# Local boundedness and Harnack inequality for solutions of linear non-uniformly elliptic equations

## TL;DR

This paper establishes local boundedness and Harnack inequalities for solutions to linear non-uniformly elliptic equations under sharp integrability conditions, advancing classical regularity results and applying them to stochastic homogenization.

## Contribution

It provides new regularity results for non-uniformly elliptic equations with optimal integrability assumptions, and applies these to stochastic homogenization.

## Key findings

- Proved local boundedness and Harnack inequality under sharp integrability conditions.
- Improved classical regularity results for non-uniformly elliptic equations.
- Established sublinearity of the corrector in stochastic homogenization.

## Abstract

We study local regularity properties for solutions of linear, non-uniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon classical results by Trudinger [ARMA 1971]. We then apply the deterministic regularity results to the corrector equation in stochastic homogenization and establish sublinearity of the corrector.