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 {text {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.