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

doi:10.3233/ASY-151283

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

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

24 Citations (Scopus)

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 u_ε the corresponding solution. If p is a point of the domain, then for ε small we write u_ε(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 u_ε(p) in the case of a ring domain.

Original language	English
Pages (from-to)	339-361
Number of pages	23
Journal	Asymptotic Analysis
Volume	92
Issue number	3-4
DOIs	https://doi.org/10.3233/ASY-151283
Publication status	Published - 2015

Keywords

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

Access to Document

10.3233/ASY-151283

Permanent link

Persistent link

Cite this

@article{f9da97b95a874bdb8f9cd9e5c5172fae,

title = "Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole",

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.",

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

author = "{Dalla Riva}, M. and P. Musolino and Rogosin, {S. V.}",

year = "2015",

doi = "10.3233/ASY-151283",

language = "English",

volume = "92",

pages = "339--361",

journal = "Asymptotic Analysis",

issn = "0921-7134",

publisher = "IOS Press BV",

number = "3-4",

}

TY - JOUR

T1 - Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole

AU - Dalla Riva, M.

AU - Musolino, P.

AU - Rogosin, S. V.

PY - 2015

Y1 - 2015

N2 - 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.

AB - 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.

KW - Dirichlet problem for the Laplace equation

KW - doubly connected domain

KW - potential theory

KW - real analytic continuation in Banach space

KW - singularly perturbed perforated domain

UR - http://www.scopus.com/inward/record.url?scp=84928530215&partnerID=8YFLogxK

U2 - 10.3233/ASY-151283

DO - 10.3233/ASY-151283

M3 - Article

AN - SCOPUS:84928530215

SN - 0921-7134

VL - 92

SP - 339

EP - 361

JO - Asymptotic Analysis

JF - Asymptotic Analysis

IS - 3-4

ER -

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

Abstract

Keywords

Access to Document

Permanent link

Other files and links

Fingerprint

Cite this