*Published Paper*

**Inserted:** 29 sep 2012

**Last Updated:** 29 sep 2012

**Journal:** Rend. Lincei, Mat. Appl.

**Volume:** 19

**Pages:** 265-277

**Year:** 2008

**Abstract:**

Let $u\colon \Omega \to \mathcal R^N$ be any given solution to the Dirichlet variational problem \[ \min_{w} \int_{\Omega} F(x,w,Dw)\ dx\qquad w\equiv u_0 \ \ \mbox{on}\ \ \partial \Omega \,, \] where the integrand $F(x,w,Dw)$ is strongly convex in the gradient variable $Dw$, and suitably H\"older continuous with respect to $(x,u)$. 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 H\"older 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.