A-quasiconvexity:: weak-star convergence and the gap

Lower semicontinuity results with respect to weak-* convergence in the sense of measures and with respect to weak convergence in L-P are obtained for functionals nu is an element of L-1 (Omega; R-m) --> integral(Omega) f(x, nu (x)) dx, where admissible sequences {nu(n)} satisfy a first order system of PDEs Anu(n) = 0. We suppose that A has constant rank, f is A-quasiconvex and satisfies the non standard growth conditions 1/C(\nu\(P) - 1) less than or equal to f(nu) less than or equal to C(1 + \nu\(q)) with q is an element of [p, pN/(N - 1)) for p less than or equal to N - 1, q is an element of [p, p + 1) for p > N - 1. In particular, our results generalize earlier work where Anu = 0 reduced to nu = del(s)u for some s is an element of N. (C) 2003 Elsevier SAS. All rights reserved.