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

Abstract

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