Delaunay hypersurfaces with constant nonlocal mean curvature

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 ${\mathbb{R}}^N$, $N\ge 2$, 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.

Minor changes have been made. To appear in "J. Math. Pures Appl."

References in corpus (3)

• Nonlocal diffusion and applications
• Gamma-convergence of nonlocal perimeter functionals
• Near-sphere lattices with constant nonlocal mean curvature

Cited by in corpus (2)

• Asymptotics for the fractional Allen-Cahn equation and stationary nonlocal minimal surfaces
• Near-sphere lattices with constant nonlocal mean curvature