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

Article type: Research Article

Authors: Dalla Riva, M.^; | Musolino, P. | Rogosin, S.V.

Affiliations: Centro de Investigação e Desenvolvimento em Matemática e Aplicações (CIDMA), Universidade de Aveiro, Aveiro, Portugal | Dipartimento di Matematica, Università degli Studi di Padova, Padova, Italy | Institute of Mathematics, Physics and Computer Science (IMPACS), Aberystwyth University, Aberystwyth, UK

Note: [] Corresponding author. Tel.: +351 234372543, Fax: +351 234370066; E-mail: [email protected]

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)−1log |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.

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

DOI: 10.3233/ASY-151283

Journal: Asymptotic Analysis, vol. 92, no. 3-4, pp. 339-361, 2015