Regularity for minimizers of scalar integral functionals

TL;DR

This paper establishes local Lipschitz regularity for minimizers of scalar integral functionals with $(p,q)$-growth conditions, without requiring specific structure in the energy density, assuming only regularity in the spatial variable.

Contribution

It proves regularity results for minimizers under minimal structural assumptions, notably without special structure in the energy density.

Findings

Minimizers are locally Lipschitz continuous.

Regularity holds under $(p,q)$-growth conditions.

No special structure needed for the energy density.

Abstract

We prove the local Lipschitz regularity of the local minimizers of scalar integral functionals of the form \begin{equation*} \mathcal{F}(v;\Omega)= \int_{\Omega} 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.