Boundary Regularity in Variational Problems

We prove that,if u:Ω ⊂ ℝn → ℝN is a solution to the Dirichlet variational problem involving a regular boundary datum (u0, ∂Ω) and a regular integrand F(x, w, Dw) strongly convex in Dw and satisfying suitable growth conditions, then Hn-1 -almost every boundary point is regular for u in the sense that Du is Hölder 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 and 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. The methods 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. © 2010 Springer-Verlag