TY - JOUR

T1 - A singularly perturbed Neumann problem for the Poisson equation in a periodically perforated domain. A functional analytic approach

AU - Lanza de cristoforis, Massimo

AU - Musolino, Paolo

PY - 2016/2/1

Y1 - 2016/2/1

N2 - 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.

AB - 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.

KW - Neumann problem

KW - Periodically perforated domain

KW - Real analytic continuation in Banach space

KW - Singularly perturbed domain

UR - http://www.scopus.com/inward/record.url?scp=84923882049&partnerID=8YFLogxK

U2 - 10.1002/zamm.201400035

DO - 10.1002/zamm.201400035

M3 - Article

AN - SCOPUS:84923882049

SN - 0044-2267

VL - 96

SP - 253

EP - 272

JO - ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

JF - ZAMM - Journal of Applied Mathematics and Mechanics / Zeitschrift für Angewandte Mathematik und Mechanik

IS - 2

ER -