Regularity theorems for random elliptic operators on domains

TL;DR

This paper extends regularity results for random elliptic operators to smooth domains and convex polytopes, enhancing the theoretical foundation of stochastic homogenization.

Contribution

It establishes $C^{1,eta}$ regularity for random elliptic operators on smooth domains and convex polytopes, filling a gap in the existing theory.

Findings

Proves $C^{1,eta}$ regularity on smooth domains.

Extends regularity results to convex polytopes.

Provides auxiliary estimates like weighted Meyers estimate.

Abstract

Regularity theorems \`a la Avellaneda-Lin are an indispensable part of the modern quantitative theory of stochastic homogenization. While interior regularity results for random elliptic operators have been available for a while, on general smooth domains the existing theory has until recently remained limited to Lipschitz estimates. We establish $C^{1,\alpha}$ regularity results for random elliptic operators on bounded sufficiently smooth domains, as well as for scalar problems on convex polytopes. We, furthermore, prove a number of auxiliary results typically employed in the derivation of fluctuation bounds, such as a weighted Meyers estimate.