Lipschitz continuity for energy integrals with variable exponents

Abstract

A regularity result for integrals of the Calculus of Variations with variable exponents is presented. Precisely, we prove that any vector-valued minimizer of an energy integral over an open set $\Omega \subset {\mathbb{R}}^n$, with variable exponent $p(x)$ in the Sobolev class $W_{\mathrm{loc}}^{1,r}(\Omega )$ for some $r>n$, is locally Lipschitz continuous in $\Omega$ and an a priori estimate holds.