Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole

M. Dalla Riva*, P. Musolino, S. V. Rogosin

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

24 Citations (SciVal)

Abstract

We consider the Dirichlet problem for the Laplace equation in a planar domain with a small hole. The diameter of the hole is proportional to a real parameter ε and we denote by uε the corresponding solution. If p is a point of the domain, then for ε small we write uε(p) as a convergent power series of ε and of 1/(r0 + (2π)-1 log |ε|), with r0 ∈ ℝ. The coefficients of such series are given in terms of solutions of recursive systems of integral equations. We obtain a simplified expression for the series expansion of uε(p) in the case of a ring domain.

Original languageEnglish
Pages (from-to)339-361
Number of pages23
JournalAsymptotic Analysis
Volume92
Issue number3-4
DOIs
Publication statusPublished - 2015

Keywords

  • Dirichlet problem for the Laplace equation
  • doubly connected domain
  • potential theory
  • real analytic continuation in Banach space
  • singularly perturbed perforated domain

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