On the global regularity for minimizers of variational integrals: splitting-type problems in 2D and extensions to the general anisotropic setting

Abstract

We mainly discuss superquadratic minimization problems for splitting-type variational integrals on a bounded Lipschitz domain $\Omega \subset {\mathbb{R}}^2$ and prove higher integrability of the gradient up to the boundary by incorporating an appropriate weight-function measuring the distance of the solution to the boundary data. As a corollary, the local Hölder coefficient with respect to some improved Hölder continuity is quantified in terms of the function $\mathrm{dist}(\cdot ,\partial \Omega )$. The results are extended to anisotropic problems without splitting structure under natural growth and ellipticity conditions. In both cases we argue with variants of Caccioppoli's inequality involving small weights