First-order expansion of homogenized coefficients under Bernoulli perturbations

TL;DR

This paper derives a first-order expansion of homogenized coefficients for divergence-form operators with stationary random coefficients, analyzing the impact of rare but significant local perturbations on the effective medium properties.

Contribution

It introduces a novel first-order expansion formula for homogenized coefficients considering rare Bernoulli-like perturbations in the medium.

Findings

Derived a first-order expansion formula for homogenized coefficients

Quantified the effect of rare local perturbations on effective properties

Provided theoretical insights into perturbation impacts in random media

Abstract

Divergence-form operators with stationary random coefficients homogenize over large scales. We investigate the effect of certain perturbations of the medium on the homogenized coefficients. The perturbations that we consider are rare at the local level, but when occurring, have an effect of the same order of magnitude as the initial medium itself. The main result of the paper is a first-order expansion of the homogenized coefficients, as a function of the perturbation parameter.