Annealed Calder\'on-Zygmund estimates for elliptic operators with random coefficients on $C^{1}$ domains

TL;DR

This paper develops new annealed Calderón-Zygmund estimates for elliptic operators with random coefficients on $C^{1}$ domains, utilizing advanced homogenization and functional analysis techniques to achieve scaling-invariant results.

Contribution

It introduces novel annealed Calderón-Zygmund estimates for elliptic operators with random coefficients, employing a non-perturbation method and boundary analysis in UMD spaces.

Findings

Established new weighted annealed Calderón-Zygmund estimates

Proved scaling-invariant properties of the estimates

Derived resolvent estimates as a by-product

Abstract

Concerned with elliptic operators with stationary random coefficients governed by linear or nonlinear mixing conditions and bounded (or unbounded) $C^1$ domains, this paper mainly studies (weighted) annealed Calder\'on-Zygmund estimates, some of which are new even in a periodic setting. Stronger than some classical results derived by a perturbation argument in the deterministic case, our results own a scaling-invariant property, which additionally requires the non-perturbation method (based upon a quantitative homogenization theory and a set of functional analysis techniques) recently developed by M. Joisen and F. Otto \cite{Josien-Otto22}. To handle boundary estimates in certain UMD (unconditional martingale differences) spaces, we hand them over to Shen's real arguments \cite{Shen05, Shen23} instead of using Mikhlin's theorem. As a by-product, we also established ``resolvent estimates''. The potentially attractive part is to show how the two powerful kernel-free methods work together to make the results clean and robust.