Homogenization for the Poisson equation in randomly perforated domains under minimal assumptions on the size of the holes

TL;DR

This paper establishes homogenization results for the Poisson equation in randomly perforated domains with minimal assumptions on hole sizes, showing that overlapping holes do not affect the homogenized limit.

Contribution

It extends homogenization theory to domains with random, potentially overlapping holes under minimal size assumptions, including non-periodic and correlated configurations.

Findings

Homogenization occurs despite overlapping holes.

The 'strange term' appears in the limit similar to periodic cases.

Minimal assumptions on hole radii are sufficient for homogenization.

Abstract

This paper deals with the homogenization of the Poisson equation in a bounded domain of $\mathbb{R}^d$, $d>2$, which is perforated by a random number of small spherical holes with random radii and positions. We show that for a class of stationary short-range correlated measures for the centres and radii of the holes, we recover in the homogenized limit an averaged analogue of the "strange term" obtained by Cioranescu and Murat in the periodic case [D. Cioranescu and F. Murat, \textit{Un term \'etrange venu d'ailleurs} (1986)]. We stress that we only require that the random radii have finite $(d-2)$-moment, which is the minimal assumption in order to ensure that the average of the capacity of the balls is finite. Under this assumption, there are holes which overlap with probability one. However, we show that homogenization occurs and that the clustering holes do not have any effect in the resulting homogenized equation.