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

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.