On an electrostatic problem and a new class of exceptional subdomains of ${\mathbb{R}}^3$

Abstract

We study the existence of nontrivial unbounded surfaces $S\subset {\mathbb{R}}^3$ with the property that the constant charge distribution on $S$ is an electrostatic equilibrium, i.e. the resulting electrostatic force is normal to the surface at each point on $S$. Among bounded regular surfaces $S$, only the round sphere has this property by a result of Reichel $[23]$ (see also Mendez and Reichel $[16]$) confirming a conjecture of P. Gruber. In the present paper, we show the existence of nontrivial exceptional domains $\Omega \subset {\mathbb{R}}^3$ whose boundaries $S=\partial \Omega$ enjoy the above property.

Minor corrections made. To Appear in SIAM Journal on Mathematical Analysis