Real analytic families of harmonic functions in a planar domain with a small hole

M. Dalla Riva*, P. Musolino

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

12 Citations (SciVal)

Abstract

We consider a Dirichlet problem in a planar domain with a hole of diameter proportional to a real parameter ε and we denote by uε the corresponding solution. The behavior of uε for ε small and positive can be described in terms of real analytic functions of two variables evaluated at (ε, 1/log ε). We show that under suitable assumptions on the geometry and on the boundary data one can get rid of the logarithmic behavior displayed by uε for ε small and describe uε by real analytic functions of ε. Then it is natural to ask what happens when ε is negative. The case of boundary data depending on ε is also considered. The aim is to study real analytic families of harmonic functions which are not necessarily solutions of a particular boundary value problem.

Original languageEnglish
Pages (from-to)37-55
Number of pages19
JournalJournal of Mathematical Analysis and Applications
Volume422
Issue number1
Early online date22 Aug 2014
DOIs
Publication statusPublished - 01 Feb 2015

Keywords

  • Harmonic functions
  • Real analytic continuation in Banach space
  • Singularly perturbed perforated planar domains

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