On an inverse problem in potential theory

Abstract

The Newtonian potential of a Euclidean ball $B$ of ${\mathbb{R}}^n$ centered at $x_0$ is proportional, outside $B$, to the Newtonian potential of a mass concentrated at $x_0$. Vice-versa, as proved by Aharonov, Schiffer and Zalcman, if $D$ is a bounded open set in ${\mathbb{R}}^n$, containing $x_0$, whose Newtonian potential is proportional, outside $D$, to the one of a mass concentrated at $x_0$, then $D$ is a Euclidean ball with center $x_0$. In this paper we generalize this last result to more general measures and domains.