A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach
Massimo Lanza de Cristoforis, Паоло Мусоліно · ZAMM ‐ Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik · 2015
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 a suitably normalized solution. Then we are interested to analyze the behavior of when ε 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 ε and that if then can be expanded into a convergent double series expansion of powers of ε and . Our approach is based on potential theory and functional analysis and is alternative to those of asymptotic analysis.