Homogenization of the stochastic double-porosity model

TL;DR

This paper advances the understanding of stochastic homogenization in double-porosity models by proving qualitative results under weak conditions and providing sharp error estimates under stronger assumptions.

Contribution

It addresses two open problems by establishing qualitative stochastic homogenization in high-contrast media and deriving precise error bounds for the two-scale expansion.

Findings

Proved qualitative stochastic homogenization under weak conditions.

Provided sharp error estimates for the two-scale expansion.

Addressed open problems in high-contrast media homogenization.

Abstract

This work is devoted to the homogenization of elliptic equations in high-contrast media in the so-called 'double-porosity' resonant regime, for which we solve two open problems of the literature. First, we prove qualitative stochastic homogenization under very weak conditions, which cover the case of inclusions that are not uniformly bounded or separated. Second, under stronger assumptions, we provide sharp error estimates for the two-scale expansion. The main difficulty is related to the loss of integrability of the control in the resonant zones.