Corrector estimates for higher-order linearizations in stochastic homogenization of nonlinear uniformly elliptic equations

TL;DR

This paper develops optimal-order corrector estimates for higher-order linearizations in the stochastic homogenization of nonlinear elliptic equations, improving understanding of convergence rates and regularity in random media.

Contribution

It provides the first optimal-order estimates for higher-order linearized correctors in stochastic homogenization of nonlinear elliptic equations, extending previous work with quantitative bounds.

Findings

Optimal-order corrector estimates established

Higher-order regularity of the homogenized operator proven

Advances the quantitative theory of stochastic homogenization

Abstract

Corrector estimates constitute a key ingredient in the derivation of optimal convergence rates via two-scale expansion techniques in homogenization theory of random uniformly elliptic equations. The present work follows up - in terms of corrector estimates - on the recent work of Fischer and Neukamm (arXiv:1908.02273) which provides a quantitative stochastic homogenization theory of nonlinear uniformly elliptic equations under a spectral gap assumption. We establish optimal-order estimates (with respect to the scaling in the ratio between the microscopic and the macroscopic scale) for higher-order linearized correctors. A rather straightforward consequence of the corrector estimates is the higher-order regularity of the associated homogenized monotone operator.