Boundary regularity of minima

Let u: Omega -> R-N be any given solution to the Dirichlet variational problem min(w) integral(Omega) F(x, w, Dw)dx, w u(0) on partial derivative Omega where the integrand F (x; w; Dw) is strongly convex in the gradient variable Dw, and suitably Holder continuous with respect to. x; w /. We prove that almost every boundary point, in the sense of the usual surface measure of partial derivative Omega, is a regular point for u. This means that D u is Holder continuous in a relative neighbourhood of the point. The existence of even one such regular boundary point was an open problem for the general functionals considered here, and known only under certain very special structure assumptions.