Lipschitz regularity of minimizers of variational integrals with variable exponents

TL;DR

This paper establishes Lipschitz regularity for minimizers of convex variational integrals with variable exponents, under near-critical regularity conditions in the Orlicz Sobolev framework, advancing understanding of regularity in variable growth problems.

Contribution

It proves Lipschitz regularity for minimizers with energy densities exhibiting almost critical regularity in the Orlicz Sobolev setting, a novel extension in variable exponent calculus.

Findings

Lipschitz regularity of minimizers is achieved under new regularity assumptions.

The results extend regularity theory to energy densities with near-critical regularity.

The approach advances the understanding of variable growth conditions in calculus of variations.

Abstract

In this paper we prove the Lipschitz regularity for local minimizers of convex variational integrals of the form \[ \mathfrak{F}( v, \Omega )= \int_{\Omega} \! F(x, Dv(x)) \, dx, \] where, for ${n > 2}$ and $N\ge 1$, $\Omega$ is a bounded open set in $\mathbb{R}^n$, $u \in W^{1,1}(\Omega , \mathbb{R}^N)$ and the energy density $F:\Omega\times \mathbb{R}^{N \times n}\to \mathbb{R}$ satisfies the so called variable growth conditions. The main novelty of the paper is that we assume an almost critical regularity in the Orlicz Sobolev setting for the energy density as a function of the $x$ variable.