Boundary regularity of minima

Abstract

Let $u:\Omega \to R^N$ be any given solution to the Dirichlet variational problem

where the integrand $F(x,w,Dw)$ is strongly convex in the gradient variable $Dw$, and suitably Hölder continuous with respect to $(x,w)$. We prove that almost every boundary point, in the sense of the usual surface measure of $\partial \Omega$, is a regular point for $u$. This means that $Du$ is Hölder continuous in a relative neighborhood of the point. The existence of even one of such regular boundary points was an open problem for the general functionals considered here, and known only under certain very special structure assumptions.