# Homogenization of the Poisson equation in a non-periodically perforated domain

## TL;DR

This paper extends the homogenization theory of the Poisson equation to non-periodically perforated domains, providing new results on correctors, convergence, and two-scale expansions under geometric assumptions.

## Contribution

It introduces geometric assumptions for non-periodic perforations and proves homogenization results similar to the periodic case, including corrector existence and convergence.

## Key findings

- Existence of a corrector in non-periodic perforated domains
- Convergence of solutions to the homogenized problem
- Development of a two-scale expansion in the non-periodic setting

## Abstract

We study the Poisson equation in a perforated domain with homogeneous Dirichlet boundary conditions. The size of the perforations is denoted by $\epsilon$ > 0, and is proportional to the distance between neighbouring perforations. In the periodic case, the homogenized problem (obtained in the limit $\epsilon$ $\rightarrow$ 0) is well understood (see [21]). We extend these results to a non-periodic case which is defined as a localized deformation of the periodic setting. We propose geometric assumptions that make precise this setting, and we prove results which extend those of the periodic case: existence of a corrector, convergence to the homogenized problem, and two-scale expansion.