A functional analytic approach for a singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain

TL;DR

This paper develops a functional analytic framework to analyze how solutions to a Dirichlet problem for the Laplace operator behave in a domain with periodically distributed small holes, showing solutions depend analytically on the hole size parameter.

Contribution

It introduces a novel functional analytic approach to study the asymptotic behavior of solutions in periodically perforated domains as the perforation size tends to zero.

Findings

Solutions depend analytically on the perforation size parameter.

The method applies to a class of singularly perturbed elliptic problems.

Provides a basis for asymptotic analysis in perforated domains.

Abstract

We consider a sufficiently regular bounded open connected subset $\Omega$ of $\mathbb{R}^n$ such that $0 \in \Omega$ and such that $\mathbb{R}^n \setminus \cl\Omega$ is connected. Then we choose a point $w \in ]0,1[^n$. If $\epsilon$ is a small positive real number, then we define the periodically perforated domain $T(\epsilon) \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\cl(w+\epsilon \Omega +z)$. For each small positive $\epsilon$, we introduce a particular Dirichlet problem for the Laplace operator in the set $T(\epsilon)$. More precisely, we consider a Dirichlet condition on the boundary of the set $w+\epsilon \Omega$, and we denote the unique periodic solution of this problem by $u[\epsilon]$. Then we show that (suitable restrictions of) $u[\epsilon]$ can be continued real analytically in the parameter $\epsilon$ around $\epsilon=0$.