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

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 $D^{\prime }:={\mathbb{R}}^3\setminus \overline{D}$; $g(x,y)=\frac{e^{ik|x-y|}}{4\pi |x-y|}$, $w:=w(x,\mu ):={\int }_S{g}_N(x,s)\mu (s)ds$ 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. This result is new, although the potential theory has more than 150 years of history.

Necessary and sufficient conditions are given for the existence of $u$ and the relation $w=u$ in $D^{\prime }$, given $w$ in $D^{\prime }$, and for the existence of $w$ and the relation $w=u$ in $D^{\prime }$, given $u$ in $D^{\prime }$.