Homogenization of the random Neumann sieve problem under minimal assumptions on the size of the perforations

TL;DR

This paper investigates the stochastic homogenization of the Neumann sieve problem with randomly distributed perforations, identifying minimal conditions on perforation sizes for homogenization despite clustering effects.

Contribution

It establishes the conditions under which stochastic homogenization occurs for the Neumann sieve problem with minimal assumptions on perforation sizes.

Findings

Homogenization occurs under minimal integrability conditions.

Clustering of holes does not prevent homogenization.

Optimal stochastic integrability for perforation radii is identified.

Abstract

We study the limit behavior of the solutions to the Neumann sieve problem for the Poisson equation when the sieve-holes are randomly distributed according to a stationary marked point process. We determine the optimal stochastic integrability for the random radii of the perforations for which stochastic homogenization takes place despite the presence of clustering holes.