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A characterization of balls through optimal concavity for potential functions


Author: Paolo Salani
Journal: Proc. Amer. Math. Soc. 143 (2015), 173-183
MSC (2010): Primary 35N25, 35R25, 35R30, 35B06, 52A40
DOI: https://doi.org/10.1090/S0002-9939-2014-12196-4
Published electronically: August 28, 2014
MathSciNet review: 3272742
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Abstract: In this short note two unconventional overdetermined problems are considered. Let $ p\in (1,n)$; first, the following is proved: if $ \Omega $ is a bounded domain in $ \mathbb{R}^n$ whose $ p$-capacitary potential function $ u$ has two homotetic convex level sets, then $ \Omega $ is a ball. Then, as an application, we obtain the following: if $ \Omega $ is a convex domain in $ \mathbb{R}^n$ whose $ p$-capacitary potential function $ u$ is $ (1-p)/(n-p)$-concave (i.e. $ u^{(1-p)/(n-p)}$ is convex), then $ \Omega $ is a ball.


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Additional Information

Paolo Salani
Affiliation: DiMaI - Departimento di Matematica e Informatica “U. Dini”, Università degli Studi di Firenze, Viale Morgagni 67/A, 50134 Firenze, Italy
Email: paolo.salani@unifi.it

DOI: https://doi.org/10.1090/S0002-9939-2014-12196-4
Keywords: Capacity, convex sets, power-concavity, Brunn-Minkowski inequality
Received by editor(s): October 28, 2012
Received by editor(s) in revised form: March 6, 2013
Published electronically: August 28, 2014
Communicated by: Joachim Krieger
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.