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/(r 0 + (2π) −1 log |ε|), with r 0 ∈ 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.

Asymptotic behavior of the solution of singularly perturbed transmission problems in a periodic domain

“…Our results imply that solutions can be expanded into absolutely convergent power series. The author plans to compute explicitly such power series and to adapt the technique of the present paper (see also Dalla Riva, Musolino, and Rogosin) to compute expansions for the effective properties of periodic composite materials (see, eg, Berlyand and Mityushev; Ammari, Kang, and Touibi; and Dalla Riva and Musolino) and other quantities relevant in the applications, as, for example, the longitudinal permeability (see, eg, Musolino and Mityushev).…”

“…The Functional Analytic Approach presents some advantages: Indeed, if, for example, we know that the solution equals a real analytic function of ε , then we can deduce the possibility to expend the solution into convergent power series of ε when ε is small. Moreover, the coefficients of such power series can be explicitly solved by computing the solutions of recursive systems of integral equations (cf Dalla Riva, Musolino, and Rogosin). By this approach, Dalla Riva and Musolino have investigated a nonideal singularly perturbed linear transmission problem and have applied the results to the effective conductivity of a periodic dilute two‐phase composite with interfacial thermal resistance.…”

Asymptotic behavior of the solutions of a transmission problem for the Helmholtz equation: A functional analytic approach

“…In this paper, we do not provide the algorithms to compute the coefficients of the power series expansion of u o ( ϵ , x ). For this type of computations, we mention the work of Dalla Riva et al 30 for the Laplace operator.…”

A nonlinear problem for the Laplace equation with a degenerating Robin condition

“…To construct the sequences and , we wish to exploit the integral equation formulation of problem and the approach of Dalla Riva et al…”