*Published Paper*

**Inserted:** 29 sep 2012

**Last Updated:** 29 sep 2012

**Journal:** Arch. Rat. Mech. Anal.

**Volume:** 198

**Pages:** 369-455

**Year:** 2010

**Abstract:**

We prove that, if $u\colon \Omega \to \mathcal R^N$ is a solution to the Dirichlet variational problem \[ \min_{w} \int_{\Omega} F(x,w,Dw)\ dx\ \ \ \mbox{ subject to } \ \ w\equiv u_0 \ \ \mbox{on} \ \ \partial \Omega \,, \] with regular boundary datum $(u_0, \partial \Omega)$ and regular integrand $F(x,w,Dw)$, strongly convex in $Dw$, then almost every boundary point, in the sense of the usual surface measure of $\partial \Omega$, is regular for $u$ in the sense that $Du$ is H\"older continuous in a relative neighborhood of the point. The question of existence of even one such regular boundary point was open except for very special cases. The result is a consequence of new up-to-the-boundary higher differentiability results that we establish for minima of the functionals in question. Our 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.