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

TL;DR

This paper investigates special unbounded surfaces in three-dimensional space where a uniform charge distribution results in electrostatic forces normal to the surface, revealing new classes of such exceptional domains beyond the known spherical case.

Contribution

The paper demonstrates the existence of nontrivial unbounded surfaces with electrostatic equilibrium properties, extending the known classification of such surfaces beyond the sphere.

Findings

Existence of nontrivial exceptional domains in ^3

Bounded regular surfaces with this property are only spheres

New classes of unbounded surfaces with electrostatic equilibrium

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.