A Functional Analytic Approach for a Singularly Perturbed Dirichlet Problem for the Laplace Operator in a Periodically Perforated Domain

We consider a sufficiently regular bounded open connected subset $Omega$ of $mathbbR^n$ such that $0 in Omega$ and such that $mathbbR^n setminus clOmega$ is connected. Then we choose a point $w in ]0,1[^n$. If $epsilon$ is a small positive real number, then we define the periodically perforated domain $T(epsilon) equiv mathbbR^nsetminus cup_z in mathbbZ^ncl(w+epsilon Omega +z)$. For each small positive $epsilon$, we introduce a particular Dirichlet problem for the Laplace operator in the set $T(epsilon)$. More precisely, we consider a Dirichlet condition on the boundary of the set $w+epsilon Omega$, and we denote the unique periodic solution of this problem by $u[epsilon]$. Then we show that (suitable restrictions of) $u[epsilon]$ can be continued real analytically in the parameter $epsilon$ around $epsilon=0$.