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

TL;DR

This paper investigates a Dirichlet problem for the Laplace operator in a periodically perforated domain with small holes, analyzing how solutions depend analytically on the size of the holes and boundary data.

Contribution

It introduces a novel functional analytic approach to study the analytic dependence of solutions on small perturbations in a perforated domain.

Findings

Established real analytic continuation of solutions with respect to perturbation parameter

Analyzed the energy integral as a functional of the domain perturbation and boundary data

Provided insights into the asymptotic behavior as the perforation size tends to zero

Abstract

Let $\Omega$ be a sufficiently regular bounded open connected subset of $\mathbb{R}^n$ such that $0 \in \Omega$ and that $\mathbb{R}^n \setminus \mathrm{cl}\Omega$ 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 $\epsilon$ is a small positive number, then we define the periodically perforated domain $\mathbb{S}[\Omega_\epsilon]^{-} \equiv \mathbb{R}^n\setminus \cup_{z \in \mathbb{Z}^n}\mathrm{cl}\bigl(p+\epsilon \Omega +\sum_{j=1}^n (q_{jj}z_j)e_j\bigr)$, where $\{e_1,...,e_n\}$ is the canonical basis of $\mathbb{R}^n$. For $\epsilon$ small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set $\mathbb{S}[\Omega_\epsilon]^{-}$. Namely, we consider a Dirichlet condition on the boundary of the set $p+\epsilon \Omega$, 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 $\epsilon$ and of the Dirichlet datum on $p+\epsilon \partial \Omega$, around a degenerate pair with $\epsilon=0$.