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arXiv · 2203.15713

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

Mouhamed Moustapha Fall·Ignace Aristide Minlend·Tobias Weth
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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.

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Mouhamed Moustapha Fall, Ignace Aristide Minlend, Tobias Weth. 2022-03-29. On an electrostatic problem and a new class of exceptional subdomains of $\mathbb{R}^3$. https://arxiv.org/abs/2203.15713

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