By Andres Braides
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Extra resources for A-quasiconvexity relaxation and homogenization
VII 33 (1987), pp. 157–177.  A. Braides and A. Defranceschi, Homogenization of Multiple Integrals, Clarendon Press, Oxford, 1998.  A. Braides, A. Defranceschi and E. Vitali, Homogenization of free–discontinuity problems, Arch. Rat. Mech. , 135 (1996), pp. 297–356.  G. Buttazzo, Semicontinuity, relaxation and integral representation problems in the Calculus of Variations, Pitman Res. Notes in Math. 207, Longman, Harlow, 1989.  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  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,  I. Fonseca and S. Mu to appear in SIAM J. Math. Anal.  N. , 29 (1980), pp. 307–323.  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 → ∞.