Calderon-Zygmund estimates for stochastic elliptic systems on bounded Lipschitz domains

TL;DR

This paper establishes Calderón-Zygmund estimates for stochastic elliptic systems on Lipschitz domains, providing optimal homogenization error bounds and introducing a new minimal radius concept for boundary value problems.

Contribution

It introduces a novel minimal radius concept and extends Calderón-Zygmund estimates to stochastic elliptic systems on Lipschitz domains, with applications to homogenization error analysis.

Findings

Optimal homogenization error bounds up to logarithmic factors

A new minimal radius concept for boundary value problems

Quantitative stochastic homogenization results in Lipschitz domains

Abstract

Concerned with elliptic operators with stationary random coefficients of integrable correlations and bounded Lipschitz domains, arising from stochastic homogenization theory, this paper is mainly devoted to studying Calder\'on-Zygmund estimates. As an application, we obtain the homogenization error in the sense of oscillation and fluctuation, respectively. These results are optimal up to a quantity $O(\ln(1/\varepsilon))$, which is caused by the quantified sublinearity of correctors in dimension two and the less smoothness of the boundary. In this paper, we find a novel form of \emph{minimal radius}, which is proved to be a suitable tool for quantitative stochastic homogenization on boundary value problems, when we adopt Gloria-Neukamm-Otto's strategy originally inspired by the pioneering work of Naddaf and Spencer.