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 \u3b5. For each positive and small \u3b5, we denote by v(\u3b5,\ub7) a suitably normalized solution. Then we are interested to analyze the behavior of v(\u3b5, \ub7) when \u3b5 is close to the degenerate value \u3b5 = 0, where the holes collapse to points. In particular we prove that if n 65 3, then v(\u3b5, \ub7) can be expanded into a convergent series expansion of powers of \u3b5 and that if n = 2 then v(\u3b5, \ub7) can be expanded into a convergent double series expansion of powers of \u3b5 and \u3b5 log \u3b5. Our approach is based on potential theory and functional analysis and is alternative to those of asymptotic analysis