Poisson equation in domains with concentrated holes

TL;DR

This paper studies the behavior of solutions to Poisson equations in domains with non-periodically distributed holes, showing convergence to a modified problem with an effective potential, extending previous periodic results.

Contribution

It generalizes earlier periodic hole distribution results to non-periodic distributions, demonstrating convergence of solutions to a Poisson problem with an effective potential.

Findings

Solutions converge to a Poisson problem with a simple potential

Extends previous periodic hole results to non-periodic cases

Provides a generalized framework for heterogeneous perforated domains

Abstract

We consider solutions $u^\varepsilon$ of Poisson problems with the Dirichlet condition on domains $\Omega_\varepsilon$ with holes concentrated at subsets of a domain $\Omega$ non-periodically. We show $u^\varepsilon$ converges to a solution of a Poisson problem with a simple function potential. This is a generalized result of a sample model given by Cioranescu and Murat (1997). They showed a result for case that holes are distributed at $\Omega$ periodically.