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 Ω ⊂ ℝ2 and prove higher integrability of the gradient up to the boundary by incorporating an appropriate weightfunction measuring the distance of the solution to the boundary data. As a corollary, the local Hölder coefcient with respect to some improved Hölder continuity is quantifed in terms of the function dist(⋅, 휕Ω).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.