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Advances in Calculus of Variations

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

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Source Normalized Impact per Paper (SNIP) 2017: 1.138

Mathematical Citation Quotient (MCQ) 2017: 1.15

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Volume 8, Issue 4


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


Let (,g) be a compact Riemannian manifold of dimension N, N ≥ 2. In this paper, we prove that there exists a family of domains (Ωε)ε(0,ε0) and functions uε such that -Δguε=1 in Ωε, uε=0 on Ωε, g(guε,νε)=-εN on Ωε, where νε is the unit outer normal of Ωε. The domains Ωε 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 (Ωε)ε(0,ε0) constitutes a smooth foliation of a neighborhood of p0. By considering a family of domains Ωε in which the above problem is satisfied, we also prove that if this family converges to some point p0 in a suitable sense as ε0, then p0 is a critical point of the scalar curvature. A Taylor expansion of the energy rigidity for the torsion problem is also given.

Keywords: Over-determined problem; foliation; isochoric profile

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

About the article

Received: 2014-06-07

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, DOI: https://doi.org/10.1515/acv-2014-0017.

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