New regularity theorems for non-autonomous anisotropic variational integrals

Abstract

In the calculus of variations one prominent problem is minimizing anisotropic integrals with a (p,q)-elliptic density F:\mathbb{R}^{n}\supset\Omega\rightarrow\mathbb{R}^{N}. The best known sufficient bound for regularity of solutions is q < p (n + 2)/n. On the other hand, if we allow an additional x-dependence of the density we have the much weaker result q < p (n + 1)/n. If one additionally imposes the local boundedness of the minimizer, then these bounds can be improved to q < p+2 and q < p + 1. In this paper we give natural assumptions for F closing the gap between the autonomous and non-autonomous situation.