N \{0 Ax, λ ψλ p p p C A y xψy p L p ;R d, where we used the fact that Ax, λ ψλ C Ay xψyλ 4.5 for every x and λ Z N \ {0, by the definition of the Fourier coefficients, and where both constants in 4.4 and 4.5 are independent of λ and x. Consider now the case in which p 2. By 4.3 we have ψy Πxψy p 4.6

12 12 E. DAVOLI AND I. FONSECA max λ Z N, λ =1 C C λ Z N \{0 sup λ Z N \{0 p x, λ Ax, λ ψλ p λ Z N \{0 Ax, λ ψλ p 2 1 p p λ p λ Z N \{0 λ Z N \{0 Ax, λ ψλ 2. Ax, λ ψλ p By the definition of Fourier coefficients and by Hölder s inequality we have Ax, λ ψλ C A y xψy Lp ;R l for every x and λ Z N \ {0. In view of [12, Theorem 2.9], Ax, λ ψλ 2 = A y xψy 2 L 2 ;R l. λ Z N \{0 Therefore by 4.6, applying again Hölder s inequality, ψy Πxψy p 4.7 C A y xψy p 2 L p ;R l A yxψy 2 L 2 ;R l C A yxψy p L p ;R l where the constant C is independent of x and y. Property P3 follows by 4.4 and 4.7 via a density argument. P4 follows directly from P3, arguing as in the proof of [12, Lemma 2.14 iv]. Let now ϕ C 1 ; CperR N ; R d. The regularity of the map ϕ Π is a direct consequence of Proposition 2.1, the definition of Π and the regularity of A. Indeed, ϕ Π x, y := Px, λ ϕx, λe 2πiy λ, 4.8 λ Z N \{0 for every x and y R N, where ϕx, λ := ϕx, ξe 2πiξ λ dξ for every x and λ Z N \ {0. By the regularity of ϕ and by [12, Theorem 2.9] we obtain the estimate 4π 2 λ 2 ϕx, λ N ϕ x, y C y i L 2 ;R d λ Z N \{0 for every x, hence by Proposition 2.1 and Cauchy-Schwartz inequality there holds Px, λ ϕx, λe 2πiy λ C ϕx, λ 4.9 λ Z N \{0, λ n C λ Z N \{0, λ n C λ Z N \{0, λ n λ Z N \{0, λ n ϕx, λ 2 λ λ λ Z N \{0, λ n 1 λ By 4.9 the series in 4.8 is uniformly convergent, and hence ϕ Π is continuous. The differentiability of ϕ Π follows from an analogous argument.

13 HOMOGENIZATION FOR A x UASICONVEXITY 13 For every v L p ; R d, let { S v := {u ε L p ; R d 2 s : u ε v weakly two-scale in L p ; R d 4.10 and A u ε 0 strongly in W 1,p ; R l. Let also S := { S v : v L p ; R d We provide a characterization of the set S. Lemma 4.2. Let v L p ; R d. Let A be a first order differential operator with variable coefficients, satisfying 2.1. The following conditions are equivalent: C1 v F see 4.1; C2 S v is nonempty. Proof. We first show that C2 implies C1. Let v L p ; R d and let {u ε S v. Consider a test function ψ W 1,p 0 ; R l. Then On the other hand, A u ε, ψ := A u ε, ψ 0 as ε 0. A i x u ε x ψx + A i xu ε x ψx dx 4.12 and by Proposition 2.3, up to the extraction of a not relabeled subsequence, u ε vx, y dy weakly in L p ; R d Passing to the limit in 4.12 yields A x vx, y dy = 0 in W 1,p ; R l. In order to deduce the second condition in the definition of F, we consider a sequence of test functions { εϕ x ε ψx, where ϕ W 1,p 0 ; R d and ψ Cc. Since this sequence is uniformly bounded in W 1,p 0 ; R d, we have A u ε, εϕ ψ 0 ε as ε 0, where A u ε, εϕ ε = ε ψ A i x x x ψx u ε x ϕ ψx + A i xu ε x ϕ dx ε ε A i xu ε x ϕ x ψx dx. ε

14 14 E. DAVOLI AND I. FONSECA Passing to the subsequence of {u ε extracted in 4.13, the first line of the previous expression converges to zero. By the definition of two-scale convergence, the second line converges to and thus for a.e. x, that is A i xvx, y ϕy ψx dy dx, y i A i vx, x, ϕ y = 0 W 1,p ;R l,w 1,p i 0 ;R l This completes the proof of C1. A y v = 0 in W 1,p ; R l for a.e. x. Assume now that C1 holds true, i.e., v F. In order to construct the sequence {u ε, set v 1 x, y = vx, y vx, z dz. We first assume that v 1 C 1 ; CperR 1 N ; R d. Defining u ε x := vx, y dy + v 1 x, x for a.e. x, ε 2 s by Proposition 2.4 we have u ε v strongly two-scale in L p ; R d. Moreover, by the definition of F and Propositions 2.3 and 2.4, A i x u εx = = A i x vx, y dy + v 1 x, x ε A i x v x, x ε A i x vx, y dy = 0 weakly in L p ; R l. Hence v satisfies C2. In the general case in which v 1 L p ; R d, we first need to approximate v 1 in order to keep the periodicity condition during the subsequent regularization. To this purpose, we extend v 1 to 0 outside, we consider a sequence {ϕ j Cc such that 0 ϕ j 1 and ϕ j 1 pointwise, and we define the maps v j 1 x, y := ϕ jyv 1 x, y for a.e. x and y. Extend these maps to R N by periodicity. It is straightforward to see that v j 1 v 1 strongly in L p ; L p perr N ; R d 4.14 by the dominated convergence theorem. Moreover Indeed, by 4.14, and since A y v = 0, and A y v j 1 W 1,p ;R l 0 strongly in L p A y xv j 1 x, W 1,p ;R l 0 for a.e. x, A y xv j 1 x, p W 1,p ;R l C ϕ j p L v 1x, y p L p ;R d C v 1x, y p L p ;R d

15 HOMOGENIZATION FOR A x UASICONVEXITY 15 for a.e. x. Thus 4.15 follows by the dominated convergence theorem. Convolving first with respect to y and then with respect to x we construct a sequence {v δ,j 1 C ; C perr N ; R d such that v δ,j 1 v j 1 strongly in L p ; L p perr N ; R d, 4.16 and A y v δ,j 1 W 1,p ;R l 0 strongly in L p, 4.17 as δ 0. In view of , a diagonal argument provides a subsequence {δj such that {v δj,j 1 satisfies v δj,j 1 v 1 strongly in L p ; L p perr N ; R d 4.18 and A y v δj,j 1 W 1,p ;R l 0 strongly in L p Set w j x, y := Πx v δj,j 1 x, y v δj,j 1 x, y dy for a.e. x and y. By Lemma 4.1 we have w j C ; CperR N ; R d, and A y w j = 0 in W 1,p ; R l for a.e. x, 4.20 w j v 1 p L p ;R d C w j v δj,j + v δj,j 1 v 1 C x, y dy v δj,j v δj,j 1 x, y dy p L p ;R d A y xv δj,j 1 x, y p W 1,p ;R l dx + v δj,j 1 v 1 Therefore, in view of 4.18 and 4.19, v δj,j 1 x, y dy p L p ;R d. p L p ;R d w j v 1 strongly in L p ; R d We set u j εx := vx, y dy + w j x, x ε By Proposition 2.4 and 4.21, u j ε for a.e. x. 2 s v strongly two-scale in L p ; R d 4.22 as ε 0 and j +, in this order. Moreover, by 4.20 and since v F, A u j ε = A i x x, wj x. ε By 4.21, Proposition 2.4, and the compact embedding of W 1,p into L p, we conclude that A u j ε A v 1 x, y dy =

16 16 E. DAVOLI AND I. FONSECA strongly in W 1,p ; R l, as ε 0 and j +, in this order. By 4.22, 4.23, and Theorem 2.7 it follows in particular that lim lim T ε u j ε v Lp ;R j + ε 0 d + A u j ε W 1,p ;R l = 0. Attouch s diagonalization lemma [3, Lemma 1.15 and Corollary 1.16] provides us with a subsequence {jε such that, setting u ε := u jε ε, there holds and T ε u ε 2 s v strongly two-scale in L p ; R d A u ε 0 strongly in W 1,p ; R l. The thesis follows applying again Theorem 2.7. and In order to state the main result of this section we introduce the classes W := U := {u L p ; R d : A u = { w L p ; R d : wx, y dy = 0 and A y w = It is clear that v F if and only if vx, y dy U and v vx, y dy W. Let Ẽhom : L p ; R d L p ; R d be the functional { lim infn + inf w W fx, ny, ux + wx, y dy dx if u U, Ẽ hom u := + otherwise in L p ; R d We now provide a first characterization of 4.2. Theorem 4.3. Under the assumptions of Theorem 1.2, for every u L p ; R d there holds { inf lim inf f x, x ε 0 ε, u εx dx : u ε u weakly in L p ; R d and A u ε 0 strongly in W 1,p ; R l { = inf lim sup f x, x ε 0 ε, u εx dx : u ε u weakly in L p ; R d and A u ε 0 strongly in W 1,p ; R l = Ẽhomu We subdivide the proof of Theorem 4.3 into the proof of a limsup inequality Corollary 4.5 and a liminf inequality Propositions 4.6 and 4.7. We first show how an adaptation of the construction in Lemma 4.2 yields an outline for proving the limsup inequality in Proposition 4.4. Under the assumptions of Theorem 1.2, for every n N, u U and w W there exists a sequence {u ε S u+w see 4.10 such that u ε u weakly in L p ; R d, 4.28

17 HOMOGENIZATION FOR A x UASICONVEXITY 17 lim sup f x, x ε 0 ε, u εx dx f x, ny, ux + wx, y dy dx Proof. Step 1 : We first assume that u C; R d and w C 1 ; C 1 perr N ; R d. Arguing as in [11, Proof of Proposition 2.7] we introduce the auxiliary function gx, y := fx, ny, ux + wx, y for every x and for a.e. y R N. By definition, g C; L p perr N. Hence, setting g ε x := g x, x for a.e. x, nε Proposition 2.4 yields lim f x, x ε 0 ε, ux + w x, x dx = lim g ε x dx nε ε 0 = gx, y dy dx = fx, ny, ux + wx, y dy dx. Define u ε x := ux + w x, x for a.e. x. nε By the periodicity of w in the second variable and by the definition of W, u ε u + wx, y dy = u weakly in L p ; R d. 2 s By Proposition 2.4, u ε u + w strongly two-scale in L p ; R d. Finally recalling the definitions of the classes U and W by the regularity of w and by Proposition 2.4, A u ε = A i x w x, x A i x wx, y dy = 0 nε xi weakly in L p ; R l and hence strongly in W 1,p ; R l, due to the compact embedding of L p into W 1,p. Step 2 : Consider the general case in which u U and w W. Arguing as in the second part of the proof of Lemma 4.2 up to 4.21, we construct a sequence {w j C 1 ; C 1 perr N ; R d such that and Set w j u + w strongly in L p ; L p perr N ; R d, 4.30 A y w j = 0 in W 1,p ; R l for a.e. x, for every j. By Proposition 2.4, there holds u j ε u j εx := w j x, x nε for a.e. x. 2 s u + w strongly two-scale in L p ; R d, 4.31 as ε 0 and j +, in this order. In addition, arguing as in the proof of 4.23, we have A u j ε 0 strongly in W 1,p ; R l 4.32 as ε 0 and j +, in this order.

18 18 E. DAVOLI AND I. FONSECA To conclude, it remains to study the asymptotic behavior of the energies associated to the sequence {u j ε. Consider the functions g j x, y := fx, ny, w j x, y and gεx j := g j x, x. nε Arguing as in Step 1, we obtain lim lim f x, x j + ε 0 ε, uj εx dx = lim lim gεx j dx 4.33 j + ε 0 = lim g j x, y dy dx = fx, ny, ux + wx, y dy dx j + where we used the periodicity of gε, j together with 1.9 and In view of , Attouch s diagonalization lemma [3, Lemma 1.15 and Corollary 1.16], and Theorem 2.7, we obtain a subsequence {jε such that u ε := u jε ε satisfies both 4.28 and Proposition 4.4 yields the following limsup inequality. Corollary 4.5. Under the assumptions of Theorem 1.2, for every u L p ; R d { inf lim sup f x, x ε 0 ε, u εx dx : u ε u weakly in L p ; R d and A u ε 0 strongly in W 1,p ; R l Ẽhomu. We now turn to the proof of the liminf inequality in Theorem 4.3. For simplicity, we subdivide it into two intermediate results. Proposition 4.6. Under the assumptions of Theorem 1.2, for every sequence ε n 0 +, u U and {u n L p ; R d with u n 0 weakly in L p ; R d and A u n 0, there exists a p-equiintegrable family of functions V := {v ν,n : ν, n N such that V is a bounded subset of L p ; R d, for every ν N and as n + v ν,n 0 weakly in L p ; R d, A v ν,n 0 strongly in W 1,q ; R l for every 1 < q < p. Furthermore, lim inf f x, x, ux + u n x n + ε n { dx sup ν N lim inf n + fx, νnx, ux + v ν,n x dx. Proof. The proof follows the argument of [11, Proof of Proposition 3.8]. We sketch the main steps for the convenience of the reader. Step 1 : We first truncate our sequence in order to achieve p equiintegrability. Without loss of generality, up to translations and dilations we can assume that. Arguing as in [12, Proof of Lemma 2.15], we construct a p-equiintegrable sequence {ũ n L p ; R d such that ũ n u n 0 strongly in L q ; R d for every 1 < q < p, ũ n 0 weakly in L p ; R d,

19 HOMOGENIZATION FOR A x UASICONVEXITY 19 and lim inf n + A ũ n 0 strongly in W 1,q ; R l for every 1 < q < p, f x, x, ux + u n x dx lim inf f x, x, ux + ũ n x dx. ε n n + ε n Step 2 : we consider the sequence k ν,n := 1 νε n. If { k ν,n is a sequence of integers without loss of generality we can assume that it is increasing as n increases, then there is nothing to prove and we simply set { ũ n if k = v ν,k := k ν,n, 0 otherwise. In the case in which { k ν,n is not a sequence of integers, we define where In particular k ν,n := θ ν,n νε n, θ ν,n := νε n 1 νε n. θ ν,n 1 as n An adaptation of [11, Lemma 2.8] applied to {ũ n yields a p equiintegrable sequence {ū n L p ; R d such that ũ n ū n 0 strongly in L p ; R d, ũ n 0 weakly in L p \ ; R d, A ū n 0 strongly in W 1,q ; R l for every 1 < q < p Arguing as in [11, Proof of Proposition 3.8] we obtain lim inf f x, x, u + u n x dx lim inf fx, νk ν,n x, ux + v ν,kν,n x dx, n + ε n n + where v ν,kν,n x := ū n θ ν,n x for a.e. x, ν N and n N are large enough so that θ ν,n. Setting { v ν,kν,n if n = k ν,n, v ν,n := 0 otherwise, the sequence {v ν,n is uniformly bounded in L p ; R d, p-equiintegrable, and satisfies v ν,n 0 weakly in L p ; R d as n +. To conclude, it remains only to show that A v ν,n 0 strongly in W 1,q ; R l for every 1 < q < p 4.36 as n +. Let q as above be fixed, and let ϕ W 1,q 0 ; R l. A change of variables yields A v ν,kν,n, ϕ

20 20 E. DAVOLI AND I. FONSECA = = 1 θ N ν,n + N θ ν,n A i xū n θ ν,n x ϕx + N A i y θ ν,n ū n y ϕ y θ ν,n A i y y ū n y ϕ dy. θ ν,n θ ν,n A i x ū n θ ν,n x ϕx dx For n big enough θ ν,n. Hence, by 4.34, adding and subtracting the quantity 1 θ N ν,n θ ν,n N we deduce the upper bound A i yū n y ϕ y θ ν,n + A i y yū n y ϕ dy, θ ν,n A v ν,n, ϕ C A i y A i y C ū n θ ν,n 0 ;M Lq l d ;R ϕ d W 1,q 0 ;R l + C Ai y Ai y C ū n y i θ ν,n 0 ;M L l d q ;R ϕ d W 1,q 0 ;R l + C A ū n W 1,q ;R l ϕ W 1,q 0 ;R l. Property 4.36 follows now by 4.34 and To complete the proof of the liminf inequality in 4.27 we apply the unfolding operator see Subsection 2.3 to the set V constructed in Proposition 4.6. Proposition 4.7. Under the assumptions of Theorem 1.2, for every u U and every family V = {v ν,n : ν, n N as in Proposition 4.6 there holds lim inf lim inf fx, νnx, ux + v ν,n x dx Ẽhomu. ν + n + Proof. Fix u U and let {v ν,n : ν, n N be p equiintegrable and bounded in L p ; R d, with and v ν,n 0 weakly in L p ; R d 4.37 A v ν,n 0 strongly in W 1,q ; R l for every 1 < q < p, 4.38 as n +, for every ν N. Fix and for z Z N and n N, define and We consider the maps ν,z := z ν + 1 ν, Z ν := {z Z N : ν,z. T 1 ν v ν,nx, y := v ν,n 1 ν νx + 1 ν y for a.e. x, y,

21 HOMOGENIZATION FOR A x UASICONVEXITY 21 where we have extended the sequence {v ν,n to zero outside. A change of variables yields fx, νnx, ux + v ν,n x dx fx, νnx, ux + v ν,n x dx z Z ν ν,z = ν z N f z Z ν ν + y z ν, ny, u ν + y z + v ν,n ν ν + y dy ν = νx f + y z Z ν ν,z ν ν, ny, T 1 ux, y + T 1 v ν,nx, y dy dx ν ν νx f + y ν ν, ny, T 1 ux, y + T 1 v ν,nx, y dy dx, ν ν z Z ν ν,z where the last inequality is due to 1.9. By [11, Proposition 3.6 i] and Proposition 2.8 we conclude that fx, νnx, ux + v ν,n x dx 4.39 σ ν + fx, ny, ux + v ν,z,n y dy dx, where z Z ν ν,z v ν,z,n y := T 1 ν v ν,n z ν, y for a.e. y, and σ ν 0 as ν +. The sequence {v ν,z,n is p-equiintegrable by [11, Proposition A.2], and is uniformly bounded by 4.37 and Proposition 2.6, since v ν,z,n y p dy = 1 ν N v ν,n x p dx. ν,z By the boundedness of {v ν,n : ν, n N in L p ; R d, and by 4.37 there holds v ν,z,n 0 weakly in L p ; R d 4.40 as n +, for every z Z ν, ν N. Denoting by χ ν,z the characteristic functions of the sets ν,z, we claim that A y x = L q lim sup ν + lim sup n + ν Z ν χ ν,z xv ν,z,ny W 1,q ;R l for every 1 < q < p. Indeed, fix 1 < q < p, and let ψ W 1,q 0 ; R l. Then z A y v ν,z,n, ψ = A i z z v ν,n ν ν ν + y ψy dy ν y i = ν N A i z v ν,n x ψ νx z dx. ν y i ν,z Adding and subtracting to the previous expression the quantity ν N A i xv ν,n x ψ νx z dx, y i ν,z

22 22 E. DAVOLI AND I. FONSECA and setting φ ν zx := ψνx z for a.e. x, we obtain the estimate z A y v ν,z,n, ψ ν ν N N A i z A i x v ν,n x ψ νx z ν L q ν,z;r l y i + ν N 1 A i xv ν,n x W 1,q ν,z;r φ ν z l W 1,q. 0 ν,z;r l A change of variables yields the upper bound ψ φ ν z νx z y + W 1,q i L q ν,z;r l ν Thus, by the regularity of the operators A i, z A y v ν,z,n, ψ ν Cν N q 1 N 0 ν,z;r l L q ν,z;r l C ν N q ψ W 1,q 0 ;R l. A i L v ν,n x L i ;M l d q ν,z;r ψ d W 1,q 0 ;R l + Cν N q A i xv ν,n x W 1,q ν,z;r ψ l W 1,q 0 ;R l Using again the Lipschitz regularity of the operators A i, i = 1,, N, we deduce A y xv ν,z,n y W 1,q ;R l z + A y v ν,z,n y ν C ν A i x A i z L v ν,z,n ν Lq ;M l d ;R l W 1,q ;R l A i L v ν,z,n x Lq i ;M l d ;R l + for a.e. x ν,z. Hence, by 4.42 and 4.43, we obtain A y xv ν,z,n y q W 1,q ;R l dx ν,z z Z ν C ν q z Z ν z Z ν ν,z A i q ν,z xv ν,z,n L ;M l d z A y v ν,z,n y ν v ν,z,n q L q ;R d dx W 1,q ;R l + z A y v ν,z,n y q z Z ν ν,z ν W 1,q ;R l dx C ν q χ ν,z xv ν,z,ny q + C z Z ν L q ;R d ν q v ν,n L q ;R d + C A i v ν,n W ν q 1,q ν,z;r l C ν q χ ν,z xv ν,z,ny q + C z Z ν L q ;R d ν q v ν,n Lq ;R d 4.43

23 + Cν N q HOMOGENIZATION FOR A x UASICONVEXITY 23 A i v ν,n W 1,q ;R l Property 4.41 follows now by 4.37 and 4.38, and by the compact embedding of L p into W 1,p. Consider the maps Πx v ν,z,n y v ν,z,nξ dξ v Πx ν,z,n y v ν,z,nξ dξ dy w ν,n x, y := for x ν,z, z Z ν, y, 0 otherwise in. By Lemma 4.1 the sequence {w ν,n is p-equiintegrable, and A y w ν,n = 0 in W 1,p ; R l for a.e. x, for all ν, n N. In particular, {w ν,n W. We claim that w ν,n x, y χ xv ν,z ν,z,ny Lq z Z ν ;R d as n +, ν +, for every 1 < q < p. In fact, by Lemma 4.1 there holds Πx v ν,z,n y v ν,z,n ξ dξ v ν,z,n y q L q ;R d C A y xv ν,z,n y qw 1,q;R l + v ν,z,n y dy q. Therefore χ x ν,z Πx v ν,z,n y z Z ν C z Z ν + z Z ν A y xv ν,z,n y q W 1,q ;R l dx ν,z v ν,z,n y dy q dx. ν,z q v ν,z,n ξ dξ v ν,z,n y L q ;R d 4.45 The first term in the right-hand side of 4.45 converges to zero as n + and ν +, in this order, owing to The second term in the right-hand side of 4.45 converges to zero as n + and ν +, in this order, by the dominated convergence theorem, owing to 4.40 and the uniform boundedness in L p of {v ν,z,n. Hence, both the left-hand side of 4.45 and the quantity { χ xπx ν,z v ν,z,n y v ν,z,n ξ dξ dy 0, z Z ν converge to zero as n + and ν +, and we obtain Up to the extraction of a not relabeled subsequence, we can assume that lim inf lim inf fx, ny, ux + v ν,z,n y dy dx 4.46 ν + n + z Z ν ν,z

24 24 E. DAVOLI AND I. FONSECA = lim lim inf ν + n + z Z ν ν,z fx, ny, ux + v ν,z,n y dy dx. Hence, in view of 4.44 and 4.46 we can extract a subsequence {nν such that lim lim inf fx, ny, ux + v ν,z,n y dy dx 4.47 and ν + n + z Z ν = lim ν + z Z ν w ν,nν x, y ν,z ν,z fx, nνy, ux + v ν,z,nν y dy dx, 4.48 z Z ν χ ν,z xv ν,z,nνy 0 strongly in L q ; R d, 4.49 for every 1 < q < p. Going back to 4.39, by [11, Proposition 3.5 ii], 4.47 and 4.49, lim inf lim inf fx, νnx, ux + v ν,n x dx ν + n + lim inf fx, nνy, ux + w ν,nν x, y dy dx. ν + By the p-equiintegrability of {w ν,nν and by 1.9, letting \ tend to zero, we conclude lim inf lim inf fx, νnx, ux + v ν,n x dx ν + n + lim inf fx, nνy, ux + w ν,nν x, y dy dx ν + lim inf inf fx, nνy, wx, y dy dx ν + w W lim inf inf fx, ny, wx, y dy dx = E hom u. n + w W Proof of Theorem 4.3. The proof follows by combining Corollary 4.5 with Propositions 4.6 and 4.7. Corollary 4.8. Under the same assumptions of Theorem 4.3, for every u U F Ẽ hom u = E hom u := f hom x, ux dx, where f hom x, ux = lim inf inf fx, ny, ux + vy dy, n + v C x and C x is the class defined in 1.5. Proof. We omit the proof of this corollary as it follows from [11, Remark 3.3 ii] and by adapting the arguments in [11, Corollary 3.2] and Lemma 4.9 below. Proof of Theorem 1.2. The thesis results from Theorem 4.3 and Corollary 4.8.

25 HOMOGENIZATION FOR A x UASICONVEXITY 25 We conclude this section by showing that Theorem 4.3 yields a relaxation result in the framework of A quasiconvexity with variable coefficients. Before stating the corollary, we prove a preliminary lemma which guarantees the measurability of the function x A fx, ux for every u L p ; R d. Lemma 4.9. Let 1 < p < +, u L p ; R d, let A be as in Theorem 1.2, and let f : R d [0, + be a Carathéodory function satisfying Then the map 0 fx, ξ C1 + ξ p for a.e. x R d, and for all ξ R d. is measurable in. Proof. We first remark that A fx, ux = where { r A fx, ux := inf x A fx, ux and C x is the class defined in 1.5. Clearly inf r 0,+ r A fx, ux for a.e. x, 4.50 fx, ux + wy dy : w C x and w L p ;R d r, r A fx, ux A fx, ux for a.e. x, for every r N. Moreover, for every ε > 0 there exists w ε C x such that A fx, ux fx, ux + w ε y dy ε wε L p ;R d A fx, ux ε inf r N r A fx, ux ε, which in turn implies the second inequality in By 4.50 it is enough to show that x r A fx, ux is measurable for every r N. We claim that where r A fx, ux = sup r,n A fx, ux for a.e. x, 4.51 n N { r,n A fx, ux = inf fx, ux + wy dy + n A y xwx, W 1,p ;R l : w L p ; R d, wy dy = 0 and w Lp ;R d r. Clearly, r,n A fx, ux r A fx, ux for a.e. x, for all n N. To prove the opposite inequality, fix x, and for every n N, let w n L p ; R d, with w ny dy = 0 and w n Lp ;R d r, be such that fx, ux + w n y dy + n A y xw n W 1,p ;R l 4.52 r,n A fx, ux + 1 n fx, ux + 1 n

26 26 E. DAVOLI AND I. FONSECA the last inequality holds because 0 C x for every x. Since {w n is uniformly bounded in L p ; R d and by 4.52 A y xw n W 1,p ;R l 0 as n +, there exists a map w L p ; R d, with wy dy = 0, w L p ;R d r and A y xw = 0 such that w n w weakly in L p ; R d. By [5, Lemma 3.1] we can construct a sequence { w n such that w ny dy = 0, A y x w n = 0 for every n N, and lim inf fx, ux + w n y dy lim inf fx, ux + w n y dy n + n + In view of 4.50 we have r A fx, ux A fx, ux sup n N r,n A fx, ux. fx, ux + w n y dy for every n N, and we obtain the second inequality in By the measurability of x r,n A fx, ux for every r, n N we can reduce it to a countable pointwise infimum of measurable functions, we deduce the measurability of x r A fx, ux for every r N, which in turn implies the thesis. For every D O and u L p ; R d, define { Iu, D := inf lim inf fx, u n x : u n u weakly in L p ; R m 4.53 n + D and A u n 0 strongly in W 1,p ; R l. Corollary 4.8 provides us with the following integral representation of I. Corollary Let 1 < p < + and let A be as in Theorem 1.2. Let f : R d [0, + be a Carathéodory function satisfying Then 0 fx, ξ C1 + ξ p for a.e. x, and for all ξ R d. D A fx, ux dx = Iu, D for all D O and u L p ; R d with A u = 0.

27 HOMOGENIZATION FOR A x UASICONVEXITY 27 acknowledgements The authors thank the Center for Nonlinear Analysis NSF Grant No. DMS , where this research was carried out, and also acknowledge support of the National Science Foundation under the PIRE Grant No. OISE The research of I. Fonseca and E. Davoli was funded by the National Science Foundation under Grant No. DMS The research of E. Davoli was also supported by the Austrian FWF project Global variational methods for nonlinear evolution equations. E. Davoli is a member of the INdAM-GNAMPA Project 2015 Critical phenomena in the mechanics of materials: a variational approach. The research of I. Fonseca was further partially supported by the National Science Foundation under Grant No. DMS References [1] Emilio Acerbi and Nicola Fusco. Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal., 862: , [2] Grégoire Allaire. Homogenization and two-scale convergence. SIAM J. Math. Anal., 236: , [3] H. Attouch. Variational convergence for functions and operators. Applicable Mathematics Series. Pitman Advanced Publishing Program, Boston, MA, [4] John M. Ball. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal., 634: , 1976/77. [5] Andrea Braides, Irene Fonseca, and Giovanni Leoni. A-quasiconvexity: relaxation and homogenization. ESAIM Control Optim. Calc. Var., 5: electronic, [6] D. Cioranescu, A. Damlamian, and G. Griso. The periodic unfolding method in homogenization. SIAM J. Math. Anal., 404: , [7] Doina Cioranescu, Alain Damlamian, and Georges Griso. Periodic unfolding and homogenization. C. R. Math. Acad. Sci. Paris, 3351:99 104, [8] Bernard Dacorogna. Weak continuity and weak lower semicontinuity of nonlinear functionals, volume 922 of Lecture Notes in Mathematics. Springer-Verlag, Berlin-New York, [9] Bernard Dacorogna and Irene Fonseca. A-B quasiconvexity and implicit partial differential equations. Calc. Var. Partial Differential Equations, 142: , [10] Elisa Davoli and Irene Fonseca. Homogenization of integral energies under periodically oscillating differential constraints. Calc. Var. Partial Differential Equations, to appear. [11] Irene Fonseca and Stefan Krömer. Multiple integrals under differential constraints: two-scale convergence and homogenization. Indiana Univ. Math. J., 592: , [12] Irene Fonseca and Stefan Müller. A-quasiconvexity, lower semicontinuity, and Young measures. SIAM J. Math. Anal., 306: electronic, [13] Carolin Kreisbeck and Stefan Krömer. Heterogeneous thin films: combining homogenization and dimension reduction with directors. SIAM J. Math. Anal., 482: , [14] Dag Lukkassen, Gabriel Nguetseng, and Peter Wall. Two-scale convergence. Int. J. Pure Appl. Math., 21:35 86, [15] Paolo Marcellini. Approximation of quasiconvex functions, and lower semicontinuity of multiple integrals. Manuscripta Math., 511-3:1 28, [16] José Matias, Marco Morandotti, and Pedro M. Santos. Homogenization of functionals with linear growth in the context of A-quasiconvexity. Appl. Math. Optim., 723: , [17] Charles B. Morrey, Jr. Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften, Band 130. Springer-Verlag New York, Inc., New York, [18] François Murat. Compacité par compensation: condition nécessaire et suffisante de continuité faible sous une hypothèse de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4, 81:69 102, [19] Gabriel Nguetseng. A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal., 203: , [20] Pedro M. Santos. A-quasi-convexity with variable coefficients. Proc. Roy. Soc. Edinburgh Sect. A, 1346: , 2004.

28 28 E. DAVOLI AND I. FONSECA [21] Augusto Visintin. Some properties of two-scale convergence. Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei 9 Mat. Appl., 152:93 107, [22] Augusto Visintin. Towards a two-scale calculus. ESAIM Control Optim. Calc. Var., 123: electronic, Elisa Davoli Department of Mathematics, University of Vienna, Oskar-Morgenstern- Platz 1, 1090 Vienna Austria address, E. Davoli: elisa.davoli@univie.ac.at Irene Fonseca Department of Mathematics, Carnegie Mellon University, Forbes Avenue, Pittsburgh PA address, I. Fonseca: fonseca@andrew.cmu.edu