# Boundary regularity results for variational integrals

created by beck on 11 Nov 2009
modified on 27 May 2013

[BibTeX]

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.