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Journal für die reine und angewandte Mathematik

Managing Editor: Weissauer, Rainer

Ed. by Colding, Tobias / Huybrechts, Daniel / Hwang, Jun-Muk / Williamson, Geordie

IMPACT FACTOR 2018: 1.859

CiteScore 2018: 1.14

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Source Normalized Impact per Paper (SNIP) 2018: 1.411

Mathematical Citation Quotient (MCQ) 2018: 1.55

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1435-5345
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Volume 2018, Issue 745

Curves and surfaces with constant nonlocal mean curvature: Meeting Alexandrov and Delaunay

Xavier Cabré
• ICREA and Universitat Politècnica de Catalunya, Departament de Matemàtiques, Diagonal 647, 08028 Barcelona, Spain
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• Other articles by this author:
• De Gruyter OnlineGoogle Scholar
/ Mouhamed Moustapha Fall
/ Joan Solà-Morales
/ Tobias Weth
Published Online: 2016-04-16 | DOI: https://doi.org/10.1515/crelle-2015-0117

Abstract

We are concerned with hypersurfaces of ${ℝ}^{N}$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter under a volume constraint. Our results are twofold. First we prove the nonlocal analogue of the Alexandrov result characterizing spheres as the only closed embedded hypersurfaces in ${ℝ}^{N}$ with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in ${ℝ}^{2}$ with constant nonlocal mean curvature and bifurcating from a straight band. These are Delaunay-type bands in the nonlocal setting. Here we use a Lyapunov–Schmidt procedure for a quasilinear type fractional elliptic equation.

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Revised: 2015-12-14

Published Online: 2016-04-16

Published in Print: 2018-12-01

Funding Source: MINECO

Award identifier / Grant number: MTM2011-27739-C04-01

Funding Source: MINECO

Award identifier / Grant number: MTM2014-52402-C3-1-P

The first and third authors are supported by MINECO grants MTM2011-27739-C04-01 and MTM2014-52402-C3-1-P, and they are part of the Catalan research group 2014 SGR 1083. The second author’s work is supported by the Alexander von Humboldt foundation.

Citation Information: Journal für die reine und angewandte Mathematik (Crelles Journal), Volume 2018, Issue 745, Pages 253–280, ISSN (Online) 1435-5345, ISSN (Print) 0075-4102,

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© 2019 Walter de Gruyter GmbH, Berlin/Boston.

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