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

Abstract

Let $Ω$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in Ω$ and that $\mathbb{R}^n \setminus \mathrm{cl}Ω$ is connected. Then we take $q_{11},..., q_{nn}\in ]0,+\infty[$ and $p \in Q\equiv \prod_{j=1}^{n}]0,q_{jj}[$. If $ε$ is a small positive number, then we define the periodically perforated domain $\mathbb{S}[Ω_ε]^{-} \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\mathrm{cl}\bigl(p+εΩ+\sum_{j=1}^n (q_{jj}z_j)e_j\bigr)$, where $\{e_1,...,e_n\}$ is the canonical basis of $\mathbb{R}^n$. For $ε$ small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set $\mathbb{S}[Ω_ε]^{-}$. Namely, we consider a Dirichlet condition on the boundary of the set $p+εΩ$, together with a periodicity condition. Then we show real analytic continuation properties of the solution and of the corresponding energy integral as functionals of the pair of $ε$ and of the Dirichlet datum on $p+ε\partial Ω$, around a degenerate pair with $ε=0$.