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

Mouhamed Moustapha FallIgnace Aristide MinlendTobias Weth
摘要
We study the existence of nontrivial unbounded surfaces $S\subset\mathbb{R}^3$ with the property that the constant charge distribution on $S$ isan electrostatic equilibrium, i.e. the resulting electrostatic force is normalto the surface at each point on $S$. Among bounded regular surfaces $S$, onlythe round sphere has this property by a result of Reichel $[23]$ (see alsoMendez and Reichel $[16]$) confirming a conjecture of P. Gruber. In the presentpaper, we show the existence of nontrivial exceptional domains $\Omega \subset\mathbb{R}^3$ whose boundaries $S=\partial \Omega$ enjoy the above property.
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