Boundary Regularity in Variational Problems

We prove that, if u : Omega subset of R(n) -> R(N) is a solution to the Dirichlet variational problem min(w) integral(Omega) F(x, w, Dw) dx subject to w u(0) on partial derivative Omega, involving a regular boundary datum (u(0), partial derivative Omega) and a regular integrand F(x, w, Dw) strongly convex in Dwand satisfying suitable growth conditions, thenH(n-1)-almost every boundary point is regular for u in the sense that Du is Holder continuous in a relative neighborhood of the point. The existence of even one such regular boundary point was previously not known except for some very special cases treated by Jost & Meier ( Math Ann 262: 549- 561, 1983). Our results are consequences of new up- to- the- boundary higher differentiability results that we establish for minima of the functionals in question. Themethods also allow us to improve the known boundary regularity results for solutions to non- linear elliptic systems, and, in some cases, to improve the known interior singular sets estimates for minimizers. Moreover, our approach allows for a treatment of systems and functionals with " rough" coefficients belonging to suitable Sobolev spaces of fractional order.