A singularly perturbed Dirichlet problem for the Laplace operator in a periodically perforated domain. A functional analytic approach

Let Ω be a sufficiently regular bounded connected open subset of R ⁿ such that 0 ∈ Ω and that R ⁿ\clΩ is connected. Then we take q ₁₁, ⋯ ,q _nn ∈ ]0,+ ∞ [and p∈Q≡∏ _j=1 ⁿ]0,q _jj[. If ε is a small positive number, then we define the periodically perforated domain S[Ω _ε]-≡R ⁿ\ ∪ _z∈Zn/cl(p+εΩ+∑ _j=1 ⁿ(q _jjz _j)e _j, where {e ₁, ⋯ ,e _n} is the canonical basis of R ⁿ. For ε small and positive, we introduce a particular Dirichlet problem for the Laplace operator in the set S[Ωε]-. Namely, we consider a Dirichlet condition on the boundary of the set p + εΩ, together with a periodicity condition. Then we show real analytic continuation properties of the solution and of the corresponding energy integral as functionals of the pair of ε and of the Dirichlet datum on p + ε∂Ω, around a degenerate pair with ε = 0.