Riemannian aspects of potential theory

2014/03/23 by Virginia Agostiniani, Agostiniani, Virginia, Lorenzo Mazzieri +1
Mathematics · #31B15 #35B06 #35N25 #53C21 #Analysis of PDEs (math.AP) #Differential Geometry (math.DG) #FOS: Mathematics #math.AP #math.DG #msc:31B15 #msc:35B06 #msc:35N25 #msc:53C21

paper · pdf · doi:10.48550/arxiv.1403.5796

minor corrections

arxiv created 2015/02/17 · arxiv updated 2015/02/19

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.

Related

• Two symmetry problems in potential theory
• Splitting theorems, symmetry results and overdetermined problems for Riemannian manifolds
• Conformal geometry and half-integrable spacetimes
• A note on an overdetermined problem for the capacitary potential
• Physical Principles Based on Geometric Properties