Cauchy–Dirichlet Problem for First Order Nonlinear Systems

LetΩ⊂Rnbe open,u : Ω→Rmand thus the gradient matrixDu∈Rm×n. We letE⊂Rm×nbe compact and denote byRcoEandPcoEthe rank one convex and polyconvex hull ofE, respectively. We show that ifRcoE=PcoE(and two other hypotheses, named the segment property and the extreme points property) and ifϕ∈C1(Ω; Rm) is such thatDϕ(x)∈E∪int PcoE, x∈Ωthen there exists (a dense set of)u∈W1, ∞(Ω; Rm) such that[formula]We apply this existence theorem to some relevant examples studied in the literature, as well as to problems with (x, u) dependence.