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Journal für die reine und angewandte Mathematik

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

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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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/ Mouhamed Moustapha Fall / Joan Solà-Morales / Tobias Weth
Published Online: 2016-04-16 | DOI: https://doi.org/10.1515/crelle-2015-0117


We are concerned with hypersurfaces of 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 N with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in 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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About the article

Received: 2015-08-24

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, DOI: https://doi.org/10.1515/crelle-2015-0117.

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