Show Summary Details
More options …

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang

IMPACT FACTOR 2017: 1.676

CiteScore 2017: 1.30

SCImago Journal Rank (SJR) 2017: 2.045
Source Normalized Impact per Paper (SNIP) 2017: 1.138

Mathematical Citation Quotient (MCQ) 2017: 1.15

Online
ISSN
1864-8266
See all formats and pricing
More options …

Serrin's over-determined problem on Riemannian manifolds

Mouhamed Moustapha Fall
/ Ignace Aristide Minlend
Published Online: 2014-09-18 | DOI: https://doi.org/10.1515/acv-2014-0017

Abstract

Let $\left(ℳ,g\right)$ be a compact Riemannian manifold of dimension N, N ≥ 2. In this paper, we prove that there exists a family of domains ${\left({\Omega }_{\epsilon }\right)}_{\epsilon \in \left(0,{\epsilon }_{0}\right)}$ and functions ${u}_{\epsilon }$ such that $-{\Delta }_{g}{u}_{\epsilon }=1$ in ${\Omega }_{\epsilon }$, ${u}_{\epsilon }=0$ on $\partial {\Omega }_{\epsilon }$, $g\left({\nabla }_{g}{u}_{\epsilon },{\nu }_{\epsilon }\right)=-\frac{\epsilon }{N}$ on $\partial {\Omega }_{\epsilon }$, where ${\nu }_{\epsilon }$ is the unit outer normal of $\partial {\Omega }_{\epsilon }$. The domains ${\Omega }_{\epsilon }$ are smooth perturbations of geodesic balls of radius ε. If, in addition, p0 is a non-degenerate critical point of the scalar curvature of g, then the family ${\left(\partial {\Omega }_{\epsilon }\right)}_{\epsilon \in \left(0,{\epsilon }_{0}\right)}$ constitutes a smooth foliation of a neighborhood of p0. By considering a family of domains ${\Omega }_{\epsilon }$ in which the above problem is satisfied, we also prove that if this family converges to some point p0 in a suitable sense as $\epsilon \to 0$, then p0 is a critical point of the scalar curvature. A Taylor expansion of the energy rigidity for the torsion problem is also given.

MSC: 58J05; 58J32; 58J37; 35N10; 35N25

Accepted: 2014-08-27

Published Online: 2014-09-18

Published in Print: 2015-10-01

Citation Information: Advances in Calculus of Variations, Volume 8, Issue 4, Pages 371–400, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

Export Citation