Andrea Braides 1, Irene Fonseca 2 and Giovanni Leoni 3

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1 ESAIM: Control, Optimisation and Calculus of Variations URL: Will be set by the publisher A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION Andrea Braides, Irene Fonseca 2 and Giovanni Leoni 3 Abstract. Integral representation of relaxed energies and of Γ-limits of functionals (u, v) f(x, u(x), v(x))dx are obtained when sequences of fields v may develop oscillations and are constrained to satisfy a system of first order linear partial differential equations. This framework includes the treatement of divergence-free fields, Maxwell s equations in micromagnetics, and curl-free fields. In the latter case classical relaxation theorems in W,p are recovered. AMS Subject Classification. 3599, 35E99, 49J45. Received March 6, Revised September 26, Introduction In a recent paper Fonseca and Müller [22] have proved that A-quasiconvexity is a necessary and sufficient condition for (sequential) lower semicontinuity of a functional (u, v) f(x, u(x), v(x)) dx, whenever f : R m R d [0, ) is a Carathéodory integrand satisfying 0 f(x, u, v) a(x, u)( + v q ), for a.e. x and all (u, v) R m R d, where q <, a L loc ( R; [0, )), RN is open, bounded, u n u in measure, v n v in L q (; R d ) and Av n 0 in W,q (; R l ) (see also [4]). Here, and in what follows, following [32] A : L q (; R d ) W,q (; R l ), Av := N i= (i) v A, x i Keywords and phrases: A-quasiconvexity, equi-integrability, Young measure, relaxation, Γ-convergence, homogenization. SISSA, Trieste, Italy. 2 epartment of MathematicalSciences, Carnegie-Mellon University, Pittsburgh, PA, U.S.A.; Fonseca@anderw.cmu.edu 3 ipartimento di Scienze e Tecnologie Avanzate, Università del Piemonte Orientale, Alessandria, Italy. c EP Sciences, SMAI 2000

2 2 A. BRAIES, I. FONSECA AN G. LEONI is a constant rank, first order linear partial differential operator, with A (i) : R d R l linear transformations, i =,..., N. We recall that A satisfies the constant-rank property if there exists r N such that where rank Aw = r for all w S N, (.) Aw := N w i A (i), w R N. A function f : R d R is said to be A-quasiconvex if f(v) f(v + w(y))dy i= for all v R d and all w C-per(R N ; R d ) such that Aw = 0 and w(y)dy = 0. Here denotes the unit cube in R N, and the space C-per (RN ; R d ) is introduced in Section 2. The relevance of this general framework, as emphasized by Tartar (see [32, 34 39]), lies on the fact that in continuum mechanics and electromagnetism PEs other than curl v = 0 arise naturally, and this calls for a relaxation theory which encompasses PE constraints of the type Av = 0. Some important examples included in this general setting are given by: (a) [Unconstrained Fields] Av 0. Here, due to Jensen s inequality A-quasiconvexity reduces to convexity. (b) [ivergence Free Fields] where v : R N R N (see [33]). (c) [Magnetostatics Equations] Av = 0 if and only if div v = 0, ( m A := h) ( ) div(m + h) = 0, curl h where m : R 3 R 3 is the magnetization and h : R 3 R 3 is the induced magnetic field (see [7,38]); often these are also called Maxwell s Equations in the micromagnetics literature. (d) [Gradients] Av = 0 if and only if curl v = 0. Note that w C-per (RN ; R d ) is such that curl w = 0 and w(y)dy = 0 if and only if there exists ϕ C-per(R N ; R n ) such that ϕ = v, where d = n N. Thus in this case we recover the well-known notion of quasiconvexity introduced by Morrey [30]. (e) [Higher Order Gradients] Replacing the target space R d by an appropriate finite dimensional vector space Es n, it is possible to find a first order linear partial differential operator A such that v L p (; Es n ) and Av = 0 if and only if there exists ϕ W s,q (; R n ) such that v = s ϕ (see Th..3). This paper is divided into two parts. In the first part we give an integral representation formula for the relaxed energy in the context of A quasiconvexity. Precisely, let p < and < q <, and consider the functional F : L p (; R m ) L q (; R d ) O() [0, ) defined by F((u, v); ) := f(x, u(x), v(x)) dx,

3 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 3 where O() is the collection of all open subsets of, and the density f satisfies the following hypothesis: (H) f : R m R d [0, ) is Carathéodory function satisfying 0 f(x, u, v) C ( + u p + v q ) for a.e. x and all (u, v) R m R d, and where C > 0. For O() and (u, v) L p (; R m ) ( L q (; R d ) ker A ) define F((u, v); ) := inf lim inf F((u n, v n ); ) : (u n, v n ) L p (; R m ) L q (; R d ), u n u in L p (; R m ), v n v in L q (; R d ), Av n 0 in W,q (; R l ) (.2) It turns out that the condition Av n 0 imposed in (.2) may be replaced by requiring that v n do satisfy the homogeneous PE Av = 0. Precisely, and in view of Lemma 3. and Corollary 3.2 below, it can be shown that F((u, v); ) = inf lim inf F((u, v n); ) : v n L q (; R d ), v n v in L q (; R d ), Av n = 0, and thus F((u, v); ) = inf lim inf F((u n, v n ); ) : (u n, v n ) L p (; R m ) L q (; R d ), u n u in L p (; R m ), v n v in L q (; R d ), Av n = 0 =: F 0 ((u, v); ). (.3) The first main result of the paper is given by the following theorem: Theorem.. Under condition (H) and the constant-rank hypothesis (.), for all O(), u L p (; R m ), and v L q (; R d ) ker A, we have F((u, v); ) = A f(x, u(x), v(x))dx where, for each fixed (x, u) R m, the function A f(x, u, ) is the A-quasiconvexification of f(x, u, ), namely for all v R d. A f(x, u, v) := inf f(x, u, v + w(y))dy : w C-per(R N ; R d ) ker A, w(y)dy = 0 Remarks.2. (i) Note that in the degenerate case where A = 0, A-quasiconvex functions are convex and Theorem. together with condition (.4) yield a convex relaxation result with respect to L p L q (weak) convergence. See the monograph of Buttazzo [2] for related results in this context. (ii) If the function f also satisfies a growth condition of order q from below in the variable v, that is f(x, u, v) C v q C (.4)

4 4 A. BRAIES, I. FONSECA AN G. LEONI for a.e. x and all (u, v) R m R d, then a simple diagonalization argument shows that (u, v) F((u, v); ) is L p (L q -weak) lower semicontinuous, i.e., A f(x, u(x), v(x))dx lim inf A f(x, u n (x), v n (x))dx (.5) whenever u n L p (; R m ), v n L q (; R d ) ker A, u n u in L p (; R m ), v n v in L q (; R d ). In particular A f is A-quasiconvex if f is continuous and C v q C f(v) C( + v q ) for some C > 0, and all v R d (see the proof of Cor. 5.7). The lower semicontinuity result (.5) is not covered by Theorem 3.7 in [22], where it is assumed that the integrand be A-quasiconvex and continuous in the v variable. However, as remarked in [22], in the realm of general A-quasiconvexity the function A f(x, u, ) may not be continuous, even if f(x, u, ) is. Indeed in the degenerate case ker A = 0 all functions are A-quasiconvex. Also, when N =, d = 2, and v = (v, v 2 ), consider Av := ( 0 ) ( ) v v 2. Then for w R Aw = ( 0 w ) and thus when w = the matrix Aw has constant rank. For any given function f(v) the A-quasiconvex envelope of f is obtained by convexification in the first component, so that by considering e.g. (cf. [22,28]) one gets A f (v) = f (v) := e v v2 2, f2 (v) := ( + v ) v2, 0 if v 2 0 if v 2 = 0, ( + v ) v2 if v 2 Af (v) = if v 2 <. (iii) The continuity of f with respect to v is essential to ensure the representation of F provided in Theorem., in contrast with the case where Av = 0 if and only if curl v = 0. In fact, if f : R n N [0, ) is a Borel function satisfying the growth condition 0 f(v) C( + v q ) for C > 0, q <, v R n N, then it can be shown easily that F(w; ) = f( w(x)) dx (.6) for all O(), w W,q (; R n ), where f is the quasiconvex envelope of f. Indeed, f is a (continuous) quasiconvex function satisfying (H) (see [8, 8] Th. 4.3); therefore by Theorem. w is W,q -sequentially weakly lower semicontinuous, and so f( w(x)) dx f( w(x))dx F(w; ).

5 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 5 Conversely, under hypothesis (H) it is known that F(v; ) admits an integral representation (see Th. 9. in [0], Th. 20. in [5]) F(w; ) = ϕ( w(x)) dx, where ϕ is a quasiconvex function, and ϕ(v) f(v) for all v R n N. Hence ϕ f and we conclude that (.6) holds. For general constant-rank operators A, and if f is not continuous with respect to v, it may happen that F 0 ((u, v); ) is not even the trace of a Radon measure in O() and thus (.3) fails. As an example, consider d = 2, N =, := (0, ), v = (v, v 2 ), and let A(v) = 0 if and only if v 2 = 0 as in (ii) above. Let (v ) 2 + v2 2 f(v) :=, if v 2 (v + ) 2 + v2, 2 if v 2 /. Although f satisfies a quadratic growth condition of the type (H), and (A 3 ) holds with q = 2, it is easy to see that for all intervals (a, b) (0, ), b b F 0 ((u, v); (a, b)) = F 0 (v; (a, b)) = min ((v ) 2 + v2)dx, 2 ((v + ) 2 + v2)dx 2 a which is not the trace of a Radon measure on O(). On the other hand, it may be shown that (see the Appendix below for a proof) where ψ (v ) is the convex envelope of F((u, v); (a, b)) = F(v; (a, b)) = b a a (ψ (v ) + v 2 2 )dx, ψ(v ) := min (v ) 2, (v + ) 2 (iv) Using the growth condition (H), a mollification argument, and the linearity of A, it can be shown that (see Rem. 3.3 in [22]) A f(x, u, v) = inf f(x, u, v + w(y))dy : w L q -per (RN ; R d ) ker A, w(y)dy = 0 We write w L q -per (RN ; R d ) ker A when w L q -per (RN ; R d ) and Aw = 0 in W,q (; R l ). (v) We may also treat the cases q =, and p =. See Theorem 3.6 below. The proof of Theorem. relies heavily on the use of Young measures (see [5, 40]). However, instead of applying directly the arguments of Fonseca and Müller [22] (based on Balder s [4] and Kristensen s [26] approach in the curl free case), we use these together with the blow-up method introduced by Fonseca and Müller in [20]. Although in Theorem. the functions u and v are not related to each other, the arguments of the proof work equally well when u and v are not independent. Indeed as a corollary, we can prove the following two theorems: Theorem.3. Let p, s N, and suppose that f : E n [s ] En s [0, ) is a Carathéodory function satisfying 0 f(x,u, v) C ( + u p + v p ), p <, for a.e. x and all (u, v) E n [s ] En s, where C > 0, and f L loc ( En [s ] En s ; [0, )) if p =.

6 6 A. BRAIES, I. FONSECA AN G. LEONI Then for every u W s,p (; R n ) we have s f(x, u,..., s u)dx = inf lim inf f(x, u k,..., s u k )dx : u k W s,p (; R n ), u k u in W s,p (; R n ) ( if p = ), where, for a.e. x and all (u, v) E n [s ] En s, s f(x,u, v) := inf f(x,u, v + s w(y))dy : w C-per(R N ; R n ) Remarks.4. (i) Here E n s stands for the space of n-tuples of symmetric s-linear maps on RN, E n [s ] := Rn E n E n s, and ( l l ) u u := x α.., l.. xαn N α +...+α N=l (ii) When s = we recover classical relaxation results (see e.g. the work of Acerbi and Fusco [], acorogna [3], Marcellini and Sbordone [28] and the references contained therein). When s > lower semicontinuity results related to Theorem.3 are due to Meyers [29], Fusco [23] and Guidorzi and Poggiolini [25], while we are not aware of any integral representation formula for the relaxed energy, when the integrand depends on the full set of variables, that is f = f(x, u,..., s u). This is due to the fact that classical truncation methods for s = cannot be extended in a simple way to truncate higher order derivatives. The results of Fonseca and Müller (see the proof of Lem. 2.5 in [22]), where the truncation is only on the highest order derivative s u, and Corollary 3.2 below, allows us to overcome this difficulty. Note however that this technique relies heavily on p-equi-integrability, and thus cannot work in the case p =, if one replaces weak convergence in W s, (; R n ) with the natural convergence, which is strong convergence in W s, (; R n ). In this context, a relaxation result has been given by Amar and e Cicco [2], but only when f = f( s u), so that truncation is not needed. The general case where f depends also on lower order derivatives has been addressed by Fonseca et al. [9]. Theorem.5. Let p, let R N be an open, bounded, connected set, and suppose that f : R N R N2 [0, ) is a Carathéodory function satisfying 0 f(x, u, v) C ( + u p + v p ), p <, for a.e. x and all (u, v) R N R N2, where C > 0, and f L loc( R N R N2 ; [0, )) if p =. Then for every u W,p (; R N ) such that div u = 0, we have f(x, u(x), u(x))dx = inf lim inf f(x, u n (x), u n (x))dx : u n W,p (; R N ), div u n = 0, u n u in W,p (; R N ) ( if p = ), (.7)

7 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 7 where, for a.e. x and all (u, v) R N R N2, f(x, u, v) := inf f(x, u, v + w(y))dy : w C-per(R N ; R N ), div w = 0 Remark.6. To the authors knowledge, this result is new in this generality (for a different proof, with additional smoothness assumptions, see [9]). A related problem was addressed by al Maso et al. in [6], where it was shown that the Γ-limit of a family of functionals of the type (.7) may be non local if (H) is violated. In the second part of the paper we present (Γ-convergence) homogenization results for periodic integrands in the context of A-quasiconvexity. Let ε > 0 and < q <, and consider a family of functionals defined by F ε : ( L q (; R d ) ker A ) O() [0, ) F ε (v; ) := where the density f satisfies the following hypotheses: ( x ) f ε, v(x) dx, (A ) f : R N R d [0, ) is a continuous function, -periodic in the first argument, that is f(x+e i, v) = f(x, v) for every i =,..., N, where e i are the elements of the canonical basis of R N ; (A 2 ) there exists C > 0 such that for all (x, v) R N R d ; (A 3 ) there exists C > 0 such that for all (x, v) R N R d. Let ε n 0 +. We say that a functional 0 f(x, v) C( + v q ) f(x, v) C v q C J : ( L q (; R d ) ker A ) O() [0, + ] is the Γ lim inf (resp. Γ lim sup) of the sequence of functionals F εn with respect to the weak convergence in L q (; R d ) if for every v L q (; R d ) ker A J (v; ) = inf lim inf (resp. lim sup) F εn (v n ; ) : v n L q (; R d ) ker A, v n v in L q (; R d ), (.8) and we write ( J = Γ lim inf F ε n resp. J = Γ lim sup F εn ). When finite energy sequences are L q -equibounded then the infimum in the definition of Γ lim inf (resp. Γ lim sup) is attained. We say that the sequence F εn Γ-converges to J if the Γ lim inf and Γ lim sup coincide, and we write J = Γ lim F ε n.

8 8 A. BRAIES, I. FONSECA AN G. LEONI The functional J is said to be the Γ lim inf (resp. Γ lim sup) of the family of functionals F ε with respect to the weak convergence in L q (; R d ) if for every sequence ε n 0 + we have that ( J = Γ lim inf F ε n resp. J = Γ lim sup F εn ), and we write ( ) J = Γ lim inf F ε resp. J = Γ lim sup F ε. ε 0 ε 0 Finally, we say that J is the is the Γ-limit of the family of functionals F ε, and we write J = Γ lim F ε n, if Γ lim inf and Γ lim sup coincide. In the sequel we will also consider functionals J given by (.8) where we replace the weak convergence v n v with the convergence v n v with respect to some metric d. In order to highlight this dependence on the metric d these functionals will be denoted as as it is customary (see [0,5]). J = Γ(d) lim inf F ε n ( ) resp. J = Γ(d) lim sup F εn, Theorem.7. Under hypotheses (A ) (A 2 ) and the constant-rank hypothesis (.), where for all v L q (; R d ) ker A and O(), and f hom (v) := inf k N for all v R d. Moreover, if (A 3 ) holds then F hom = Γ lim inf ε 0 F ε, F hom (v; ) := f hom (v)dx k N inf f(x, v + w(x))dx : w L q k-per (RN ; R d ) ker A, k F hom = Γ lim ε 0 F ε. For the definition of the space L q k-per (RN ; R d ), we direct the reader to Section 2. k (.9) w(x)dx = 0 Remarks.8. (i) Using the growth condition (A 2 ), a mollification argument, and the linearity of A, it can be shown that f hom (v) = inf k N k N inf f(x, v + w(x))dx : w L k-per(r N ; R d ) ker A, k w(x)dx = 0 See also Corollary 5.7 below. k

9 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 9 (ii) When f satisfies the q-lipschitz condition f(x, v ) f(x, v 2 ) C( v q + v 2 q + ) v v 2 (.0) for all x R N, v, v 2 R d, and for some C > 0, then the continuity of f(, v) can be weakened to measurability, namely f can be assumed to be simply Carathéodory. Note that (.0) is not restrictive when A = curl, that is when v = u for some u W,q (; R m ), d = N m. Indeed, in this case in the definition of Γ-convergence we may replace the weak convergence of the gradients in L q (; R d ) with the strong convergence in L q (; R m ) of the potentials normalized to have zero average over, and thus ( x ) ( x ) Γ lim f ε 0 ε, v(x) dx = Γ(L q (; R m )) lim f ε 0 ε, u(x) dx ( x ) = Γ(L q (; R m )) lim f ε 0 ε, u(x) dx, by Proposition 7.3 in [0]. As shown in [27], if f(x, v) is a Borel function which satisfies the growth condition (A 2 ) then its quasiconvex envelope f satisfies (.0). A similar argument fails for general A-quasiconvexity, since the function A f(x, ) may not even be continuous, see Remark.2(i) above. In Section 2 we collect preliminary results on Young measures and Γ-convergence. The general relaxation results (see Th.. and its exstension Th. 3.6) are proved in Section 3, and Section 4 is devoted to the applications of the general relaxation principle to Theorems.3 and.5. Finally, in Section 5 we address homogenization of functionals of A-constrained vector fields. 2. Preliminaries We start with some notation. Here is an open, bounded subset of R N, L N is the N dimensional Lebesgue measure, S N := x R N : x = is the unit sphere, and := ( /2, /2) N the unit cube centered at the origin. We set (x 0, ε) := x 0 +ε for ε > 0 and x 0 R N. A function w L q loc (RN ; R d ) is said to be periodic if w(x + e i ) = w(x) for a.e. all x R N and every i =,..., N, where (e,..., e N ) is the canonical basis of R N. We write w L q -per (RN ; R d ). More generally, w L q loc (RN ; R d ) is said to be k periodic, k N, if w(k ) is periodic. We write w L q k-per (RN ; R d ). Also C-per(R N ; R d ) will stands for the space of periodic functions in C (R N ; R d ). We recall briefly some facts about Young measures which will be useful in the sequel (see e.g. [5, 33]). If is an open set (not necessarily bounded), we denote by C c (; R d ) the set of continuous functions with compact support in, endowed with the supremum norm. The dual of the closure of C c (; R d ) may be identified with the set of R d -valued Radon measures with finite mass M(; R d ), through the duality ν, f := f(y)dν(y), ν M(; R d ), f C c (; R d ). A map ν : M(; R d ) is said to be weak- measurable if x ν x, f are measurable for all f C c (; R d ). The following result is a corollary of the Fundamental Theorem on Young Measures (see [5,7,34]) Theorem 2.. Let z n : R d be measurable functions such that z n q dx <, sup n N for some q > 0. Then there exists a subsequence z nk of z n and a weak- measurable map ν : M(R d ; R d ) such that

10 0 A. BRAIES, I. FONSECA AN G. LEONI (i) ν x 0, ν x M = R d dν x = for a.e. x ; (ii) if f : R d R is a normal function bounded from below then lim inf f(x, z nk (x))dx f(x)dx <, where f(x) := ν x, f(x, ) = f(x, y)dν x (y); R d (iii) for any Carathéodory function f : R d R bounded from below one has if and only if f(, z nk ( )) is equi-integrable. lim f(x, z nk (x)dx = f(x)dx < The map ν : M(R d ; R d ) is called theyoung measure generated by the sequence z nk. Proposition 2.2. If z n generates a Young measure ν and v n 0 in measure, then z n + v n still generates the Young measure ν. If < q then W,q (; R l ) is the dual of W,q 0 (; R l ), where q is the Hölder conjugate exponent of q, that is /q + /q =. It is well known that F W,q (; R l ) if and only if there exist g,..., g N L q (; R l ) such that N F, w = g i w dx for all w W,q 0 (; R l ). x i i= Consider a collection of linear operators A (i) : R d R l, i =,..., N, and define the differential operator A : L q (; R d ) W,q (; R l ) v Av as follows: Av, w := N i= (i) v A, w = x i N i= A (i) v w x i dx for all w W,q 0 (; R l ). Even though the operator A so defined depends on, we will omit reference to the underlying domain whenever it is clear from the context. In particular, if v L q -per (RN ; R d ) then we will say that v ker A if Av = 0 in W,q (; R l ). Throughout the paper we assume that A satisfies the constant-rank property (.). The following proposition is due to Fonseca and Müller [22]. Proposition 2.3. (i) ( < q < + ) Let < q < +, let V n be a bounded sequence in L q (; R d ) such that AV n 0 in W,q (; R l ), V n V in L q (; R d ), and assume that V n generates a Young measure ν. Then there exists a q-equi-integrable sequence v n L q (; R d ) ker A such that v n dx = V dx, v n V n Ls () 0 for all s < q, and, in particular, v n still generates ν. Moreover, if = then v n V L q -per (RN ; R d ) ker A.

11 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION (ii) (q = ) Let V n be a sequence converging weakly in L (; R d ) to a function V, AV n 0 in W,r (; R l ) for some r (, N/(N )), and assume that V n generates a Young measure ν. Then there exists an equi-integrable sequence v n L (; R d ) ker A such that v n dx = V dx, v n V n L () 0, and, in particular, v n still generates ν. Moreover, if = then v n V L -per(r N ; R d ) ker A. (iii) (q = + ) Let V n be a sequence that satisfies V n V in L (; R d ), AV n 0 in L r () for some r > N, and assume that V n generates a Young measure ν. Then there exists a sequence v n L (; R d ) ker A such that v n dx = V dx, v n V n L () 0, and, in particular, v n still generates ν. Moreover, if = then v n V L -per(r N ; R d ) ker A. In the second part of the paper we will need the following classical results from Γ-convergence. For a proof see [0]. Proposition 2.4. Let (X, d) be a separable metric space and let f n : X [, ]. Then (i) there exists an increasing sequence of integers n k such that Γ(d) lim f n k (x) exists for all x X. (ii) Moreover f = Γ(d) lim f n if and only if for every subsequence f nk there exists a further subsequence f nkj which Γ(d)-converges to f. 3. Relaxation In this section we prove Theorem. and its generalization to the case where q, and p = (see Th. 3.6). Lemma 3.. Let f : R m R d [0, ) be a Carathéodory function satisfying (H), with p < and < q <. Let (u, v) L p (; R m ) ( L q (; R d ) ker A ), where O(), and consider a sequence of functions (u k, ˆv k ) L p (; R m ) L q (; R d ) such that u k u in L p (; R m ), ˆv k v in L q (; R d ) Aˆv k 0 in W,q (; R l ). (3.) Then we can find a q-equi-integrable sequence v k L q (; R d ) ker A such that v k v in L q (; R d ), v k dx = v dx, and lim inf f(x, u(x), v k (x))dx lim inf f(x, u k (x), ˆv k (x))dx.

12 2 A. BRAIES, I. FONSECA AN G. LEONI Proof of Lemma 3.. Consider a subsequence (u n, ˆv n ) of (u k, ˆv k ) such that lim f(x, u n (x), ˆv n (x))dx = lim inf f(x, u k (x), ˆv k (x))dx and (u n, ˆv n ) generates the Young measure δ u(x) ν x x. For i N let F i := x : dist(x, ) <, i and consider cut-off functions θ i with compact support in and such that θ i in \ F i. Set w i,n := θ i (ˆv n v) L q (; R d ) and fix ϕ L q (; R d ), where q is the Hölder conjugate exponent of q. Then lim lim ϕ(x)w i,n (x)dx = lim lim ϕ(x)θ i (x)(ˆv n (x) v(x))dx = 0, (3.2) i i where we have used the fact that ˆv k v in L q (; R d ). Hence w i,n 0 in L q (; R d ) as n and i. Moreover, in view of the compact embedding L q (; R l ) W,q (; R l ) and the assumption that Aˆv k 0 in W,q (; R l ), we have that lim lim Aw i,n = 0 in W,q (; R l ). i Let G be a countable dense subset of L q (; R d ). By means of a diagonalization process we obtain subsequences u i := u ni and ŵ i := w i,ni = θ i (ˆv ni v) such that u i u L p 0, (3.2) holds for each ϕ G, and Aŵ i 0 in W,q (; R l ). Hence ŵ i 0 in L q (; R d ), by the density of G in L q (; R d ). By Proposition 2.3(i) there exists a q-equiintegrable sequence w i L q (; R d ) ker A such that w i 0 in L q (; R d ), and w i dx = 0, ŵ i w i Ls () 0 for all s < q. (3.3) Set v i := v + w i. Then v i dx = v dx, v i v in L q (; R d ). By Hölder s inequality and by (3.3), for s < q ˆv ni v i Ls () ˆv ni v ŵ i Ls () + ŵ i w i Ls () ( θ i )(ˆv ni v) Ls () + ŵ i w i Ls () ˆv ni v L q() F i r + ŵ i w i L s() 0 (3.4) as i and where r := (q s)/sq. By (3.4) and Proposition 2.2, the two sequences (u(x), v i (x)) and (u i (x), ˆv i (x)) generate the same Young measure δ u(x) ν x x. Hence by Theorems 2.(ii) and (iii) lim f(x, u(x), v i (x))dx = f(x, u(x), V )dν x (V )dx lim inf f(x, u i (x), ˆv i (x))dx i R d i = lim inf f(x, u k (x), ˆv k (x))dx, (3.5)

13 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 3 where we have used the fact that f(x, u(x), v i (x)) is equi-integrable over, which follows from (H) and the q-equi-integrability of v i over. It follows immediately from Lemma 3. that under its assumptions on f it holds: Corollary 3.2. For O() and (u, v) L p (; R m ) ( L q (; R d ) ker A ) F((u, v); ) = inf lim inf g(x, v n (x))dx : v n L q (; R d ) ker A is q-equi-integrable and v n v in L q (; R d ), where g is the Carathéodory function defined by g(x, v) := f(x, u(x), v). Note that, by (H), the function g satisfies the growth condition 0 g(x, v) C ( + u(x) p + v q ) (3.6) for a.e. x and all v R d. Moreover, since g is a Carathéodory function, by the Scorza-ragoni theorem for each j N there exists a compact set K j, with \ K j /j, such that g : K j R d [0, ) is continuous. Let K j be the set of Lebesgue points of χ Kj, and set ω := (K j Kj ) L(u, v), (3.7) j where L(u, v) is the set of Lebesgue points of (u, v). Then \ ω \ K j j 0 as j. Corollary 3.3. Assume that x 0 ω, let v L q (; R d ) ker A, and consider r k 0 + and a sequence of functions such that ˆv k L q (; R d ) ker A ˆv k v in L q (; R d ). Then we can find a q-equi-integrable sequence w k L q -per (RN ; R d ) ker A such that w k 0 in L q (; R d ), w k dx = 0, and lim inf g(x 0, v(y) + w k (y))dy lim inf g(x 0 + r k y, ˆv k (y))dy.

14 4 A. BRAIES, I. FONSECA AN G. LEONI Proof of Corollary 3.3. We proceed as in the proof of Lemma 3. up to (3.4). Since the sequence v i is q-equiintegrable, for any η > 0 there exists δ > 0 such that sup i C( + u(x 0 ) p + v i (y) q )dy < η (3.8) for any measurable set, with < δ, and where C is the constant given in (H). Fix η > 0 and let δ > 0 be given according to (3.8). By the Biting Lemma (see [6]) we may find a further subsequence ˆv nj ˆv ni and a set E such that \ E < δ and ˆv nj is q-equi-integrable over E. Hence there exists 0 < δ < δ such that sup C( + u(x 0 ) p + ˆv nj (y) q )dy < η (3.9) j for any measurable set E, with < δ. Moreover, as ˆv nj, v j are bounded in L q (; R d ), we may find L > 0 such that E \ E j δ, where E j := y E : ˆv nj (y) L, v j (y) L (3.0) Note that by construction of v i and by Proposition 2.3, v i = v + w i where w i L q -per (RN ; R d ) ker A. From the definition of the set ω there exists an integer j 0 such that x 0 K j0 K j 0. Since is uniformly continuous, there exists ρ > 0 such that g : K j0 B d (0, L) [0, ) g(x, v) g(x, v) η (3.) for all (x, v), (x, v) K j0 B d (0, L), with x x ρ. By (3.0) and (3.) lim g(x 0 + r ni y, ˆv ni (y))dy lim inf g(x 0 + r nj y, ˆv nj (y))dy i j E j lim inf g(x, ˆv j rn N nj ((x x 0 )/r nj ))dx j (x 0+r nj E j) K j0 η + lim inf g(x 0, ˆv nj ((x x 0 )/r nj ))dx. j (x 0+r nj E j) K j0 r N n j (3.2) Using, once again, the fact that ˆv nj (y) L for y E j, by (3.6) we have that r N n j (x 0+r nj E j)\k j0 g(x 0, ˆv nj ((x x 0 )/r nj ))dx C( + u(x 0 ) p + L q ) (x 0, r nj ) \ K j0 r N n j 0 as j, because x 0 is a Lebesgue point of χ Kj0. Consequently, from (3.2) we get lim g(x 0 + r ni y, ˆv ni (y))dy η + lim inf g(x i j rn N 0, ˆv nj ((x x 0 )/r nj ))dx j x 0+r nj E j = η + lim inf g(x 0, ˆv nj (y))dy j E j 2η + lim inf g(x 0, ˆv nj (y))dy, j E

15 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 5 where we have used (3.6, 3.9) and the fact that E \ E j δ. We may now proceed as in the previous lemma, using the Carathéory function h(x, v) := χ E (x)g(x 0, v), to obtain lim i g(x 0 + r ni y, ˆv ni (y))dy 2η + lim inf j E by (3.8). It now suffices to let η 0 +. g(x 0, v j (y))dy 3η + lim inf g(x 0, v j (y))dy j Theorem. follows from Lemmas 3.4 and 3.5 below. We will use the notation µ A to denote the restriction of a Radon measure µ to the Borel set A, i.e., µ A(X) := µ(x A) where X is an arbitrary Borel set in the domain of µ. Lemma 3.4. F((u, v); ) is the trace of a Radon measure absolutely continuous with respect to L N. Proof of Lemma 3.4. As it is usual, it suffices to prove subadditivity (see e.g. [3, 2]), i.e. F((u, v); ) F((u, v); \ B) + F((u, v); C) if B C. Fix η > 0. By Corollary 3.2 there exist two q-equi-integrable sequences v k L q ( \ B; R d ) ker A, w k L q (C; R d ) ker A, such that and v k v in L q ( \ B; R d ), w k v in L q (C; R d ), lim lim \B C g(x, v k (x))dx F((u, v); \ B) + η, g(x, w k (x))dx F((u, v); C) + η. Let θ j be smooth cut-off functions, θ j C c (C; [0, ]), θ j (x) = for all x B, and 0 < θ j < 0 as j. Set ˆV j,k := ( θ j )v k + θ j w k. Then, for j fixed, AˆV j,k = ( θ j )Av k + θ j Aw k N i= A (i) v k θ j x i + N i= A (i) w k θ j x i 0 as k in W,q (; R l ) strong. Using a diagonalization procedure such as that adopted in the proof of Lemma 3., we get ˆV j v in L q (; R d ), AˆV j 0 in W,q (; R l ), where ˆV j := ˆV j,kj. By Lemma 3. we can find a q-equi-integrable sequence V j L q (; R d ) ker A such that V j v in L q (; R d ) and lim inf j g(x, V j (x))dx lim inf g(x, ˆV j (x))dx. j

16 6 A. BRAIES, I. FONSECA AN G. LEONI Consequently, in view of Corollary 3.2 F((u, v); ) lim inf j lim sup j + lim sup j g(x, V j (x))dx lim inf j θ j=0 0<θ j< g(x, v kj (x))dx + lim sup j 2η + F((u, v); \ B) + F((u, v); C). It suffices to let η 0 +. Finally, note that by (H) we have that g(x, ˆV j (x))dx g(x, w kj (x))dx θ j= C( + u(x) p + w kj (x) q + v kj (x) q )dx F((u, v), ) C( + u p + v q )L N. Lemma 3.5. For L N a.e. x 0 we have df((u, v); ) dl N (x 0 ) = A f(x 0, u(x 0 ), v(x 0 )). Proof of Lemma 3.5. Fix x 0 ω, where ω is defined as in (3.7), and such that and lim u(x) u(x r 0 + r N 0 ) p dx = lim v(x) v(x (x 0,r) r 0 + r N 0 ) q dx = 0 (3.3) (x 0,r) df((u, v); ) F((u, v); (x 0, r)) dl N (x 0 ) = lim r 0 + r N <, where, by virtue of Lemma 3.4, we have chosen the radii r 0 + such that F((u, v); ((x 0, r))) = 0. By Corollary 3.2 and for r > 0 fixed, let v n,r L q ((x 0, r); R d ) ker A be such that v n,r v in L q ((x 0, r); R d ) as n and Then df((u, v); ) dl N (x 0 ) lim inf r 0 + lim g(x, v n,r (x))dx F((u, v); (x 0, r)) + r N+. (x 0,r) lim r N (x 0,r) g(x, v n.r (x))dx = lim inf r 0 + lim g(x 0 + ry, v(x 0 ) + w n,r (y))dy where w n,r (y) := v n,r (x 0 + ry) v(x 0 ). We claim that w n,r 0 in L q (; R d ) if we first let n and then r 0 +. Indeed let ϕ L q (; R d ), where q is the Hölder conjugate exponent of q. Using Hölder s inequality

17 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 7 and then making a change of variables, we get ϕ(y)w r,n (y)dy ϕ(y)(v n,r (x 0 + ry) v(x 0 + ry))dy + ϕ(y)(v(x 0 + ry) v(x 0 ))dy r N ϕ((x x 0 )/r)(v n,r (x) v(x))dx (x 0,r) ( /q + ϕ L q () v(x) v(x r N 0 ) dx) q. (x 0,r) If we now let n the first integral tends to zero, since v n,r v in L q ((x 0, r); R d ). The claim then follows by letting r 0 + and by using (3.3). iagonalize to get ŵ k L q (; R d ) ker A such that ŵ k 0 in L q (; R d ) and df((u, v); ) dl N (x 0 ) lim g(x 0 + r k y, v(x 0 ) + ŵ k (y))dy where r k 0. By Corollary 3.3 there is a q-equi-integrable sequence w k L q -per (RN ; R d ) ker A such that and w k 0 in L q (; R d ), w k dy = 0, df((u, v); ) dl N (x 0 ) lim g(x 0 + r k y, v(x 0 ) + ŵ k (y))dy lim inf f(x 0, u(x 0 ), v(x 0 ) + w k (y))dy A f(x 0, u(x 0 ), v(x 0 )). To conclude the proof of the lemma it remains to show that df((u, v); ) dl N (x 0 ) A f(x 0, u(x 0 ), v(x 0 )) for L N a.e. x 0. Fix η > 0 and let w C-per(R N ; R d ) ker A be such that w dy = 0 and f(x 0, u(x 0 ), v(x 0 ) + w(y))dy A f(x 0, u(x 0 ), v(x 0 )) + η. (3.4) For any fixed r > 0 set w n,r (x) := w(n(x x 0 )/r). Then w n,r 0 in L q ((x 0, r); R d ) as n. Hence, by Corollary 3.2, df((u, v); ) F((u, v); (x 0, r)) (x dl N 0 ) = lim lim inf lim inf g(x, v(x) + w r 0 + r N r 0 + r N n,r (x))dx. (3.5) (x 0,r) Fix L > v(x 0 ) + w L +, and let j be such that x 0 K j Kj L(u, v), where we are using the notation introduced in (3.7). Since g : K j B d (0, L) [0, ) is uniformly continuous, there exists 0 < ρ < such that g(x, v) g(x, v ) η (3.6)

18 8 A. BRAIES, I. FONSECA AN G. LEONI for all (x, v), (x, v ) K j B d (v(x 0 ), L), with x x ρ and v v ρ. Let E r,ρ := x (x 0, r) : v(x) v(x 0 ) ρ We claim that lim sup r 0 r N C( + u(x) p + v(x) q + w q L )dx = 0. (3.7) + (x 0,r)\(E r,ρ K j) Since v(x) v(x 0 ) ρ for x (x 0, r) \ E r,ρ, we have and r N C( + u(x) p + v(x) q + w q L )dx C (x 0, r) \ (E r,ρ K j ) (x 0,r)\(E r,ρ K j) r N + C r N ( u(x) u(x 0 ) p + v(x) v(x 0 ) q ) dx. (x 0,r) (x 0, r) \ (E r,ρ K j ) (x 0, r) \ K j + (x 0, r) \ E r,ρ (x 0, r) \ K j r N r N r N r N + C ρ q r N v(x) v(x 0 ) q dx 0 as r 0 +, (x 0,r) where we have used (3.3) and the fact that x 0 is a Lebesgue point of χ Kj. Then by (3.6, ) and (3.4), df((u, v); ) dl N (x 0 ) lim inf lim inf r 0 + r N g(x, v(x) + w n,r (x))dx E r,ρ K j + lim sup C( + u(x) p + v(x) q + w q r 0 r + N L )dx (x 0,r)\(E r,ρ K j) η + lim inf lim inf g(x r 0 + r N 0, v(x 0 ) + w n,r (x))dx (x 0,r) = η + lim inf g(x 0, v(x 0 ) + w(ny))dy by virtue of the equality lim inf = η + g(x 0, v(x 0 ) + w(y))dy 2η + A f(x 0, u(x 0 ), v(x 0 )), g(x 0, v(x 0 ) + w(ny))dx = g(x 0, v(x 0 ) + w(y))dy, which follows from the -periodicity of the function g(x 0, v(x 0 ) + w( )). It now suffices to let η 0 +. As mentioned in the Introduction, Theorem. continues to hold when q, and p =. Indeed, let p, q and assume that (A 4 ) f : R m R d [0, ) is a Carathéodory function satisfying the following growth conditions for a.e. x and all (u, v) R m R d : 0 f(x, u, v) C ( + u p + v q ) if p, q <, (3.8)

19 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 9 where C > 0; 0 f(x, u, v) a(x, u)( + v q ) if p = and q <, where a L loc ( Rd ; [0, )); where b L loc ( Rm ; [0, )); 0 f(x, u, v) b(x, v)( + u p ) if p < and q =, f L loc( R m R d ; [0, )) if p = q =. For O() and (u, v) L p (; R m ) ( L q (; R d ) ker A ) define F((u, v); ) := inf lim inf F((u n, v n ); ) : (u n, v n ) L p (; R m ) L (; R d ), u n u in L p (; R m ), v n v in L (; R d ), Av n 0 in W,r (; R l ) if q = and for some r (, N/(N )); as in (.2), we set F((u, v); ) := inf lim inf F((u n, v n ); ) : (u n, v n ) L p (; R m ) L q (; R d ) u n u in L p (; R m ), v n v in L q (; R d ), Av n 0 in W,q (; R l ) if < q < ; F((u, v); ) := inf lim inf F((u n, v n ); ) : (u n, v n ) L p (; R m ) L (; R d ), u n u in L p (; R m ), v n v in L (; R d ), Av n 0 in L r (; R l ) if q = and for some r > N. We can prove the following theorem: Theorem 3.6. Under condition (A 4 ) and the constant-rank hypothesis (.), for all O(), u L p (; R m ) and v L q (; R d ) ker A, we have F((u, v); ) = A f(x, u(x), v(x))dx. Proof of Theorem 3.6. Step. Assume first that p < and q =. The proof is similar to the one of Theorem., with the exceptions that in Lemma 3. condition (3.) should be replaced by that we use the compact embedding u k u in L p (; R m ), ˆv k v in L (; R d ), Aˆv n 0 in W,r (; R l ) for some r (, N/(N )), L (; R l ) W,r (; R l ), r (0, N/(N ),

20 20 A. BRAIES, I. FONSECA AN G. LEONI to diagonalize w i,n, and (3.3, 3.4) are replaced, respectively, by w i dx = 0, ŵ i w i L () 0, ˆv ni v i L () ˆv ni v ŵ i L () + ŵ i w i L () ( θ i )(ˆv ni v) L () + ŵ i w i L () ˆv ni v L (F i) + ŵ i w i L () 0, where we have used the fact that ˆv ni v L (F i) 0 as i, which is due to the equi-integrability of the original sequence ˆv k v and the fact that F i 0. Step 2. If p = and q < then in Lemma 3. the only change needed is in deriving (3.5), which now follows from the fact that, by (3.8), 0 f(x, u(x), v i (x)) A ( + v i (x) q ), where A := supa(x, u) : x, u u <, and thus equi-integrability of f(x, u, v i ) follows from the q-equi-integrability of v i over. Moreover in the remaining of the proof of Theorem., the growth condition (3.6) should be replaced by for a.e. x and all v R d. 0 g(x, v) A ( + v q ) (3.9) Step 3. If p and q = then in Lemma 3. the hypothesis (3.) should be replaced by u k u in L p (; R m ), ˆv k v in L (; R d ), Aˆv n 0 in L r (; R l ) for some r > N, the growth condition should be replaced by (3.9) if p <, q =, and by g L loc ( Rd ; [0, )) if p = q =, and we can proceed similarly to the proof of Lemma 3. to show that w i,n 0 in L (; R d ) and Aw i,n 0 in L r (; R l ), and use Proposition 2.3(iii) to get We omit the details. v i ˆv ni Proofs of Theorems.3 and.5 Proof of Theorem.3. We present the proof for p <, the case p = being very similar. Fix u W s,p (; R n ), and for O() define F(u; ) := inf lim inf f(x, u k,..., s u k )dx : u k W s,p (; R n ), u k u in W s,p (; R n ), and let g be the Carathéodory function g(x, v) := f(x, u(x),..., s u(x), v).

21 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 2 Reasoning as in Lemma 3.4, it is easy to show that F(u; ) is the trace of a Radon measure absolutely continuous with respect to L N. For any function v L p (; E n s ) set G(v; ) := inf lim inf g(x, V k (x))dx : V k L p (; E n s ) ker A and V k v in L p (; E n s ), is p-equi-integrable, where the differential operator A is given by ( Av := v i...i x hji h+2...i s ) v i...i i x hii h+2...i s. j 0 h s, i,j,i...i s N Here h = 0 and h = s correspond to the multi-indeces ji 2...i s and i...i s j. By Theorem 3.6 (and Cor. 3.2), and where the target space R d is being replaced by the finite dimensional vector space Es n, for any O() G(v; ) = A g(x, v(x))dx, where for a.e. x and for all v E n s, A g(x, v) := inf g(x, v + w(y))dy : w C-per(R N ; Es n ) ker A, w(y)dy = 0 As shown in [22], w C -per(r N ; E n s ) : Aw = 0, w dx = 0 = s ϕ : ϕ C-per(R N ; R n ) (4.) Hence In particular A g(x, v) = inf g(x, v + s ϕ(y))dy : ϕ C-per(R N ; R N ) G( s u; ) = s f(x, u,..., s u)dx. (4.2) Let u k W s,p (; R n ) be any sequence such that u k u in W s,p (; R n ). Extracting a subsequence, if necessary, we may assume that u k := (u k,..., s u k ) u := (u,..., s u) in L p (; E n [s ] ). Since s u k s u in L p (; E n s ) and A s u k = 0, by Lemma 3. there exists a p-equi-integrable sequence V k L p (; E n s ) ker A such that V k s u in L p (; E n s ) and lim inf g(x, V k (x))dx lim inf f(x, u k,..., s u k )dx.

22 22 A. BRAIES, I. FONSECA AN G. LEONI Thus G( s u; ) F(u; ). (4.3) To prove the converse inequality, fix x 0 and r > 0, and consider any p-equi-integrable sequence V k L p (B(x 0 ; r); Es n) ker A such that V k s u in L p (B(x 0 ; r); Es n ). An induction argument, similar to the one used in [22] to prove (4.) above, shows that AV k = 0 if and only if there exists ϕ k W s,p (B(x 0 ; r); R n ) such that s ϕ k = V k. By Lemmas..3 in [24], for any ϕ W s,p (B(x 0 ; r); R n ) we may find a unique function P C (R N ; R n ) whose components are polynomials of degree s such that l (ϕ P)dx = 0 0 l s, (4.4) B(x 0,r) and a constant C(n, N, s, p, r) > 0 such that the following Poincaré type inequality holds ϕ P W s,p (B(x 0;r);R n ) C s ϕ Lp (B(x 0;r);E n s ). (4.5) Let P k and P be the functions associated to ϕ k and u, respectively, and satisfying (4.4, 4.5). Since s ϕ k s u in L p (B(x 0 ; r); Es n ), we have that ϕ k P k u P in W s,p (B(x 0 ; r); R n ), so u k := ϕ k P k + P u in W s,p (B(x 0 ; r); R N ). Consider a subsequence of V k (not relabelled) such that the two sequences (u k,..., s u k, V k ) and (u,..., s u, V k ) generate the Young measure δ (u(x),..., s u) ν x x B(x0,r), and B(x 0,r) (u k,..., s u k ) (u,..., s u) pointwise and in L p (B(x 0 ; r); Es n ). Since V k is p-equi-integrable and u k converge to u strongly in W s,p (; R N ), it follows from Theorem 2. and the growth condition on f that lim f(u k,..., s u k, V k )dx = lim g(x, V k (x))dx. Thus which, together with (4.3), yields B(x 0,r) G( s u; B(x 0, r)) F(u; B(x 0, r)), G( s u; B(x 0, r)) = F(u; B(x 0, r)). (4.6) Since F(u; ) and G( s u; ) are both traces of a Radon measures absolutely continuous with respect to L N, by (4.2) and (4.6) we immediately obtain that F(u; ) = G( s u; ) = f(x, u,..., s u)dx.

23 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 23 Proof of Theorem.5. We only proof Theorem.5 for p <, the case p = being very similar. For v R N2 let v = (v (),..., v (N) ), where v (i) R N, i =,..., N, v (N) R N. Given a function v L p (; R N2 ) define the differential operator A as follows curl v () Av :=. curl v (N ). curl (v (N), v ()... v (N ) N ) A straightforward calculation shows that A satisfies the constant-rank property (.). Given a Carathéodory function f : R N R N2 [0, ), we define ˆf(x, u, v), for (x, u, v) R N R N2, as ( ( ( ))) ˆf(x, u, v) = f x, u, v (),..., v (N ), v (N) v ()... v (N ) N Let u W,p (; R N ), with div u = 0, and let u n W,p (; R N ) be such that div u n = 0 and u n u in W,p (; R N ). By Lemma 3. there exists a p-equi-integrable sequence V n L p (; R N2 ) ker A such that V n v in L p (; R N2 ) and where lim inf ˆf(x, u, V n )dx lim inf v n := u() n,..., u(n ) n, u (N) n x. u (N) n x N ˆf(x, u n, v n )dx = lim inf f(x, u n, u n )dx, (4.7), v := u(),..., u (N ),. u (N) x. u (N) x N. (4.8) efine G(v; ) := inf lim inf ĝ(x, V n (x))dx : V n L p (; R N2 ) ker A is p-equi-integrable, and V n v in L p (; R N2 ), where ĝ is the Carathéodory function defined by ĝ(x, v) := ˆf(x, u(x), v). By Theorem 3.6 (and Cor. 3.2) G(v; ) = A ĝ(x, v)dx, (4.9) where A ĝ(x, v(x)) := inf ˆf(x, u(x), v(x) + w(y))dy : w C-per(R N ; R N2 ) ker A, w(y)dy = 0

24 24 A. BRAIES, I. FONSECA AN G. LEONI Now w C -per(r N ; R N2 ) ker A if and only if there exists ϕ C -per(r N ; R N ) such that and w = ϕ(),..., ϕ (N ), ϕ (N) x. ϕ (N) x N w(y)dy = 0 and ϕ(n) x N = ϕ() x... ϕ(n ) x N. Hence A ĝ(x, v(x)) = inf f(x, u(x), u(x) + ϕ(y))dy : ϕ C-per(R N ; R N ), div ϕ = 0 = f(x, u(x), u(x)). (4.0) Thus, by (4.7, 4.9), and (4.0), f(x, u(x), u(x))dx = G(v; ) lim inf ˆf(x, u, V n )dx lim inf f(x, u n (x), u n (x))dx, and, in turn, f(x, u(x), u(x))dx inf lim inf f(x, u n (x), u n (x))dx : u n W,p (; R N ), div u n = 0, u n u in W,p (; R N ) To prove the converse inequality, fix ε > 0. By the definition of G(v; ), there exists a p-equi-integrable sequence V n L p (; R N2 ) ker A such that V n v in L p (; R N2 ) and f(x, u(x), u(x))dx+ ε > lim inf ĝ(x, V n (x))dx = lim inf ϕ() n,..., ϕ (N ) n, ˆf(x, u(x), V n (x))dx, (4.) where we used for v the notation introduced in (4.8). Now AV n = 0 if and only if there exists ϕ n W,p (; R N ) such that ϕ (N) n x V n =. ϕ (N) n x N and ϕ(n) n x N = ϕ() n x... ϕ(n ) n x N. Since ϕ n u in L p (; R N2 ), we have that ϕ n ϕ n (x)dx U in W,p (; R N ),

25 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 25 where U = u + c for some constant c R N. So u n := ϕ n ϕ n (x)dx c u in W,p (; R N ), and div u n = 0. Consider a subsequence V nk of V n such that lim ˆf(x, u(x), V nk (x))dx = lim inf ˆf(x, u(x), V n (x))dx and (u nk, V nk ) and (u, V nk ) generates the Young measure δ u(x) ν x x. Since V nk is p-equi-integrable and u nk converge to u strongly in L p (; R N ), it follows from Theorem 2. and the growth condition on f that lim ˆf(x, u(x), V nk (x))dx = lim ˆf(x, u nk (x), V nk (x))dx. By (4.) f(x, u(x), u(x))dx+ ε > lim inf ˆf(x, u nk (x), V nk (x))dx = lim lim inf f(x, u nk (x), u nk (x))dx f(x, u n (x), u n (x))dx : u n W,p (; R N ), div u n = 0, u n u in W,p (; R N ) It now suffices to let ε Homogenization In this section we will limit our analysis to the case where < q <. Lemma 5.. Let f : R N R d [0, ) be a continuous function satisfying (A )-(A 2 ). Let v L q (; R d ) ker A, where O(), ε k 0 +, and let ˆv k L q ( ; R d ) be a sequence of functions such that ˆv k v in L q ( ; R d ), Aˆv k 0 in W,q ( ; R l ), for some O(), with. Then we can find a q-equi-integrable sequence v k L q (; R d ) ker A such that v k dx = v dx, v k v in L q (; R d ), ˆv k v k L s( ) 0 for all s < q (5.) and lim inf f (x/ε k, v k (x)) dx lim inf f (x/ε k, ˆv k (x)) dx, lim sup v k (x) q dx \ v(x) q dx. \ (5.2) Moreover, if =, then v k = v + w k, with w k L q -per (RN ; R d ) ker A.

26 26 A. BRAIES, I. FONSECA AN G. LEONI Remark 5.2. Lemma 5. implies, in particular, that for every v L q (; R d ) ker A Γ lim inf F ε n (v; ) = inf lim inf F ε n (v n ; ) : v n L q (; R d ) ker A, v n v in L q (; R d ), v n dx = v dx, and if = then Γ lim inf F ε n (v; ) = inf lim inf F ε n (v + w n ; ) : w n L q per (RN ; R d ) ker A w n 0 in L q (; R d ), w n dx = 0 Proof of Lemma 5.. Let g(x) := x in and extend it periodically to R N with period. Set g k (x) := g(x/ε k ). Since g k is bounded in L and ˆv k v in L q ( ; R d ), by Theorem 2. there exists a subsequence ε n of ε k such that (g n (x), ˆv n (x)) generates a Young measure ν x and For i N let lim f (x/ε n, ˆv n (x)) dx = lim inf f (x/ε k, ˆv k (x)) dx. F i := x : dist(x, ) < i and consider cut-off functions θ i with compact support in and such that θ i in \ F i. Set w i,n := θ i (ˆv n v) L q (; R d ). Then we can proceed as in the proof of Lemma 3. to find a q-equi-integrable sequence v i := v + w i, where w i satisfies (3.3, 5.) holds, and the two sequences (g ni (x), v i (x)) and (g ni (x), ˆv ni (x)) generate the same Young measure ν x. Hence by Theorem 2. ( lim f (x/ε ni, v i (x)) dx = f(x, V )dν x (X, V ) i R N R d = lim inf f (x/ε k, ˆv k (x)) dx, ) dx lim f (x/ε ni, ˆv ni (x)) dx i where we have used (A 2 ), and the facts that v i (x) is q-equi-integrable over, and that f is a continuous function. To prove the second inequality in (5.2), we remark that by (3.3) and the fact that ŵ i = θ i (ˆv ni v) 0 outside, we have for all s < q v i v Ls (\ ) = ŵ i w i Ls (\ ) 0. Hence v i (x) generates the Young measure µ x = δ v(x) on \, and since v i is q-equi-integrable we have that lim sup v i (x) q dx = Y q dµ x (Y )dx = v(x) q dx. i \ \ \

27 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 27 To complete the proof it suffices to define v k := v ni for each n i k < n i+. Clearly lim inf f (x/ε k, v k (x)) dx lim inf f (x/ε ni, v i (x)) dx. i Lemma 5.3. Let ε n 0 + and let R() be the family of all finite unions of open cubes contained in and with vertices in N. Then there exists a subsequence ε nk of ε n such that the Γ limit Γ lim F ε nk (v; R) exists for all v L q (R; R d ) ker A and for all R R(). Proof of Lemma 5.3. Fix R R(). For simplicity set F n := F εn and let B denote the closed unit ball of L q (R; R d ). For each l N consider l B := v L q (R; R d ) : v L q l Since q > the dual of L q (R; R d ) is separable, and hence the space l B endowed with the weak topology is metrizable. Let d l be any metric which generates the L q -weak topology. Consider l = and apply Proposition 2.4 to the sequence of functionals F n ( ; R) restricted to (B ker A, d ). Then we can find an increasing sequence of integers n j such that Γ(d ) lim F n j (v; R) j exists for all v B ker A. We now proceed recursively, so that given l N we apply Proposition 2.4 to the sequence of functionals F n l ( ; R) restricted to (l B ker A, d l ) to obtain a subsequence n l j of nl j such j that Γ(d l ) lim F n (v; R) j l j exists for all v l B ker A. Let n k := n k k. Since n k is a subsequence of all n l j we have that for each l N exists for all v l B ker A. We claim that the Γ limit Γ(d l ) lim F n k (v; R) Γ lim F n k (v; R) (5.3) exists for all v L q (R; R d ) ker A. Indeed assume by contradiction that this is not the case. Then there exists v L q (R; R d ) ker A for which F (v; R) := Γ lim inf Let v k L q (R; R d ) ker A be such that v k v in L q (R; R d ) and F n k (v; R) < F + (v; R) := Γ lim sup F nk (v; R). lim inf F n k (v k ; R) = F (v; R). Since v k v in L q (R; R d ), we may find an integer l 0 such that v k, v l 0 B ker A for all k N. Consequently d l0 (v k, v) 0 as k,

28 28 A. BRAIES, I. FONSECA AN G. LEONI and thus Γ(d l0 ) lim inf F n k (v; R) lim inf F n k (v k ; R) = F (v; R) < F + (v; R) Γ(d l0 ) lim sup F nk (v; R), which contradicts the existence of the Γ-limit Γ(d l0 ) lim n k (v; R), and where we have used the fact that F + (v; R) = inf lim sup F nk (z k ; R) : z k L q (R; R d ) ker A, z k v in L q (R; R d ) Γ(d l0 ) lim sup F nk (v; R) = inf lim sup F nk (z k ; R) : z k l 0 B ker A, z k v in L q (R; R d ) Hence (5.3) holds. To conclude the proof of the lemma it suffices to observe that since the family R() is countable, with a diagonal process it is possible to extract a further subsequence for which (5.3) holds for all R R(). Remark 5.4. The previous proof asserts that for any given O() and ε n 0 + there exists a subsequence ε nk (depending on the particular set ) of ε n such that such that the Γ limit exists for all v L q (; R d ) ker A. Γ lim F ε nk (v; ) Lemma 5.5. Assume that conditions (A )-(A 2 ) hold. Given ε n 0 +, let ε nk be as in Lemma 5.3, and for any O() set F ( ; ) := Γ lim inf F ε nk ( ; ). Then F (v; ) is the trace of a Radon measure. Proof of Lemma 5.5. We start by establishing inner regularity. Precisely, we claim that for any v L q (; R d ) ker A and O() F (v; ) = sup F (v; R) : R R(), R = lim Rր F (v; R), (5.4) where the limit is taken over all finite unions of cubes R R() with R. For fixed η > 0 there exists δ > 0 such that 0 C( + v(x) q )dx < η (5.5) for any measurable set 0, with 0 < δ, and where C is the constant given in (A 2 ). Let R R(), with R and \ R < δ, and, in light of Lemma 5.3, consider a sequence ˆv k L q (R; R d ) ker A, with ˆv k v in L q (R; R d ), and such that lim F ε nk (ˆv k ; R) = F (v; R). By Lemma 5. there exists a q-equi-integrable sequence v k L q (; R d ) ker A such that v k v in L q (; R d ), v k dx = v dx,

29 A-UASICONVEXITY: RELAXATION AN HOMOGENIZATION 29 and lim inf R lim sup f (x/ε nk, v k (x)) dx lim f (x/ε nk, ˆv k (x)) dx, R v k (x) q dx v(x) q dx. \R \R Hence F (v; ) lim inf f (x/ε nk, v k (x)) dx lim f (x/ε nk, ˆv k (x)) dx + lim sup C( + v k (x) q )dx R \R F (v; R) + C( + v(x) q )dx F (v; R) + η, \R where we have used (A 2 ) and (5.5). Consequently F (v; ) sup F (v; R) : R R(), R + η, and letting η 0 + we obtain one inequality in (5.4). To show the opposite inequality, note that if v k L q (; R d ) ker A, with v k v in L q (; R d ), then the restriction of v k to R belongs to L q (R; R d ) ker A, and v k v in L q (R; R d ). Therefore F (v; R) lim inf R and by taking the infimum over all such sequences we get that f (x/ε nk, v k (x)) dx lim inf f (x/ε nk, v k (x)) dx F (v; R) F (v; ), (5.6) and in turn (5.4) holds. In order to prove that F (v; ) is the trace of a Radon measure, as it is usual it suffices to prove subadditivity for nested sets (see [3,2]). Let B C. By (5.4) for fixed η > 0 we find R R() such that R and Construct R, R 2 R() with By (5.6) we have F (v; ) η + F (v; R). R R R 2, R \ B and R 2 C. F (v; ) η + F (v; R) η + F (v; R R 2 ). (5.7) By the definition of Γ-convergence and Lemma 5. there exist v k L q (R ; R d ) ker A and w k L q (R 2 ; R d ) ker A, with v k v in L q (R ; R d ) and w k v in L q (R 2 ; R d ), such that F (v; R ) = lim F ε nk (v k ; R ), F (v; R 2 ) = lim F ε nk (w k ; R 2 ), (5.8)