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A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach
Author(s) -
Lanza de Cristoforis Massimo,
Musolino Paolo
Publication year - 2016
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.201400035
Subject(s) - mathematics , neumann series , mathematical analysis , degenerate energy levels , domain (mathematical analysis) , euclidean space , asymptotic expansion , poisson's equation , neumann boundary condition , poisson kernel , series (stratigraphy) , singular perturbation , boundary value problem , physics , paleontology , quantum mechanics , biology
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.