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

摘要

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 perforateddomain $T(\epsilon) \equiv \mathbb{R}^n\setminus \cup_{z \in\mathbb{Z}^n}\cl(w+\epsilon \Omega +z)$. For each small positive $\epsilon$, weintroduce a particular Dirichlet problem for the Laplace operator in the set$T(\epsilon)$. More precisely, we consider a Dirichlet condition on theboundary of the set $w+\epsilon \Omega$, and we denote the unique periodicsolution of this problem by $u[\epsilon]$. Then we show that (suitablerestrictions of) $u[\epsilon]$ can be continued real analytically in theparameter $\epsilon$ around $\epsilon=0$.