Delaunay hypersurfaces with constant nonlocal mean curvature

Xavier Cabre; Mouhamed Moustapha Fall; Tobias Weth

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arXiv · 1602.02623

Delaunay hypersurfaces with constant nonlocal mean curvature

Xavier Cabre·Mouhamed Moustapha Fall·Tobias Weth

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Abstract

We study hypersurfaces of $\mathbb{R}^N$ with constant nonlocal (or fractional) mean curvature. This is the equation associated to critical points of the fractional perimeter functional under a volume constraint. We establish the existence of a smooth branch of periodic cylinders in $\mathbb{R}^N$, $N\geq 2$, all of them with the same constant nonlocal mean curvature, and bifurcating from a straight cylinder. These are Delaunay type cylinders in the nonlocal setting. The proof uses the Crandall-Rabinowitz theorem applied to a quasilinear type fractional elliptic equation.

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BibTeX RIS

Xavier Cabre, Mouhamed Moustapha Fall, Tobias Weth. 2016-02-08. Delaunay hypersurfaces with constant nonlocal mean curvature. https://arxiv.org/abs/1602.02623

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