On an electrostatic problem and a new class of exceptional subdomains of r3*

We study the existence of nontrivial unbounded surfaces S \subset R3 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 [Arch. Ration. Mech. Anal., 137 (1997), pp. 381--394] (see also Mendez and Reichel [Forum Math., 12 (2000), pp. 223--245]) confirming a conjecture of Gruber. In the present paper, we show the existence of nontrivial exceptional domains \Omega \subset R3 whose boundaries S = \partial \Omega enjoy the above property.