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VII 33 (1987), pp. 157–177. [10] A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Clarendon Press, Oxford, 1998. [11] A. Braides, A. Defranceschi and E. Vitali, Homogenization of free–discontinuity problems, Arch. Rat. Mech. , 135 (1996), pp. 297–356. [12] G. Buttazzo, Semicontinuity, relaxation and integral representation problems in the Calculus of Variations, Pitman Res. Notes in Math. 207, Longman, Harlow, 1989. [13] B. Dacorogna, Direct Methods in the Calculus of Variations, Springer-Verlag, Berlin, 1989.

Math. , 23 (1992), pp. 1081–1098. ¨ller, Relaxation of quasiconvex functionals in BV (Ω, Rp ) for [21] I. Fonseca and S. Mu integrands f (x, u, ∇u), Arch. Rat. Mech. , 123 (1993), pp. 1-49. ¨ller, A-quasiconvexity, lower semicontinuity and Young measures, [22] I. Fonseca and S. Mu to appear in SIAM J. Math. Anal. [23] N. , 29 (1980), pp. 307–323. [24] M. Giaquinta and G. Modica, Regularity results for some classes of higher order non linear elliptic systems, J. f¨ ur reine and angew. , 311/312 (1979), pp.

4) By letting R F − (v; D) ≥ F − (v(· − x0 ); D + x0 ). The converse inequality follows in a similar way. e. x0 ∈ Ω. 5, and where we have chosen the radii r → 0+ such that F − (v(·+x0 ); ∂(rQ)) = 0. Here vε,r ∈ Lq (rQ; Rd )∩ker A and vε,r v(·+x0 ) in Lq (rQ; Rd ) + as ε → 0 . Then dF − (v; ·) (x0 ) = lim+ lim inf dLN r→0 ε→0+ f Q r y, v(x0 ) + wε,r (y) dy ε where wε,r (y) := vε,r (ry) − v(x0 ). 5, we have that wε,r 0 in Lq (Q; Rd ) if we first let ε → 0 and then r → 0+ . Diagonalize to get w ˆk ∈ Lq (Q; Rd ) ∩ ker A such that w ˆk 0 in Lq (Q; Rd ), dF − (v; ·) (x0 ) = lim k→∞ dLN f (sk y, v(x0 ) + w ˆk (y)) dy, Q and where sk := 1/εk → ∞.

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