Computation of volume integrals in potential theory

Abstract

It is well known that the solution of the Laplace equation in a plane domain D with Dirichlet data φ on ∂D can be represented as a double layer potential [1]

$$W\left( x \right)=\frac{1}{\pi }\int\limits_{\partial D}{\mu \left( y \right)\frac{\partial }{\partial {{n}_{y}}}\log r\left( x,y \right)d{{S}_{y}}}$$

(12.1)

where r(x, y) = |x − y|, x ∈ D, n _y is the outward normal to ∂D at y, and μ satisfies the integral equation

$$\mu \left( {{x}_{0}} \right)+\frac{1}{\pi }\int\limits_{\partial D}{\mu \left( y \right)}\frac{\partial }{\partial {{n}_{y}}}\log r\left( {{x}_{0}},y \right)d{{S}_{y}}=\varphi \left( {{x}_{0}} \right),{{x}_{0}}\in \partial D.$$

(12.2)