Rigidity and sharp stability estimatesfor hypersurfaces with constant and almost-constant nonlocal mean curvature

摘要

We prove that the boundary of a (not necessarily connected) bounded smooth set with constant nonlocal mean curvature is a sphere. More generally, and in contrast with what happens in the classical case,we show that the Lipschitz constant of the nonlocal mean curvature of such a boundary controlsits C 2 {C^{2}} -distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate.