# A Singularly Perturbed Dirichlet Problem for the Poisson Equation in a Periodically Perforated Domain. A Functional Analytic Approach

Conference paper
Part of the Operator Theory: Advances and Applications book series (OT, volume 229)

## 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.

### Keywords

Dirichlet problem singularly perturbed domain Poisson equation periodically perforated domain real analytic continuation in Banach space.

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## Copyright information

© Springer Basel 2013

## Authors and Affiliations

1. 1.Dipartimento di Matematica Pura ed ApplicataUniversità degli Studi di PadovaPadovaItaly