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 ⊂ 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 Ω ⊂ R 3 whose boundaries S = ∂Ω enjoy the above property.