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 Ω of Rⁿ such that 0εΩ and such that Rⁿ/clΩ is connected. Then we choose a point wε]0, 1 [ⁿ. If e is a small positive real number, then we define the periodically perforated domain T(ε)≡Rⁿ/∪ _zεZncl(w+εΩ+z). For each small positive ε, we introduce a particular Dirichlet problem for the Laplace operator in the set T(ε). More precisely, we consider a Dirichlet condition on the boundary of the set w+εΩ, and we denote the unique periodic solution of this problem by u[ε]. Then we show that (suitable restrictions of) u[ε] can be continued real analytically in the parameter ε around ε=0.