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

TL;DR

This paper investigates the regularity of minimizers for splitting-type variational integrals in 2D, establishing higher integrability and extending results to anisotropic problems with natural growth conditions.

Contribution

It proves higher integrability of gradients for splitting-type problems and extends the regularity results to anisotropic variational integrals without splitting structure.

Findings

Higher integrability of gradients up to the boundary.

Quantification of local Hölder continuity in terms of boundary distance.

Extension of regularity results to anisotropic problems.

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 H\"older coefficient with respect to some improved H\"older continuity is quantified in terms of the function ${\rm dist}(\cdot,\partial \Omega)$. 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