Series expansions for the solution of the Dirichlet problem in a planar domain with a small hole
Маттео Далла Ріва, Паоло Мусоліно, Sergei Rogosin · Asymptotic Analysis · 2015
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.