When does a double-layer potential equal to a single-layer one?

Abstract

Let $D$ be a bounded domain in $\mathbb{R}^3$ with a closed, smooth, connected boundary $S$, $N$ be the outer unit normal to $S$, $k>0$ be a constant, $u_{N^{\pm}}$ are the limiting values of the normal derivative of $u$ on $S$ from $D$, respectively $D':=\mathbb{R}^3\setminus D$; $g(x,y)=\frac{e^{ik|x-y|}}{4\pi |x-y|}$, $w:=w(x,\mu):=\int_S g_{N}(x,s)\mu(s)ds$ be the double-layer potential, $u:=u(x,\sigma):=\int_S g(x,s)\sigma(s)ds$ be the single-layer potential. In this paper it is proved that for every $w$ there is a unique $u$, such that $w=u$ in $D$ and vice versa. Necessary and sufficient conditions are given for the existence of $u$ and the relation $w=u$ in $D'$, given $w$ in $D'$, and for the existence of $w$ and the relation $w=u$ in $D'$, given $u$ in $D'$.