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

摘要

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 epsilon. For each positive and small epsilon, we denote by a suitably normalized solution. Then we are interested to analyze the behavior of when epsilon is close to the degenerate value , where the holes collapse to points. In particular we prove that if , then can be expanded into a convergent series expansion of powers of epsilon and that if then can be expanded into a convergent double series expansion of powers of epsilon and . Our approach is based on potential theory and functional analysis and is alternative to those of asymptotic analysis. (C) 2015 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim