Regularity for minimizers of scalar integral functionals

Abstract

We prove the local Lipschitz regularity of the local minimizers of scalar integral functionals of the form \begin{equation*} \mathcal{F}(v;Ω)= \int_Ω f (x, Dv) dx \end{equation*} under $(p,q)$-growth conditions. The main novelty is that, beside a suitable regularity assumption on the partial map $x\mapsto f(x,\xi )$, we do not assume any special structure for the energy density as a function of the $\xi$-variable.