On the interplay between $(p,q)$-growth and $x$-dependence of the energy integrand: a limit case

TL;DR

This paper proves local Lipschitz regularity for minimizers of certain non-autonomous energy functionals with $(p,q)$-growth, especially in the critical limit case, extending previous results.

Contribution

It introduces a unified approach to handle the limit case of $(p,q)$-growth conditions with $x$-dependence, covering and extending prior work.

Findings

Established local Lipschitz regularity of minimizers.

Unified approach covers the critical limit case.

Extends previous regularity results.

Abstract

We establish the local Lipschitz regularity of the local minimizers of non autonomous integral funtionals of the form \[ \int_\Omega F(x, Dz)\,dx, \] where $\Omega$ is a bounded open set of $\mathbb{R}^n$, $n \ge 2$. The energy density $F(x,\xi)$ satisfies $(p,q)-$growth conditions with respect to the gradient variable and belongs to the Sobolev class $W^{1,\phi}$, with $\phi(t)=t^r\log^\alpha(e+t),$ $r\ge n$, $\alpha\ge 0$, as a function of the $x$ variable, under the condition $$ 1\le\frac{q}{p} \le 1 + \frac{1}{n} - \frac{1}{r}. $$ We present a unified approach that covers the limit case $$ \frac{q}{p} = 1 + \frac{1}{n} - \frac{1}{r} $$ and retrieves the results in \cite{EMM16} and in \cite{CGHPdN20}.