In this paper we provide a new method for establishing the rotational symmetry of the solutions to a couple of very classical overdetermined problems arising in potential theory, in both the exterior and the interior punctured domain. Thanks to a conformal reformulation of the problems, we obtain Riemannian manifolds with zero Weyl tensor satisfying a quasi-Einstein type equation. Exploiting these geometric properties, we conclude via a splitting argument that the manifolds obtained are half cylinders. In turn, the rotational symmetry of the potential is implied. To the authors' knowledge, some of the overdetermining conditions considered here are new.

Riemannian aspects of potential theory

MAZZIERI, LORENZO
2015

Abstract

In this paper we provide a new method for establishing the rotational symmetry of the solutions to a couple of very classical overdetermined problems arising in potential theory, in both the exterior and the interior punctured domain. Thanks to a conformal reformulation of the problems, we obtain Riemannian manifolds with zero Weyl tensor satisfying a quasi-Einstein type equation. Exploiting these geometric properties, we conclude via a splitting argument that the manifolds obtained are half cylinders. In turn, the rotational symmetry of the potential is implied. To the authors' knowledge, some of the overdetermining conditions considered here are new.
Settore MAT/05 - Analisi Matematica
31B15; 35B06; 35N25; 53C21; Conformal geometry; Electrostatic potential; Overdetermined boundary value problems; Splitting theorem;
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Utilizza questo identificativo per citare o creare un link a questo documento: http://hdl.handle.net/11384/57222
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