# On the regularity of minima of non-autonomous functionals

## TL;DR

This paper investigates the regularity of minima for non-autonomous variational functionals with non-uniform ellipticity, providing sharp results and techniques to handle obstructions like the Lavrentiev phenomenon.

## Contribution

It offers new regularity results for minima of non-autonomous functionals, including Lipschitz estimates and higher gradient integrability, using a novel Moser iteration approach.

## Key findings

- Established local Lipschitz regularity for scalar problems.
- Proved higher gradient integrability for vector-valued problems.
- Connected regularity results to approximation-in-energy conditions.

## Abstract

We consider regularity issues for minima of non-autonomous functionals in the Calculus of Variations exhibiting non-uniform ellipticity features. We provide a few sharp regularity results for local minimizers that also cover the case of functionals with nearly linear growth. The analysis is carried out provided certain necessary approximation-in-energy conditions are satisfied. These are related to the occurrence of the so-called Lavrentiev phenomenon that that non-autonomous functionals might exhibit, and which is a natural obstruction to regularity. In the case of vector valued problems we concentrate on higher gradient integrability of minima. Instead, in the scalar case, we prove local Lipschitz estimates. We also present an approach via a variant of Moser's iteration technique that allows to reduce the analysis of several non-uniformly elliptic problems to that for uniformly elliptic ones.