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 controls its C2-distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate.
Ciraolo, G., Figalli, A., Maggi, F., Novaga, M. (2018). Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature. JOURNAL FÜR DIE REINE UND ANGEWANDTE MATHEMATIK, 2018(741), 275-294 [10.1515/crelle-2015-0088].
Rigidity and sharp stability estimates for hypersurfaces with constant and almost-constant nonlocal mean curvature
CIRAOLO, Giulio;
2018-01-01
Abstract
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 controls its C2-distance from a single sphere. The corresponding stability inequality is obtained with a sharp decay rate.
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