Lipschitz regularity for elliptic equations with random coefficients

TL;DR

This paper establishes a large-scale Lipschitz regularity theory for quasilinear elliptic equations with random coefficients, providing optimal stochastic estimates under various mixing conditions.

Contribution

It introduces a new regularity framework for elliptic equations with weak to strong mixing assumptions on random coefficients, including non-integrable correlations.

Findings

Large-scale $L^ abla$-type estimates for solutions' gradients.

Optimal stochastic integrability results.

Quenched $L^2$ estimates for homogenization errors.

Abstract

We develop a higher regularity theory for general quasilinear elliptic equations and systems in divergence form with random coefficients. The main result is a large-scale $L^\infty$-type estimate for the gradient of a solution. The estimate is proved with optimal stochastic integrability under a one-parameter family of mixing assumptions, allowing for very weak mixing with non-integrable correlations to very strong mixing (e.g., finite range of dependence). We also prove a quenched $L^2$ estimate for the error in homogenization of Dirichlet problems. The approach is based on subadditive arguments which rely on a variational formulation of general quasilinear divergence-form equations.