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

Massimo Lanza de cristoforis*, Paolo Musolino

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

8 Citations (SciVal)

Abstract

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 εlogε. Our approach is based on potential theory and functional analysis and is alternative to those of asymptotic analysis.

Original languageEnglish
Pages (from-to)253-272
JournalZAMM Zeitschrift fur Angewandte Mathematik und Mechanik
Volume96
Issue number2
Early online date04 Mar 2015
DOIs
Publication statusPublished - 01 Feb 2016

Keywords

  • Neumann problem
  • Periodically perforated domain
  • Real analytic continuation in Banach space
  • Singularly perturbed domain

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