A characterization of the sphere in terms of single-layer potentials

Abstract:

Let $\Omega$ be a bounded smooth domain in ${\mathbb {R}^n}$, and suppose the single-layer potential of $\partial \Omega$ coincides for $y \notin \overline \Omega$ with the function $c|y{|^{ - 1}}(c > 0)$. Then $\partial \Omega$ is a sphere centered at the origin.