Stochastic Homogenization on Irregularly Perforated Domains

TL;DR

This paper develops a method for stochastic homogenization of quasilinear parabolic PDEs on irregular perforated domains, overcoming geometric challenges and proving the limit equation's independence from regularization, including models based on Poisson processes.

Contribution

It introduces a novel homogenization approach for irregular geometries, demonstrating the limit equation's independence from regularization and extending to Boolean models of Poisson point processes.

Findings

Homogenization on regularized geometries is possible despite irregular perforations.

The form of the homogenized equation is independent of the regularization method.

Boolean models of Poisson point processes are included in the framework.

Abstract

We study stochastic homogenization of a quasilinear parabolic PDE with nonlinear microscopic Robin conditions on a perforated domain. The focus of our work lies on the underlying geometry that does not allow standard homogenization techniques to be applied directly. Instead we prove homogenization on a regularized geometry and demonstrate afterwards that the form of the homogenized equation is independent from the regularization. Then we pass to the regularization limit to obtain the anticipated limit equation. Furthermore, we show that Boolean models of Poisson point processes are covered by our approach.