A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach

We consider a Neumann problem for the Poisson equation in the periodically perforated Euclidean space. Each periodic perforation has a size proportional to a positive parameter ε. For each positive and small ε, we denote by v(ε,·) a suitably normalized solution. Then we are interested to analyze the behavior of v(ε, ·) when ε is close to the degenerate value ε = 0, where the holes collapse to points. In particular we prove that if n ≥ 3, then v(ε, ·) can be expanded into a convergent series expansion of powers of ε and that if n = 2 then v(ε, ·) can be expanded into a convergent double series expansion of powers of ε and ε log ε. Our approach is based on potential theory and functional analysis and is alternative to those of asymptotic analysis.