Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Some sphere theorems in linear potential theory


Authors: Stefano Borghini, Giovanni Mascellani and Lorenzo Mazzieri
Journal: Trans. Amer. Math. Soc. 371 (2019), 7757-7790
MSC (2010): Primary 35N25, 31B15, 35B06, 53C21
DOI: https://doi.org/10.1090/tran/7637
Published electronically: March 19, 2019
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we analyze the capacitary potential due to a charged body in order to deduce sharp analytic and geometric inequalities, whose equality cases are saturated by domains with spherical symmetry. In particular, for a regular bounded domain $ \Omega \subset \mathbb{R}^n$, $ n\geq 3$, we prove that if the mean curvature $ H$ of the boundary obeys the condition

$\displaystyle - \, \bigg [ \frac {1}{\operatorname {Cap}(\Omega )} \bigg ]^{\fr... ..., \bigg [ \frac {1}{\operatorname {Cap}(\Omega )} \bigg ]^{\frac {1}{n-2}} \, ,$    

then $ \Omega $ is a round ball.

References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 35N25, 31B15, 35B06, 53C21

Retrieve articles in all journals with MSC (2010): 35N25, 31B15, 35B06, 53C21


Additional Information

Stefano Borghini
Affiliation: Scuola Normale Superiore di Pisa, piazza dei Cavalieri 7, 56126 Pisa (PI), Italy – and – Università degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy
Address at time of publication: Matematiska institutionen, Uppsala Universitet, Lägerhyddsvägen 1, 752 37 Uppsala, Sweden
Email: stefano.borghini@math.uu.se

Giovanni Mascellani
Affiliation: Scuola Normale Superiore di Pisa, piazza dei Cavalieri 7, 56126 Pisa (PI), Italy
Address at time of publication: Département de Mathématiques, Universitè Libre de Bruxelles, Avenue Franklin Roosevelt 50, 1050 Bruxelles, Belgium
Email: giovanni.mascellani@ulb.ac.be

Lorenzo Mazzieri
Affiliation: Università degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy
Email: lorenzo.mazzieri@unitn.it

DOI: https://doi.org/10.1090/tran/7637
Keywords: Capacity, electrostatic potential, overdetermined boundary value problems
Received by editor(s): September 28, 2017
Received by editor(s) in revised form: February 15, 2018, and March 20, 2018
Published electronically: March 19, 2019
Additional Notes: The authors are members of Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA), which is part of the Istituto Nazionale di Alta Matematica (INdAM), and were partially funded by the GNAMPA project “Principi di fattorizzazione, formule di monotonia e disuguaglianze geometriche”.
Article copyright: © Copyright 2019 American Mathematical Society