Lipschitz bounds and nonautonomous integrals

TL;DR

This paper develops a comprehensive framework for establishing Lipschitz regularity of solutions to diverse nonautonomous variational problems with nonuniform ellipticity, encompassing polynomial and exponential growth conditions, and introduces optimal regularity criteria.

Contribution

It introduces a unified approach to Lipschitz regularity for a broad class of nonautonomous variational problems with nonuniform ellipticity, including new optimal criteria.

Findings

Sharp regularity results for nonuniform elliptic problems.

Classification of ellipticity types with corresponding regularity conditions.

Extension of classical results to problems with exponential growth.

Abstract

We provide a general approach to Lipschitz regularity of solutions for a large class of vector-valued, nonautonomous variational problems exhibiting nonuniform ellipticity. The functionals considered here range amongst those with unbalanced polynomial growth conditions to those with fast, exponential type growth. The results obtained are sharp with respect to all the data considered and yield new, optimal regularity criteria even in the classical uniformly elliptic case. We give a classification of different types of nonuniform ellipticity, accordingly identifying suitable conditions to get regularity theorems.