Unified a-priori estimates for minimizers under $p,q-$growth and exponential growth

TL;DR

This paper establishes unified a-priori estimates ensuring local Lipschitz regularity for minimizers of energy functionals with various growth conditions, including polynomial and exponential, simplifying the analysis of complex variational problems.

Contribution

It introduces general growth conditions that guarantee Lipschitz regularity of minimizers, reducing non-uniform elliptic problems to uniform ellipticity for further regularity analysis.

Findings

Local minimizers are locally Lipschitz continuous under broad growth conditions.

A-posteriori boundedness of the gradient simplifies the analysis of the integrand's behavior.

Reduces complex growth conditions to standard growth, enabling classical regularity theory application.

Abstract

We propose some general growth conditions on the function $% f=f\left( x,\xi \right) $, including the so-called natural growth, or polynomial, or $p,q-$growth conditions, or even exponential growth, in order to obtain that any local minimizer of the energy integral $\;\int_{\Omega }f\left( x,Du\right) dx\,$ is locally Lipschitz continuous in $\Omega $. In fact this is the fundamental step for further regularity: the local boundedness of the gradient of any Lipschitz continuous local minimizer a-posteriori makes irrelevant the behavior of the integrand $f\left( x,\xi \right) $ as $\left\vert \xi \right\vert \rightarrow +\infty $; i.e., the general growth conditions a posteriori are reduced to a standard growth, with the possibility to apply the classical regularity theory. In other words, we reduce some classes of \textit{non-uniform} elliptic variational problems to a context of uniform ellipticity.