Some sphere theorems in linear potential theory
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- by Stefano Borghini, Giovanni Mascellani and Lorenzo Mazzieri PDF
- Trans. Amer. Math. Soc. 371 (2019), 7757-7790 Request permission
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 $\Omega \subset \mathbb {R}^n$, $n\geqslant 3$, we prove that if the mean curvature $H$ of the boundary obeys the condition \begin{equation*} - \bigg [ \frac {1}{\operatorname {Cap}(\Omega )} \bigg ]^{\frac {1}{n-2}} \!\! \leqslant \frac {H}{n-1} \leqslant \bigg [ \frac {1}{\operatorname {Cap}(\Omega )} \bigg ]^{\frac {1}{n-2}} , \end{equation*} then $\Omega$ is a round ball.References
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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
- MR Author ID: 1122157
- 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
- MR Author ID: 1073232
- Email: giovanni.mascellani@ulb.ac.be
- Lorenzo Mazzieri
- Affiliation: Università degli Studi di Trento, via Sommarive 14, 38123 Povo (TN), Italy
- MR Author ID: 835514
- Email: lorenzo.mazzieri@unitn.it
- 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”.
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 7757-7790
- MSC (2010): Primary 35N25, 31B15, 35B06, 53C21
- DOI: https://doi.org/10.1090/tran/7637
- MathSciNet review: 3955535