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