We are concerned with hypersurfaces of RN 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 RN with constant mean curvature. Here we use the moving planes method. Our second result establishes the existence of periodic bands or “cylinders” in R2 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.

Peer Reviewed