*Published Paper*

**Inserted:** 11 nov 2009

**Last Updated:** 27 may 2013

**Journal:** Q. J. Math.

**Volume:** 62

**Number:** 4

**Pages:** 791-824

**Year:** 2010

**Doi:** 10.1093/qmath/haq019

**Abstract:**

We prove partial Hölder continuity for the gradient of minimizers $u \in W^{1,p}(\Omega,R^N)$, $\Omega \subset R^n$ a bounded domain, of variational integrals of the form \[ \mathcal{F}[u;\Omega] \, := \, \int_{\Omega} [ f(x,u,Du) + h(x,u)] dx, \] where $f$ is strictly quasi-convex and satisfies standard continuity and growth conditions, but where $h$ is only a Caratheodory function of subcritical growth. The main focus is set on the presentation of a unified approach for the interior and the boundary estimates (provided that the boundary data are sufficiently regular) for all $p \in (1,\infty)$. Furthermore, a corresponding lower order Hölder regularity result for $u$ is given in dimensions $n \leq p+2$ under the stronger assumption that $f$ is strictly convex.

**Download:**