Cauchy-Dirichlet problem for first order nonlinear systems

Let Omega subset of R-n be open, u : Omega --> R-m and thus the gradient matrix Du is an element of R-mxn. We let E subset of R-mxn be compact and denote by RcoE and PcoE the rank one convex and polyconvex hull of E, respectively. We show that if RcoE = PcoE (and two other hypotheses, named the segment property and the extreme points property) and if phi is an element of C-1(<(Omega)over-bar>; R-m) is such that D-phi(x) is an element of E boolean OR Int PcoE, x is an element of Omega then there exists (a dense set of) u is an element of W-1,W- infinity(Omega; R-m) such that infinity(Omega; Rm) such that [GRAPHICS] We apply this existence theorem to some relevant examples studied in the literature, as well as to problems with (x u) dependence. (C) 1998 Academic Press.