Combined effect of homogenization and dimension-reduction in the random Neumann sieve problem

TL;DR

This paper studies how solutions to the Poisson equation behave in a thin, randomly perforated domain, revealing three limiting regimes based on the scaling of domain thickness and hole size, and identifying conditions for stochastic homogenization.

Contribution

It introduces a comprehensive analysis of the asymptotic behavior in the Neumann sieve problem, including the effects of homogenization and dimension reduction with random perforations.

Findings

Three distinct limiting regimes depending on scaling

Optimal stochastic integrability condition identified

Homogenization achieved even with clustered holes

Abstract

We investigate the asymptotic behavior of the solutions to the Neumann sieve problem for the Poisson equation in a thin, randomly perforated domain. The perforations (sieve-holes) are generated by a stationary marked point process. According to the scaling between the domain thickness and the typical hole size, three distinct limiting regimes emerge. We also identify the optimal stochastic integrability condition on the random hole radii that guarantees stochastic homogenization, even in the presence of clustering holes.