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

摘要

We consider a sufficiently regular bounded open connected subset Ω of R~n such that 0 E 52 and such that R" cl 52 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(ε)≡R~n ∪_z∈Z~ncl(w +εΩ + z). For each small positive E, 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.