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.