HOMOGENIZATION OF INTEGRAL ENERGIES UNDER PERIODICALLY OSCILLATING DIFFERENTIAL CONSTRAINTS

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1 HOMOGENIZATION OF INTEGRAL ENERGIES UNDER PERIODICALLY OSCILLATING DIFFERENTIAL CONSTRAINTS ELISA DAVOLI AND IRENE FONSECA Abstract. A homogenization result for a family of integral energies u ε fu εx dx, ε 0 +, is presented, where the fields u ε are subjected to periodic first order oscillating differential constraints in divergence form. The work is based on the theory of A -quasiconvexity with variable coefficients on twoscale convergence techniques. 1. Introduction This paper is the first step toward a complete understing of homogenization problems for oscillating energies subjected to oscillating linear differential constraints, in the framework of A quasiconvexity with variable coefficients. To be precise, we initiate the study of integral representations for limits of oscillating integral energies u ε f x, x ε α, u εx dx, where R N is an open bounded domain, ε 0 +, the fields u ε L p ; R d are subjected to periodically oscillating differential constraints such as A ε u ε := A i x uε x ε β 0 strongly in W 1,p ; R l, 1.1 or in divergence form Aε div u ε := A i x ε β u ε x 0 strongly in W 1,p ; R l, 1.2 with 1 < p < +, A i x Lin R d ; R l M l d for every x R N, i = 1,, N, d, l 1, where α, β are two nonnegative parameters. Here, in what follows, M l d sts for the linear space of matrices with l rows d columns. Oscillating divergence-type constraints as in 1.2 appear in the homogenization theory of systems of second order elliptic partial differential equations. Indeed, if u ε = v ε, with v ε W 1,p for every ε > 0, A i x = Ax M N N for i = 1,, N, then considering 1.2 reduces to the homogenization problem of finding the effective behavior of weak limits of v ε, where x div A v ε 0 strongly in W 1,p, 1 < p < +. ε 2010 Mathematics Subject Classification. 49J45; 35D99; 49K20. Key words phrases. Homogenization, two-scale convergence, A quasiconvexity. 1

2 2 E. DAVOLI AND I. FONSECA These problems have been extensively studied in the literature see e.g. [2], [5, Chapter 1, Section 6], [9], the references therein. Different regimes are expected to arise depending on the relation between α β. Here we will consider β > 0, we will assume that the energy density f is constant in the first two variables but the differential constraint in divergence form 1.2 oscillates periodically. The limit scenario α > 0, β = treated in [12] for constant coefficients, i.e., the energy density is oscillating but the differential constraint is fixed, the situation in which α > 0 β > 0, will be the subject of forthcoming papers. The key tool for our analysis is the notion of A quasiconvexity. For i = 1, N, consider matrix-valued maps A i C ; M l d, define A as the differential operator such that A vx := A i x vx, x, for v L 1 loc ; Rd, where v 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 N rank A i xw i = r for every w S n 1, 1.3 uniformly with respect to x, where S N 1 is the unit sphere in R N. The properties of A -quasiconvexity in the case of constant coefficients were first investigated by Dacorogna in [10], then studied by Fonseca Müller in [14] see also [11]. In [20] Santos extended the analysis of [14] to the case in which the coefficients of the differential operator A depend on the space variable. Definition 1.1. Let f : R d R be a continuous function, let be the unit cube in R N centered at the origin, = 1 2 2, 1 N, denote by CperR N ; R d the set of smooth maps which are -periodic in R N. Consider the set C x := w CperR N ; R d : wy dy = 0, A i x wy = 0. For a.e. x, the A quasiconvex envelope of f in x is defined as ξ A fx, ξ := inf fξ + wy dy : w C x. f is said to be A -quasiconvex if fξ = A fx, ξ for a.e. x all ξ R d. We remark that when A := curl, i.e., when v = φ for some φ W 1,1 loc ; Rm, then d = m N, then A -quasiconvexity reduces to Morrey s notion of quasiconvexity see [1, 4, 16, 17]. The following theorem was proved in [20] in the more general case when f is a Carathéodory function, generalizing the corresponding results [14, Theorems 3.6

3 HOMOGENIZATION FOR A x UASICONVEXITY 3 3.7] in the case of constant coefficients i.e. i = 1,, N. A i x A i M l d for every Theorem 1.2. Let be an open bounded domain in R N, let A i C ; M l d W 1, ; M l d, i = 1,, N, d 1, 1 < p < +, assume that the operator A satisfies the constant rank condition 1.3. Let f : R d [0, + be a continuous function satisfying i 0 fv C1 + v p, ii fv 1 fv 2 C1 + v 1 p 1 + v 2 p 1 v 1 v 2 for all v, v 1, v 2 R d, for some C > 0. Then A quasiconvexity is a necessary sufficient condition for lower semicontinuity of the functional v fvx dx for sequences v ε v weakly in L p ; R d such that A v ε 0 strongly in W 1,p ; R l. In the case of constant coefficients, in [6] Braides, Fonseca Leoni provided an integral representation formula for relaxation problems in the context of A - quasiconvexity presented via Γ-convergence homogenization results for periodic integrs evaluated along A free fields. Their homogenization results were later generalized in [12], where Fonseca Krömer worked still in the framework of constant coefficients but under weaker assumptions on the energy density f. This paper is devoted to extending the previous homogenization results to the case in which A is a differential operator with nonconstant L -coefficients, the energies under consideration are of the type u ε fu ε x dx, where u ε u weakly in L p ; R d, Aε div u ε := A i x u ε x 0 strongly in W 1,q ; R l ε for all 1 q < p. We point out that the result in Theorem 1.2 [20] covers the case q = p. Our analysis includes the case when q = p if the operator A has smooth coefficients. However, in the general situation when A has bounded coefficients, the assumption 1 q < p is required, in order to satisfy some truncation p-equiintegrability arguments see the proofs of Theorems Our starting point is a characterization of the set C A of limits of Aε div vanishing fields u ε. We show in Proposition 3.4 that a function u L p ; R d belongs to C A if only if there exists a map w L p ; L p perr N ; R d such that wx, y dy = 0 for a.e. x, 2 s u ε u + w strongly two-scale in L p ; R d see Definition 2.1, u + w satisfies the differential constraints A i yux + wx, y dy = 0 1.4

4 4 E. DAVOLI AND I. FONSECA in W 1,p ; R l, A i yux + wx, y = in W 1,p ; R l for a.e. x. This generalizes the classical characterization of 2- scale limits of solutions to linear elliptic partial differential equations in divergence form in [2, Theorem 2.3] to the case of first order linear systems. For every u C A, we denote by Cu A the class of maps w as above. We then prove that the homogenized energy is given by the functional inf r>0 inf F A u := if u C A, + otherwise in L p ; R d, lim inf F r A n u n : u n u weakly in L p ; R d where inf F r fvx + w nx, y dy dx : w Cv A n, w Lp ;R d r A n v := if v Cr A n, + otherwise in L p ; R d, the classes C A n v are defined analogously to C A v by replacing the operators A i with A i n in , C A n r := v L p ; R d : w C A n v with w Lp ;R d r, r > 0. Our main result is the following. Theorem 1.3. Let 1 < p < +. Let A i L ; M l d, i = 1,, N, let f : R d [0, + be a continuous map satisfying the growth condition 0 fv C1 + v p for every v R d, some C > 0. Then, for every u C A there holds inf lim inf fu ε x dx : u ε u weakly in L p ; R d ε 0 Aε div u ε 0 strongly in W 1,q ; R l for every 1 q < p = inf lim sup fu ε x dx : u ε u weakly in L p ; R d ε 0 Aε div u ε 0 strongly in W 1,q ; R l for every 1 q < p = F A u. Remark 1.4. i As a consequence of Theorem 1.2, we expected the homogenized energy to be related to the effective energy for an A -quasiconvex envelope of the function f, with the role of the differential constraint A to be replaced by the limit constraints We stress the fact that here the oscillatory behavior of the differential constraint as ε 0 forces the relaxation with respect to the homogenization in the differential constraint to happen somewhat simultaneously. Indeed, for every n the functional F r A n is obtained as a truncated version of a relaxation with respect to the limit differential constraints dilated by a factor n, is evaluated on a fixed element of a sequence of maps approaching u,

5 HOMOGENIZATION FOR A x UASICONVEXITY 5 whereas the limit functional F A u is deduced by a diagonal procedure, as n tends to +. ii The truncation in the definition of the functionals F r A n plays a key role in the proof of the limsup inequality inf lim sup fu ε x dx : u ε u weakly in L p ; R d ε 0 A div ε u ε 0 strongly in W 1,q ; R l for every 1 q < p F A u, because it provides boundedness of the recovery sequences thus allows us to apply a diagonalization argument see Step 3 in the proof of Proposition iii We remark that, as opposed to the case in which the operators A i are constant, we cannot expect the homogenized energy to be local, i.e., that there exists f hom : R d [0, + such that F A u = f hom x, ux dx. 1.6 We show in Example 5.4 that locality in the sense of 1.6 may fail even when the function f is convex in its second variable. As in [12], the proof of this result is based on the so-called unfolding operator, introduced in [8, 7] see also [21, 22] Subsection 2.2. A first difference with [12, Theorem 1.1] i.e., with the case in which the operators A i are constant is the fact that we are unable to work with exact solutions of the system Aε div u ε = 0, but instead we consider sequences of asymptotically Aε div vanishing fields. As pointed out in [20], in the case of variable coefficients the natural framework is pseudo-differential operators. In this setting, we do not project directly onto the kernel of a differential constraint A, but rather we 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 we refer to [20, Subsection 2.1] for a detailed explanation of this issue, to the references therein for a treatment of the main properties of pseudo-differential operators. The crucial difference with respect to the case of constant coefficients is the structure of the set C A. In the case in which the condition Aε div u ε 0 is replaced by A u ε = 0, with A being independent of the space variable, decouple see [12, Theorem 1.2] becoming separate requirements on w u. However, in our situation they can not be dealt with separately, this forces the structure of the homogenized energy to be much more involved. The oscillatory behavior of the differential constraint its ε-dependent structure render this problem quite technical due to the difficulty in obtaining a suitable projection operator on the limit differential constraint. Moreover, due to the coupling between the dependence of the operators on ε, the pseudodifferential operators method cannot be applied directly here. In order to solve this problem, in Lemma 3.2 we are led to impose a uniform invertibility requirement on the differential operator. To be precise, we require l N = d we assume that there exists a positive constant γ such that the operator Ay Lin R d ; R d,

6 6 E. DAVOLI AND I. FONSECA defined as satisfies Ayξ := A 1 yξ T. A N yξ T M N l = R d for every ξ R d, H Ayλ λ γ λ 2 for every λ R d y R N. We remark that assumption H is quite natural, as it represents a higherdimensional version of the classical uniform ellipticity assumption see e.g. [2, 2.2]. The strategy of our argument consists in first proving Theorem 1.3 in the case in which the operators A i are smooth. The general case is then deduced by means of an approximation argument of bounded operators by smooth ones, by an application of Severini-Egoroff s theorem p equiintegrability see Section 5. Our main theorem is consistent with the relaxation results obtained in [6] in the case of constant coefficients. When the linear operators A i are constant, we prove in Subsection 5.1 that the homogenized energy F A Theorem 1.3 reduce to the A quasiconvex envelope of f [6, Theorem 1.1], respectively. This article is organized as follows. In Section 2 we introduce notation recall some preliminary results on two-scale convergence on the unfolding operator. In Section 3 we provide a characterization of the limits of Aε div -vanishing fields see Proposition 3.4. Section 4 is devoted to the proof of our main result, Theorem 1.3, for smooth operators Aε div. The argument is extended to the case in which Aε div are only bounded in Section Preliminary results Throughout this paper R N is an open bounded domain O is the set of open subsets of. is the unit cube in R N centered at the origin with normals to its faces parallel to the vectors in the stard orthonormal basis of R N, e 1,, e N, i.e., := 1 2, 1 N. 2 Given 1 < p < +, we denote by p its conjugate exponent, that is 1 p + 1 p = 1. Whenever a map v L p, C,, is periodic, that is vx + e i = vx i = 1,, N for a.e. x R N, we write v L p per, C per,, respectively. We will implicitly identify the spaces L p L p perr N. We designate the Lebesgue measure of a measurable set A R N by A. We adopt the convention that C will st for a generic positive constant, whose value may change from expression to expression in the same formula.

7 HOMOGENIZATION FOR A x UASICONVEXITY Two-scale convergence. We recall the definition some properties of two-scale convergence which apply to our framework. For a detailed treatment of the topic we refer to, e.g., [2, 15, 19]. Throughout this subsection 1 < p < +. Definition 2.1. If v L p ; L p perr N ; R d 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 per R N ; R d. We say that u ε strongly two-scale converge to v in L p ; R d 2 s, u ε v, if 2 s u ε v lim u ε L ε 0 p ;R d = v L p ;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 2.2. Let u ε L p ; R d be bounded. Then there exists v L p ; L p perr N ; R d such that, up to a subsequence, u ε 2 s v weakly two-scale,, in particular, u ε u := vx, y dy weakly in L p ; R d. The following result will play a key role throughout the paper in the proofs of limsup inequalities see [2, Lemma 1.3], [22, Lemma 2.1], [12, Proposition 2.4, Lemma 2.5 Remark 2.6]. Proposition 2.3. Let v L p ; C per R N ; R d or v L p perr N ; C; R d. Then the sequence u ε, defined as u ε x := v x, x ε is p equiintegrable, u ε 2 s v strongly two-scale in L p ; R d The unfolding operator. We collect here the definition some properties of the unfolding operator see e.g. [8, 7, 21, 22]. Definition 2.4. 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 y R N, 2.1 ε where u is extended by zero outside denotes the least integer part. Proposition 2.5. T ε is a nonsurjective linear isometry from L p ; R d to L p R N ; R d. The next theorem relates the notion of two-scale convergence to L p convergence of the unfolding operator see [22, Proposition 2.5 Proposition 2.7],[15, Theorem 10].

8 8 E. DAVOLI AND I. FONSECA Theorem 2.6. Let be an open bounded domain let v L p ; L p perr N ; R d. Assume that v is extended to be 0 outside. Then the following conditions are equivalent: i u ε 2 s 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 only if T ε u ε v strongly in L p R N ; R d. The following proposition is proved in [12, Proposition A.1]. Proposition 2.7. If u L p ; R d extended by 0 outside then as ε 0. u T ε u L p R N ;R d 0 3. Characterization of limits of Aε div -vanishing fields Let 1 < p < +, for every ε > 0 denote by Aε div W 1,p ; R l the first order differential operator A div ε u := A i x ε ux : L p ; R d 3.1 for u L p ; R d. In this section we focus on the case in which the operators A i are smooth periodic, A i CperR N ; M l d, for all i = 1,, N. We will also require that N l = d, for every y R N the operator Ay Lin R d ; R d, defined as Ayξ := A 1 yξ T. A N yξ T satisfies the uniform ellipticity condition M N l = R d for every ξ R d, Ayλ λ γ λ 2 for every λ R d y R N 3.2 where γ > 0 is a positive constant. We first prove a corollary of [14, Lemma 2.14]. Lemma 3.1. Let 1 < p < + consider the differential operator defined as div R := div : L p ; R N W 1,p Then, there exists an operator R i y for every R L p ; R N. T : L p ; R N L p ; R N such that P1 T is linear bounded, vanishes on constant maps, P2 T TR = TR div TR = 0 for every R L p ; R N,

9 HOMOGENIZATION FOR A x UASICONVEXITY 9 P3 there exists a constant C = Cp > 0 such that R TR L p ;R N C divr W 1,p, for all R L p ; R N with Ry dy = 0, P4 if v n is bounded in L p ; R N p equiintegrable in, then setting w n x, := Tv n x, for a.e. x, the sequence w n is p equiintegrable in, P5 if ψ C 1 ; CperR 1 N ; R N W 2,2 ; WperR 2,2 N ; R N then setting ϕx, := Tψx, for every x R N, there holds ϕ C 1 ; CperR 1 N ; R N. Proof. We first notice that we can rewrite the operator div as div R := D i Ry for every R L p ; R N, where D i Lin R N ; R, D i m := δ i m for every i, m = 1,, N. For every λ R N \ 0, we associate to div the linear operator Dλ := D i λ i Lin R N ; R. By [14, Remark 3.3 iv] the operator div satisfies the constant rank condition 1.3. Properties P1, P2 P3 follow directly by [14, Lemma 2.14]. The proof of P4 is a straightforward adaptation of the proof of [14, Lemma 2.14 iv]. For convenience of the reader, we sketch the proof of P5. Fix ψ C 1 ; CperR 1 N ; R N W 2,2 ; WperR 2,2 N ; R N. For every λ R N \0, let P D λ : R N R N be the linear projection onto Ker Dλ. Writing the operator T explicitly see [14, Lemma 2.14], we have Tψx, y := P D λ ψx, λe 2πiy λ where ψx, λ := λ Z N \0 ψx, ze 2πiz λ dz, for λ Z N \ 0 x, are the Fourier coefficients associated to ψx,. By the regularity of ψ by Plancherel s theorem for Fourier series, we obtain that 1 4π 2 λ i 2 ψx, 2 λ 2 ψx, y L = C, for every x, 2 ;R N λ Z N \0 for i = 1,, N. Therefore, by [14, Proposition 2.7] by Cauchy-Schwartz s inequality, for every n N there holds P D λ ψx, λe 2πiy λ C ψx, λ λ Z N \0, λ n C λ Z N \0 ψx, λ 2 λ λ Z N \0, λ n λ Z N \0, λ n λ 2

10 10 E. DAVOLI AND I. FONSECA C λ Z N \0, λ n λ 2. The previous inequality implies that for every n the tail of the series of functions P D λ ψx, λe 2πiy λ 3.3 λ Z N \0 is uniformly bounded by the tail of a convergent series. This yields the uniform convergence of the series in 3.3 hence the continuity of the map Tψx, y. The differentiability of Tψx, y follows by an analogous argument. Using the previous result we can prove the following projection lemma. Lemma 3.2. Let 1 < p < +, let A i L ; M l d, i = 1,, N, with A satisfying the invertibility requirement in 3.2. Let v n, v L p ; R d be such that v n v weakly in L p ; R d, 3.4 A i yv n x, y dy 0 strongly in W 1,p ; R l, 3.5 A i yv n x, y 0 strongly in L p ; W 1,p ; R l. 3.6 Then there exists a subsequence v n k a sequence w k L p ; R d such that v n k w k 0 strongly in L p ; R d, 3.7 A i yw k x, y dy = 0 in W 1,p ; R l, 3.8 A i yw k x, y = 0 in W 1,p ; R l for a.e. x. 3.9 Proof. We first notice that imply that A i yvx, y dy = 0 in W 1,p ; R l, A i yvx, y = 0 in W 1,p ; R l for a.e. x. By linearity, it is enough to consider the case in which v = 0. Moreover, up to a translation a dilation, we can assume that is compactly contained in. By the compact embedding of L p ; R d into W 1,p ; R d, by , for every ϕ Cc ; [0, 1] there holds A i yϕxv n x, y dy = ϕx A i yv n x, y dy + ϕx A i yv n x, y dy 0

11 HOMOGENIZATION FOR A x UASICONVEXITY 11 strongly in W 1,p ; R d. On the other h, by 3.6, A i yv n W x, y 0 strongly in L p. 1,p;Rl Therefore, we may consider a sequence ϕ k Cc ; [0, 1] with ϕ k 1 such that, setting ṽk n := ϕ kv n extending ṽk n by zero to \ then periodically, there holds ṽk n 0 weakly in L p ; R d, A i yṽk n x, y dy 0 strongly in W 1,p ; R l, A i yṽ n W k x, y 0 1,p;Rl strongly in L p, as n +, k +. By a diagonal argument we extract a subsequence v k := ṽ nk k such that v k 0 weakly in L p ; R d, 3.10 A i yv k x, y dy 0 strongly in W 1,p ; R l, 3.11 Define the maps A i yv k W x, y 0 strongly in L p ,p;Rl R k i x, y := A i yv k x, y for a.e. x y, i = 1,, N, let R k L p ; R d be given by By , R k ij := R k i j, for all i = 1,, N, j = 1, l. R k 0 weakly in L p ; R d, 3.13 Ri k x, y dy 0 strongly in W 1,p ; R l, 3.14 R k i x, y 0 strongly in L p W 1,p ;R l Using Lemma 3.1, we consider the projection operators T x T y onto the kernel of the divergence operator with respect to x the divergence operator with respect to y in the set. We have T x R k x, y dy R k x, y dy C R k w, y dy dw R k w, y dy dw Lp;Rd Ri k W x, y dy, 1,p;Rl 3.16

12 12 E. DAVOLI AND I. FONSECA Ty R k x, y R k x, z dz R k x, y R k x, z dz Lp ;Rd 3.17 C R k i x, y which in turn yields T y R k x, y = [T y R k x, y R k x, z dz dy C R k i x, y W 1,p ;R l L, p Lp ;R d R k x, z dz R k x, y W 1,p ;R l Lp. Set S k x, y := T y R k x, y R k x, z dz + T x R k x, z dz R k w, z dz dw T y R k x, z ] R k x, z dz dy 3.18 L p ;R d R k x, ξ dξ dz + R k w, z dz dw for a.e. x, y. Combining , we deduce the inequality R k S k L p ;R d 3.19 T y R k x, y R k x, z dz R k x, y R k x, z dz Lp ;Rd + T x R k x, z dz R k w, z dz dw R k x, z dz R k w, z dz dw Lp;Rd + T y R k x, z R k x, ξ dξ dz + R k x, y dy dx, Lp ;R d whose right-h-side converges to zero as k +. On the other h, by Lemma 3.1 there holds Sir k x, y = 0 in W 1,p for a.e. x, 3.20 for every k, for all r = 1,, l. Finally, define w k x, y := Ay 1 Sirx, k y dy = 0 in W 1,p, 3.21 S1 k x, y.. x, y S k N for a.e. x y R N

13 HOMOGENIZATION FOR A x UASICONVEXITY 13 where the components Si k are defined analogously to the maps Ri k. Properties follow directly from Condition 3.7 is a consequence of the boundedness of A 1 due to Remark 3.3. By property P4 in Lemma 3.1, the boundedness of the operators A i, i = 1,, N, the uniform invertibility condition 3.2, it follows that if v n is p-equiintegrable, then w k is p-equiintegrable as well. In view of property P5 in Lemma 3.1 if A i CperR N ; M l d, i = 1, N, v n Cc ; CperR N ; R d, then the sequence w k constructed in the proof of Lemma 3.1 inherits the same regularity. In order to characterize the limit differential constraint, for u L p ; R d n N we introduce the classes Cu A n := w L p ; L p perr N ; R d : wx, y dy = 0 for a.e. x, 3.22 A i nyux + wx, y dy = 0 in W 1,p ; R l, A i nyux + wx, y = 0 in W 1,p ; R l for a.e. x, C A n := u L p ; R d : C A n u For simplicity we will also adopt the notation C A u := C A 1 u C A := C A 1. Lemma 3.2 allows us to provide a first characterization of the set C A in the case in which A i C perr N ; M l d, i = 1, N. Proposition 3.4. Let 1 < p < +. Let A i CperR N ; M l d, i = 1,, N, with A satisfying the invertibility requirement in 3.2. Let C A be the class introduced in 3.23 let Aε div be the operator defined in 3.1. Then C A = u L p ; R d : there exists a sequence u ε L p ; R d such that u ε u 3.24 weakly in L p ; R d Aε div u ε 0 strongly in W 1,p ; R l. Moreover, for every u C A w Cu A there exists a sequence u ε L p ; R d such that 2 s u ε u + w strongly two-scale in L p ; R d, A div ε u ε 0 strongly in W 1,p ; R l. Proof. Denote by D the set in the right-h side of We divide the proof into two steps. Step 1 : We first show the inclusion D C A. Let u D, let u ε L p ; R d be such that u ε u weakly in L p ; R d 3.25

14 14 E. DAVOLI AND I. FONSECA Aε div u ε 0 strongly in W 1,p ; R l Consider a test function ψ Cc 1 ; R l. We have A div ε u ε, ψ 0, 3.27 where, denotes the duality product between W 1,p ; R l W 1,p 0 ; R l. By definition of the operators A div Aε div u ε, ψ = ε, A i x u ε x ψx dx for every ε > 0. ε By Proposition 2.2 there exists a map w L p ; L p perr N ; R d with wx, y dy = 0 such that, up to the extraction of a not relabeled subsequence where u ε 2 s v weakly two-scale 3.28 vx, y := ux + wx, y for a.e. x, y R N. Hence, by the definition of two-scale convergence, Aε div u ε, ψ A i yvx, y ψx dy dx, by 3.27 we have that A i yux + wx, y dy = 0 in W 1,p ; R l Let now ϕ CperR 1 N ; R l, ψ Cc 1, consider the sequence of test functions x ϕ ε x := εϕ ψx for x R N. ε The sequence ϕ ε is uniformly bounded in W 1,p 0 ; R l, therefore by 3.26 with Aε div u ε, ϕ ε = A div ε u ε, ϕ ε 0, 3.31 A i x ϕ x x ψx u ε x ψx + εϕ dx ε ε ε for every ε. Passing to the subsequence of u ε extracted in 3.28, applying the definition of two-scale convergence, we obtain A i yvx, y ϕy ψx dy dx = 0 for every ϕ C 1 perr N ; R l ψ C 1 c. By density, this equality still holds for an arbitrary ϕ W 1,p 0 ; R l, so A i yux + wx, y = 0 in W 1,p ; R l for a.e. x. 3.32

15 HOMOGENIZATION FOR A x UASICONVEXITY 15 Combining , we deduce that u C A. Step 2 : We claim that C A D. Let u C A, let w C A u, set Let v δ C c vx, y := ux + wx, y for a.e x y R N. ; C perr N ; R d be such that v δ v strongly in L p ; R d The sequence v δ satisfies both , hence by Lemma 3.2 Remark 3.3 we can construct a sequence v δ C ; C perr N ; R d such that v δ v strongly in L p ; R d, 3.34 A i yv δ x, y dy = 0 in W 1,p ; R l, 3.35 A i yv δ x, y = 0 in W 1,p ; R l for a.e. x Consider now the maps u ε δx := v δ x, x ε for every x. By Proposition 2.3 we have u ε δ as ε 0, hence, by Theorem s v δ strongly two-scale in L p ; R d T ε u ε δ v δ strongly in L p R N ; R d 3.37 where T ε is the unfolding operator defined in 2.1. We observe that by 3.36, A i x v δ x, x + A i x vδ x, x = ε ε ε ε for all x, for every ε δ. Moreover, by Propositions , A i x vδ x, x A i y v δ x, y dy 3.39 ε ε = A i yv δ x, y dy = 0 as ε 0, weakly in L p ; R d, where the last equality follows by Finally, since Aε div u ε δx = A i x u ε ε δx = 1 ε + [ A i x ε v δ x, x ε A i x vδ x, x, ε ε + A i x vδ x, x ] ε ε

16 16 E. DAVOLI AND I. FONSECA by 3.38, 3.39, the compact embedding of L p into W 1,p, we conclude that A div ε u ε δ 0 strongly in W 1,p ; R l, 3.40 as ε 0. Collecting 3.34, 3.37, 3.40, we deduce that lim lim T ε u ε δ u + w Lp + Aε div u ε δ W 1,p ;R δ 0 ε 0 l = 0. By Attouch s diagonalization lemma [3, Lemma 1.15 Corollary 1.16], there exists a subsequence δε such that T ε u ε δε u + w L p + Aε div u ε δε W 1,p ;R l = 0. lim ε 0 Setting u ε := u ε δ ε, we finally obtain hence, by Proposition 2.2, A div ε u ε 0 strongly in W 1,p ; R l u ε 2 s u + w strongly two-scale in L p ; R d, u ε u weakly in L p ; R d. This yields that u D completes the proof of the proposition. Remark 3.5. The regularity of the operators A i played a key role in Step 2. In the case in which A i L perr N ; M l d, i = 1,, N, but we have no further smoothness assumptions on the operators, the argument in Step 1 still guarantees that u L p ; R d : there exists a sequence u ε L p ; R d such that 3.41 u ε u weakly in L p ; R d Aε div u ε 0 strongly in W 1,q ; R l for every 1 q < p C A. Indeed, arguing as in Step 1 we obtain that there exists w L p ; L p perr N ; R d with wx, y dy = 0, such that u ε 2 s u + w weakly two-scale in L p ; R d, A i yux + wx, y dy = 0 in W 1,q ; R l 3.42 A i yux + wx, y = 0 in W 1,q ; R l for a.e. x, for all 1 q < p. Since u + w L p ; L p perr N ; R d, it follows that 3.42 holds also for q = p. Therefore we deduce the inclusion The proof of the opposite inclusion, on the other h, is not a straightforward consequence of Proposition 3.4. In fact, in the case in which the operators A i are only bounded, the second conclusion in Remark 3.3 does not hold anymore, we are not able to guarantee that the projection operator provided by Lemma 3.2 preserves the regularity of smooth functions. Therefore, the measurability of the maps u ε δ is questionable see [2, discussion below Definition 1.4]. This difficulty

17 HOMOGENIZATION FOR A x UASICONVEXITY 17 will be overcome in Lemma 5.3 by means of an approximation of the operators A i with C operators. We recall that F r A n u := inf for every u C A n r C A n r 4. Homogenization for smooth operators fux + wx, y dy dx : w Cu A n, w L p ;R d r r > 0, where := v L p ; R d : w C A n v F r A n u := F r A n u with w Lp ;R d r, n if u CA r, + otherwise in L p ; R d, for every r > 0, inf r>0 inf lim inf F r A n u n : u n u weakly in L p ; R d F A u := if u C A, + otherwise in L p ; R d. 4.3 Remark 4.1. We observe that for every u C A there holds S := u n : u n u weakly in L p ; R d, u n C A n for every n N F A u < +. Indeed, let u C A w Cu A. Then a change of variables the periodicity of w yield immediately that wx, ny dy = 0 for a.e. x n A i nyux + wx, ny dy = 0 in W 1,p ; R l. Proving that n A i nyux + wx, ny = 0 in W 1,p ; R l for a.e. x is equivalent to showing that n A i yux + wx, y = 0 in W 1,p n; R l for a.e. x. 4.4 To this purpose, arguing as in Step 2 of the proof of Proposition 3.4, construct v δ C ; C perr N ; R d such that v δ u + w strongly in L p ; R d, 4.5 n A i nyv δ x, y dy = 0 in W 1,p ; R l, 4.6

18 18 E. DAVOLI AND I. FONSECA n A i yv δ x, y = 0 in W 1,p ; R l for a.e. x. 4.7 By the smoothness the periodicity of v δ there holds n A i yv δ x, y = 0 in W 1,p n; R l for a.e. x 4.4 follows in view of 4.5. By the previous argument, wx, ny C A n u, therefore the set S contains always the sequence u n := u for every n. Theorem 4.2. Let 1 < p < +. Let A i CperR N ; M l d, i = 1,, N, assume that the operator A satisfies the invertibility requirement in 3.2, let Aε div be the operator defined in 3.1. Let f : R d [0, + be a continuous function satisfying the growth condition 0 fv C1 + v p for every v R d, 4.8 where C > 0. Then, for every u L p ; R d there holds inf lim inf fu ε x dx : u ε u weakly in L p ; R d ε 0 Aε div u ε 0 strongly in W 1,p ; R l = inf lim sup fu ε x dx : u ε u weakly in L p ; R d ε 0 Aε div u ε 0 strongly in W 1,p ; R l = F A u. Before starting the proof of Theorem 4.2, we first state without proving a corollary of [14, Lemma 2.15] one of [12, Lemma 2.8], we prove an adaptation of [14, Lemma 2.15] to our framework. Lemma 4.3. Let 1 < p < +. Let u ε be a bounded sequence in L p ; R N such that div u ε 0 strongly in W 1,p 4.9 u ε u weakly in L p ; R N. Then there exists a p-equiintegrable sequence ũ ε such that div ũ ε = 0 in W 1,p for every ε, ũ ε u ε 0 strongly in L q ; R N for every 1 q < p, ũ ε u weakly in L p ; R N. Remark 4.4. A direct adaptation of the proof of [14, Lemma 2.15] yields also that the thesis of Lemma 4.3 still holds if we replace 4.9 with the condition for every 1 q < p. div u ε 0 strongly in W 1,q Lemma 4.5. Let 1 < p < +, let D. Let u ε L p D; R N be p- equiintegrable, with u ε 0 weakly in L p D; R N,

19 HOMOGENIZATION FOR A x UASICONVEXITY 19 div u ε 0 strongly in W 1,p D. Then there exists a p-equiintegrable sequence ũ ε L p ; R N such that ũ ε u ε 0 strongly in L p D; R N, ũ ε 0 strongly in L p \ D; R N, div ũ ε = 0 in W 1,p, ũ ε L p ;R N C u ε L p D;R N, ũ ε x dx = 0 for every ε. More generally, we have Lemma 4.6. Let 1 < p < +, u L p ; R d let u ε L p ; R d be such that u ε u weakly in L p ; R d 4.10 A div ε u ε 0 strongly in W 1,p ; R l Then there exists a p-equiintegrable sequence ũ ε such that ũ ε u weakly in L p ; R d, A div ε ũ ε 0 strongly in W 1,q ; R l for every 1 q < p, ũ ε u ε 0 strongly in L q ; R d for every 1 q < p. Proof. The proof follows by adapting [14, Lemma 2.15]. We sketch it for convenience of the reader. Consider the truncation function z if z k, τ k z := set u k ε := τ k u ε. Then by 4.10, k z z if z > k, u k ε u weakly in L p ; R d as k + ε 0, in this order. Moreover, by 4.10 u k ε u ε q L q ;R d 2 q u ε x q dx 2q k p q u ε k as k +, for every 1 q < p. Therefore, A div ε u k ε u ε 0 strongly in W 1,q ; R l as k +, for every 1 q < p, by 4.11 A div ε u k ε 0 strongly in W 1,q ; R l u ε x p dx 0 as k + ε 0 in this order, for every 1 q < p. The thesis follows now by a diagonal argument. The following propositions is a corollary of [12, Proposition 3.5 ii].

20 20 E. DAVOLI AND I. FONSECA Proposition 4.7. Let f : R d [0, + be a continuous map satisfying 4.8 for some 1 < p < +. Let u ε L p ; R d be a bounded sequence let ũ ε L p ; R d be p-equiintegrable such that Then lim inf ε 0 u ε ũ ε 0 Moreover, if g : R N R d [0, + satisfies in measure. fu ε x dx lim inf fũ ε x dx. ε 0 i g, ξ is measurable -periodic for every ξ R d, ii gy, is continuous for a.e. y R N, iii there exists a constant C such that then 0 gy, ξ C1 + ξ p for a.e. y R N ξ R d, x x lim inf g ε 0 ε, u εx dx lim inf g ε 0 ε, ũ εx dx. The next proposition is another corollary of [12, Proposition 3.5 ii]. We sketch its proof for the convenience of the reader. Proposition 4.8. Let 1 p < + λ 0, 1]. Let g : R N R d [0, + be such that i g, ξ is measurable -periodic for every ξ R d, ii gy, is continuous for a.e. y R N, iii there exists a constant C such that 0 gy, ξ C1 + ξ p for a.e. y R N ξ R d, let V be a p-equiintegrable subset of L p ; R d. Then there exists a constant C such that y y g λ, v 1x, y g λ, v 2x, y C v 1 v 2 Lp ;R L1 d for every v 1, v 2 V. Proof. Fix ε > 0. By the p-equiintegrability of V iii there exists δ 1 > 0 such that for every E measurable with E δ 1, for all v V, there holds y g λ, vx, y ε dx dy < E In view of the p-equiintegrability of V, there exists also R > 0 such that sup v R < δ 1 v V Without loss of generality, up to a translation a dilation, we can assume that. By Scorza-Dragoni Theorem see e.g. [12, Proposition B.1] there exists a compact set K such that \ K < 3 N δ 1, 4.14

21 HOMOGENIZATION FOR A x UASICONVEXITY 21 g is uniformly continuous on K B R 0. Hence, by the periodicity of g with respect to its first variable, there exists δ 2 > 0 such that ε gη, ξ 1 gη, ξ 2 < for every ξ 1, ξ 2 B R 0 such that ξ 1 ξ 2 < δ 2 η K + Z N. We observe that by Chebyshev s inequality there exists δ > 0 such that if then v 1 v 2 Lp ;R d δ, 4.16 v 1 v 2 δ 2 < δ Let v 1, v 2 V be such that 4.16 holds true. Decompose the set as = v 1 v 2 < δ 2 v 1 v 2 δ 2. By , we have y g λ, v 1x, y v 1 v 2 δ 2 y g λ, v 2ε 2x, y dy dx 7. We perform a decomposition of the set v 1 v 2 < δ 2 as In view of 4.13, v 1 v 2 < δ 2 = v 1 v 2 < δ 2, v 1 < R v 2 < R v 1 v 2 < δ 2, max v 1, v 2 R. v 1 v 2 < δ 2, max v 1, v 2 R v 1 R + v 2 R δ 1, hence by 4.12 v 1 v 2 <δ 2, max v 1, v 2 R y g λ, v 1x, y y g λ, v 2ε 2x, y dy dx 7. Defining m λ := 1 2λ + 1, by 4.14 since λ < 1 there holds 1 2m λ λ = 2λ 1 + 2λ + 1 3, 2λ \λk +Z N λ N [ m λ, m λ ] N \K +Z N λ N 2m λ N \K δ Setting E R := v 1 v 2 < δ 2, v 1 < R v 2 < R, a further decomposition yields E R = E R [ \ λk + Z N ] E R [ λk + Z N ]. By we obtain the estimate y y g λ, v 1x, y g λ, v 2ε 2x, y dy dx 7 E R [ \λk+z N ] by 4.15, E R [ λk+z N ] y g λ, v 1x, y The thesis follows combining y g λ, v ε 2x, y dy dx

22 22 E. DAVOLI AND I. FONSECA We now start the proof of Theorem 4.2. First we prove the liminf inequality. The argument relies on the use of the unfolding operator see Subsection 2.2 [12, Appendix A] the references therein. Proposition 4.9. Under the assumptions of Theorem 4.2 for every u L p ; R d there holds inf lim inf fu ε x dx : u ε u weakly in L p ; R d ε 0 Aε div u ε 0 strongly in W 1,p ; R l F A u Proof. Let C A be the class introduced in We first notice that, by Proposition 3.4, if u L p ; R d \ C A then u ε L p ; R d : u ε u weakly in L p ; R d hence 4.21 follows trivially. Define g : R N R d [0, + as A div ε u ε 0 strongly in W 1,p ; R l gy, ξ = fay 1 ξ for every y R N, ξ R d. =, By the continuity of f, g is measurable with respect to the first variable it is the composition of a continuous function with a measurable one, continuous with respect to the second variable. By 4.8, there holds 0 gy, ξ C1 + Ay 1 ξ p C1 + ξ p, 4.22 where the last inequality follows by the uniform invertibility assumption 3.2. We divide the proof of the proposition into five steps. Step 1: We first show that for every u C A there holds inf lim inf fu ε x dx : u ε u weakly in L p ; R d 4.23 ε 0 Aε div u ε 0 strongly in W 1,p ; R l x inf lim inf g, v εn x dx : ε n inf w C A u v εn Ayux + wx, y dy weakly in L p ; R d, div v εn 0 strongly in W 1,p ; R l, A x ε n 1vεn u weakly in L p ; R d Indeed let u C A let u ε be as in Up to the extraction of a subsequence ε n, lim inf fu ε x dx = lim fu εn x dx, ε 0 by Proposition 2.2 there exists w Cu A such that u εn 2 s u + w weakly two-scale in L p ; R d

23 HOMOGENIZATION FOR A x UASICONVEXITY 23 By the definition of g it is straightforward to see that x x fu εn x dx = g, A u εn x dx. ε n ε n Setting v εn that lim lim := A x ε n uεn, in view of the assumptions on u ε in 4.23, it follows div v εn = A i x u εn x 0 strongly in W 1,p ; R l, ε n A x ε n 1vεn u weakly in L p ; R d Finally, by 4.24 for every ϕ L p ; R d there holds x T lim v εn x ϕx dx = lim u εn x A ϕx dx ε n = ux + wx, y Ay T ϕx dy dx. This completes the proof of Step 2: We claim that x inf inf lim inf g, v εn x dx : 4.26 w Cu A ε n Ayux + wx, y dy weakly in L p ; R d, v εn div v εn 0 strongly in W 1,p ; R l, A x ε n 1vεn u weakly in L p ; R d inf w C A u inf lim inf ṽ εn is p-equiintegrable, ṽ εn 0 weakly in L p ; R d, div ṽ εn = 0 in W 1,p ; R l, x A 1ṽεn ux Az 1 dz ε n weakly in L p ; R d x g, Ayux + wx, y dy + ṽ εn x dx : ε n. Ayux + wx, y dy Let v εn be as in By Lemma 4.3, we construct a p-equiintegrable sequence v εn such that div v εn = 0 in W 1,p ; R l, 4.27 v εn v εn 0 strongly in L q ; R d for every 1 q < p, 4.28 Ayux + wx, y dy weakly in L p ; R d v εn

24 24 E. DAVOLI AND I. FONSECA Moreover, by Proposition 4.7, x g, v εn x ε n lim x dx lim inf g, v εn x dx. ε n By 4.29 by the uniform invertibility assumption 3.2, there exists a constant C such that A x L 1 vεn C for every n. ε n p ;R d Therefore, there exists a map φ L p ; R d such that, up to the extraction of a not relabeled subsequence, A x ε n 1 vεn φ weakly in L p ; R d By the properties of v εn in , the convergence in 4.30 holds for the entire sequence v εn φ = u. Claim 4.26 follows by setting ṽ εn x := v εn Ayux + wx, y dy for a.e. x. Indeed, by 4.29 we have since w C A u, by 4.27 ṽ εn 0 weakly in L p ; R d, div ṽ εn = 0 in W 1,p ; R l. Finally, x x x 1 A 1ṽεn x = A 1 vεn x A ε n ε n ε n Ayux + wx, y dy for a.e. x, therefore by 4.30 Riemann-Lebesgue lemma see e.g. [13], x A 1ṽεn ux Az 1 dz Ayux + wx, y dy ε n weakly in L p ; R d. Step 3: We show that for every w C A u every p-equiintegrable sequence ṽ εn satisfying ṽ εn 0 weakly in L p ; R d, 4.31 div ṽ εn = 0 in W 1,p ; R l, 4.32 x A 1ṽεn ux Az 1 dz Ayux + wx, y dy 4.33 ε n weakly in L p ; R d, there exists a p-equiintegrable family v ν,n : ν N, n N such that div v ν,n = 0 in W 1,p ; R l, 4.34 v ν,n 0 weakly in L p ; R d as n +, vν,n A ν x x u Az 1 dz Ayux + wx, y dy 4.36 νε n

25 HOMOGENIZATION FOR A x UASICONVEXITY 25 weakly in L p ; R d, as n + x lim inf g, Ayux + wx, y dy + ṽ εn x dx 4.37 ε n 1 sup lim inf g ν x, Ayux + wx, y dy + v ν,n x dx. ν N νε n To prove the claim we argue as in [12, Proposition 3.8]. Fix ν N, let 1 θ ν,n := νε n [0, 1] νε n set We notice that k ν,n := θ ν,n νε n N 0. θ ν,n 1 as n Without loss of generality, we can assume that. By Lemma 4.5 we extend every map ṽ εn to a map ṽ εn L p ; R d such that ṽ εn is p-equiintegrable, satisfies the following properties ṽ εn ṽ εn 0 strongly in L p ; R d, 4.39 ṽ εn 0 strongly in L p \ ; R d, div ṽ εn = 0 in W 1,p ; R l. In particular, by it follows that x 1ṽεn A x ux Az 1 dz ε n Ayux + wx, y dy 4.40 weakly in L p ; R d. By Proposition 4.7 the definition of θ ν,n k ν,n, x lim inf g, Ayux + wx, y dy + ṽ εn x dx 4.41 ε n lim inf g ν k ν,n θ ν,n x, Ayux + wx, y dy + ṽ εn x dx. For fixed, there holds θ ν,n for n large enough. Since g is nonnegative see 4.22, by 4.41 we have x lim inf g, Ayux + wx, y dy + ṽ εn x dx 4.42 ε n lim inf g ν k ν,n x, Ayux + wx, y dy + ṽ εn x dx θ ν,n θ ν,n = lim inf θ ν,n N g νk ν,n x, lim inf νk ν,n x, Ayux + wx, y dy + ṽ εn θ ν,n x dx g where the last inequality follows by 4.38, since Ayuθ ν,n x + wθ ν,n x, y dy + ṽ εn θ ν,n x dx uθ ν,n x + wθ ν,n x, y ux wx, y 0 strongly in L p ; R d.

26 26 E. DAVOLI AND I. FONSECA Letting tend to, by the p-equiintegrability of ṽ εn, there holds x lim inf g, Ayux + wx, y dy + ṽ εn x dx 4.43 ε n lim inf g νk ν,n x, Ayux + wx, y dy + ṽ εn θ ν,n x dx. Set v ν,n x := ṽ εn θ ν,n x for a.e. x, ν N n n 0 ν, where n 0 ν is big enough so that θ ν,n for n n 0 ν. Inequality 4.37 is a direct consequence of The p-equiintegrability of v ν,n, 4.34, 4.35 follow in view of To conclude the proof of the claim it remains to establish We first remark that, by , the sequence A ν 1 1vν,n νε n x x is uniformly bounded in L p ; R d. Therefore, there exists a map L L p ; R d such that, up to the extraction of a not relabeled subsequence, there holds Let ϕ C c A ν 1 νε n x 1 vν,n x Lx weakly in L p ; R d ; R d. Then, A ν 1 1 x vν,n x ϕx dx νε n Lx ϕx dx For n big enough, θ ν,n supp ϕ. Hence, 1 1vν,n A ν x x ϕx dx 4.46 νε n 1 y 1 y = θ ν,n N A ṽεn y ϕ dy θ ν,n supp ϕ ε n θ ν,n 1 y 1 y = θ ν,n N A ṽεn y ϕ ϕy dy ε n θ ν,n 1 y 1 + θ ν,n N A ṽεn y ϕy dy. ε n By 3.2 the first term in the right-h side of 4.46 is bounded by 1 y 1 y θ ν,n N A ṽεn y ϕ ϕy dy ε n θ ν,n y C ṽ εn Lp ;R ϕ d L ;M d N sup y, θ ν,n which is infinitesimal as n + due to By , the second term in the right-h side of 4.46 satisfies 1 y 1 lim θ ν,n N A ṽεn y ϕy dy ε n [ ] = ux Az 1 dz Ayux + wx, y dy ϕx dx. y Arguing by density, we conclude that Lx = ux Az 1 dz Ayux + wx, y dy for a.e. x

27 HOMOGENIZATION FOR A x UASICONVEXITY holds for the entire sequence. This completes the proof of Step 4: We claim that for every w C A u, every p-equiintegrable family v ν,n : ν N, n N satisfying , there exists a p-equiintegrable family w ν,n : ν N, n N such that div y w ν,n x, y := w ν,n x, y = 0 in W 1,p ; R l for a.e. x, 4.47 w ν,n 0 weakly in L p ; R d, as n +, 4.48 w ν,n x, y dy = 0 for a.e. x, wν,n A y x, y ux Az 1 dz Ayux + wx, y dy νε n 4.50 lim inf weakly in L p ; R d lim inf 1 g ν x, νε n 1 g νε n as n + ν +, in this order, Ayux + wx, y dy + v ν,n x dx 4.51 y, Azux + wx, z dz + w ν,n x, y dy dx + σ ν, where σ ν 0 as ν +. Let v ν,n be as above. We argue similarly to [12, Proof of Proposition 3.9]. We extend u, w, v ν,n to 0 outside, define ν,z := 1 ν z + 1 ν, z ZN, Z ν := z Z N : ν,z, 1 I ν,n := g ν x, Ayux + wx, y dy + v ν,n x dx. νε n By a change of variables, since g, 0 = 0 by 4.22, by the periodicity of g in its first variable, we obtain the following chain of equalities I ν,n = 1 1 z ν N g ν z Z νε n ν + η z, Ay u ν ν + η z + w ν ν + η ν, y dy ν z + v ν,n ν + η dη ν = 1 νx g η, Ay u + η νx + w + η z Z ν,z νε n ν ν ν ν, y dy ν νx + v ν,n + η dη dx ν ν = 1 g η, Ayux + wx, y dy + T 1 νε v ν,nx, η dη dx + σ n ν ν z Z ν ν,z where T 1 is the unfolding operator defined in 2.1, ν σ ν := 1 νx g η, Ay u + η νx + w + η νε n ν ν ν ν, y dy z Z ν ν,z

28 28 E. DAVOLI AND I. FONSECA 1 + T 1 v ν,nx, η g η, Ayux + wx, y dy + T 1 ν νε v ν,nx, η dη dx. n ν By Proposition 2.7, Ayux+wx, y dy as ν +. Moreover, by Proposition 2.7, the sequence Ayux + wx, y dy Ay is p-equiintegrable Ayux + wx, y dy Ay T 1 ux, η + T 1 wx, η, y dy ν ν Ay T 1 ux, η+t 1 wx, η, y dy 0 ν ν L p ;R d T 1 ν ux, η + T 1 ν wx, η, y L p z Zν ν,z\ as ν +. Hence, by Proposition 4.8, σ ν = σ ν 0 as ν +. We set z v ν,z,n y := T 1 v ν,n ν ν, y for a.e. y, for every z Z ν. dy ;R d 0 For fixed ν z Z ν, the sequence v ν,z,n is p-equiintegrable. Moreover, v ν,z,n 0 div y v ν,z,n = 0 in W 1,p ; R l weakly in L p ; R d, as n +. Setting w ν,z,n := v ν,z,n v ν,z,ny dy, the sequence w ν,z,n L p ; R d is p-equiintegrable such that div y w ν,z,n = 0 in W 1,p ; R l, w ν,z,n 0 weakly in L p ; R d, as n + w ν,z,n y dy = 0, w ν,z,n v ν,z,n 0 strongly in L p ; R d, as n By applying again Proposition 4.8 we obtain 1 z g y, Aξux + wx, ξ dξ + T 1 ν,z νε v ν,n n ν ν, y dy dx 1 g y, Aξux + wx, ξ dξ + w ν,z,n y dy dx + τ z,ν,n νε n with ν,z Therefore, by 4.22 since τ z,ν,n 0 as n +. z Z N ν,z,

29 HOMOGENIZATION FOR A x UASICONVEXITY 29 we deduce 4.51 with w ν,n x, y := z Z ν χ ν,z xw ν,z,n y for a.e. x, y. We observe that for ν fixed only a finite number of terms in the sum above are different from zero, hence properties 4.47, follow immediately. y 1wν,n 1 To prove 4.50, we notice that the sequence A νε n x, y is uniformly bounded in L p ; R d by , therefore it is enough to work with a convergent subsequence check that the limit is uniquely determined. Fix ϕ Cc ; Cper; R d set ψx := ux Az 1 dz Ayux + wx, y dy for a.e. x, ψ := 0 outside, Z ν ϕ := z Z N : ν,z supp ϕ. Then 1 1wν,n A y x, y ϕx, y dy dx νε n = 1 1wν,z,n A y y ϕx, y dy dx z Z ν ν,z supp ϕ νε n ϕ = 1 1wν,z,n A y y v ν,z,n y ϕx, y dy dx z Z ν ν,z νε n ϕ + 1 y 1vν,z,n y ϕx, y dy dx. νε n z Z ν ϕ By 3.2, we have z Z ν ϕ ν,z A ν,z A 1 1wν,z,n y y v ν,z,n y ϕx, y dy dx νε n C w ν,z,n y v ν,z,n y L p ;R N ϕ L ;R, d z Z ν ϕ which by 4.52 converges to zero as n + here we used the fact that the previous series is actually a finite sum for every ν N fixed. On the other h, 1 A y 1vν,z,n y ϕx, y dy dx 4.53 z Z ν ν,z νε n ϕ = [ 1 1T z z ] A y 1 z Z ν ν,z νε v ν,n n ν ν, y T 1 ψ ν ν, y ϕx, y dy dx ϕ + z T 1 ψ ν ν,z ν, y ϕx, y dy dx. z Z ν ϕ

30 30 E. DAVOLI AND I. FONSECA By the periodicity of A, the first term in the right-h side of 4.53 satisfies [ 1 1T z z ] lim A y 1 z Z ν ν,z νε v ν,n n ν ν, y T 1 ψ ν ν, y ϕx, y dy dx ϕ 1 N [ 1 1vν,n z = lim A y ν νε n ν + y z ψ ν ν + y ] z ϕ ν ν + η ν, y dy dη = lim z Z ν ϕ = lim z Z ν ϕ ν,z z Z ν ϕ ν,z [ 1 A νε n 1vν,n ] νx x ψx [ 1 1vν,n ] A νx x ψx νε n 1 ϕ ν νx + 1 ν η, νx dη dx ϕ 1 ν νx + 1 ν η, νx dη dx. By 4.36, recalling that ψ v ν,n have been extended to 0 outside, we conclude that 1 1wν,n lim A y x, y ϕx, y dy dx νε n = z T 1 ψ ν ν,z ν, y ϕx, y dx dy, so z Z ν 1 1wν,n A y x, y z χ ν,z xt 1 νε ψ n ν ν, y z Z ν weakly in L p ; R d, as n +. Finally, we claim that z χ ν,z xt 1 ψ ν ν, y ψx 4.54 z Z ν strongly in L p ; R d as ν +. Indeed, let ϕ L p ; R d. Then by Holder s inequality z χ ν,z xt 1 ψ ν ν, y ψx ϕx, y dy dx z Z ν = = = = z Z ν z Z ν z Z ν ν,z ν,z ν,z z T 1 ψ ν ν, y ψx z ψ ν + y ψx ν 1 ψ ν νx + y ψx ν T 1 ψx, y ψx ϕx, y dy dx ν ϕx, y dy dx ϕx, y dy dx T 1 ν ψx, y ψx L p ;R d ϕ L p ;R d. ϕx, y dy dx Property 4.54, thus 4.50, follow in view of Proposition 2.7. Step 5: by Steps 1 4 it follows that inf lim inf fu ε x dx : u ε u weakly in L p ; R d 4.55 ε 0

31 HOMOGENIZATION FOR A x UASICONVEXITY 31 Aε div u ε 0 strongly in W 1,p ; R l 1 inf inf lim inf lim inf g y, w C u ν + νε n div y w ν,n x, y = 0 in W 1,p ; R l for a.e. x, w ν,n 0 weakly in L p ; R d as n +, 1 1wν,n A y x, y ux Az 1 dz νε n weakly in L p ; R d as n + ν +, w ν,n x, y dy = 0. Azux + wx, z dz + w ν,n x, y dy dx : Ayux + wx, y dy By a diagonalization argument, given w ν,n as above we can construct nν such that, setting ε ν := ε nν w ν x, y := w ν,nν, we obtain the following inequality inf lim inf fu ε x dx : u ε u weakly in L p ; R d 4.56 ε 0 Aε div u ε 0 strongly in W 1,p ; R l 1 inf inf lim inf g y, Azux + wx, z dz + w ν x, y dy dx : w C u ν + νε ν div y w ν x, y = 0 in W 1,p ; R l for a.e. x, w ν 0 weakly in L p ; R d as ν +, 1 1wν A y x, y ux Az 1 dz Ayux + wx, y dy νε ν weakly in L p ; R d as ν +, w ν x, y dy = 0. Associating to every sequence w ν as in 4.56 the maps φ ν x, y := Azux + wx, z dz + w ν x, y for a.e. x y, inequality 4.56 can be rewritten as inf lim inf fu ε x dx : u ε u weakly in L p ; R d 4.57 ε 0 Aε div u ε 0 strongly in W 1,p ; R l 1 1φν inf inf lim inf f A y x, y dy dx : w C u ν + νε ν div y φ ν x, y = 0 in W 1,p ; R l for a.e. x, φ ν Azux + wx, z dz weakly in L p ; R d as ν +, 1 1φν A y x, y ux weakly in L p ; R d as ν +, νε ν

32 32 E. DAVOLI AND I. FONSECA φ ν x, z dz = Azux + wx, z dz. Finally, for φ ν as above, considering the maps 1 1φν v ν x, y := A y x, y for a.e. x y, νε n we deduce that inf lim inf fu ε x dx : u ε u weakly in L p ; R d 4.58 ε 0 Aε div u ε 0 strongly in W 1,p ; R l inf inf lim inf fv ν x, y dy dx : w Cu A ν + 1 div y A y v ν x, y = 0 in W 1,p ; R l for a.e. x, νε ν 1 A y v ν x, y Azux + wx, z dz weakly in L p ; R d as ν +, νε ν v ν x, y ux weakly in L p ; R d as ν +, 1 A z v ν x, z dz = Azux + wx, z dz νε ν inf lim inf fux + w ν x, y dy dx : ν + 1 div y A y ux + w ν x, y = 0 in W 1,p ; R l for a.e. x, νε ν 1 div x A y ux + w ν x, y dy = 0 in W 1,p ; R l, νε ν w ν x, y 0 weakly in L p ; R d. By 4.58 it follows, in particular, that inf lim inf fu ε x dx : u ε u weakly in L p ; R d 4.59 ε 0 Aε div u ε 0 strongly in W 1,p ; R l inf lim inf fu ν x + w ν x, y dy dx : ν + u ν u weakly in L p ; R d, w ν C A w ν x, y 0 weakly in L p ; R d. 1 νεν u, ν Fix u ν w ν as in Then there exists a constant r such that Therefore, setting sup w ν Lp ;R d r ν N n ν := 1 νε ν,

33 HOMOGENIZATION FOR A x UASICONVEXITY 33 we have lim inf ν + u n := u ν if n = n ν, u otherwise, fu ν x + w ν x, y dy dx lim inf ν + F r A 1 νεν u ν 4.61 = lim inf F r uν = lim inf F r ν + 1 A n A ν + ν u nν lim inf F r A n u n νεν inf lim inf F r A n u n : u n u weakly in L p ; R d. The thesis follows by taking the infimum with respect to r in the right-h side of 4.61 by invoking Remark We point out that the truncation by r in will be used in a fundamental way. Infact it guarantees that the sequences constructed in the proof of the limsup inequality see Proposition 4.12 are uniformly bounded in L p, hence it allows us to apply Attouch s diagonalization lemma see [3, Lemma 1.15 Corollary 1.16] in Step 3 of the proof of Proposition Remark In the case in which A i L R N ; M l d, i = 1,, N, the previous proof yields the inequality inf lim inf fu ε x dx : u ε u weakly in L p ; R d ε 0 Aε div u ε 0 strongly in W 1,q ; R l for every 1 q < p F A u. To see that, arguing as in Step 1 we get inf lim inf fu ε x dx : u ε u weakly in L p ; R d ε 0 Aε div u ε 0 strongly in W 1,q ; R l for every 1 q < p x inf lim inf g, v εn x dx : ε n inf w C A u v εn Ayux + wx, y dy weakly in L p ; R d, div v εn 0 strongly in W 1,q ; R l for every 1 q < p, A x ε n 1vεn u weakly in L p ; R d By Lemma 4.3 Remark 4.4, inequality 4.26 is replaced by x inf inf lim inf g, v εn x dx : w Cu A ε n Ayux + wx, y dy weakly in L p ; R d, v εn.