A Singularly Perturbed Dirichlet Problem for the Poisson Equation 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 ϵ Ω and that $\mathbb{R}^{n}\setminus \rm {cl}\Omega$ is connected. Then we take $(q_{11},...,q_{nn})\in]0,+\infty{[^{n}}\; \rm {and} \;p \in Q \equiv \prod\nolimits^{n}_{j=1}]0,q_{jj}[.$ If є is a small positive number, then we define the periodically perforated domain $\mathbb{S}[\Omega_{p,\epsilon}]^{-} \equiv \mathbb{R}^{n} \setminus \cup_{z\in\mathbb{Z}^{n}}\rm {cl}(p+\epsilon\Omega\;+\;\sum\nolimits^{n}_{j=1}(q_{jj}z_{j})e_{j})$, where $\left\{e_{1},...,e_{n}\right\}$ is the canonical basis of $\mathbb{R}^{n}$. For є small and positive, we introduce a particular Dirichlet problem for the Poisson equation in the set $\mathbb{S}[\Omega_{p,\epsilon}]^{-}$. . 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 as a function of є , of the Dirichlet datum on $p \; + \; \epsilon\partial\Omega$ , and of the Poisson datum, around a degenerate triple with є = 0.

Mathematics Subject Classification (2010). 35J25; 31B10; 45A05; 47H30.