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Title: A fractional Michael-Simon Sobolev inequality on convex hypersurfaces
Author: Cabré, Xavier
Cozzi, Matteo
Csató, Gyula
Keywords: Desigualtats (Matemàtica)
Espais de Sobolev
Conjunts convexos
Geometria diferencial
Inequalities (Mathematics)
Sobolev spaces
Convex sets
Differential geometry
Issue Date: 24-Jun-2022
Publisher: Elsevier Masson SAS
Abstract: The classical Michael-Simon and Allard inequality is a Sobolev inequality for functions defined on a submanifold of Euclidean space. It is governed by a universal constant independent of the manifold, thanks to an additional $L^p$ term on the righthand side which is weighted by the mean curvature of the underlying manifold. We prove here a fractional version of this inequality on hypersurfaces of Euclidean space that are boundaries of convex sets. It involves the Gagliardo seminorm of the function, as well as its $L^p$ norm weighted by the fractional mean curvature of the hypersurface. As an application, we establish a new upper bound for the maximal time of existence in the smooth fractional mean curvature flow of a convex set. The bound depends on the perimeter of the initial set instead of on its diameter.
Note: Versió postprint del document publicat a:
It is part of: Annales de l'Institut Henri Poincare-Analyse non Lineaire, 2022
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ISSN: 0294-1449
Appears in Collections:Articles publicats en revistes (Matemàtiques i Informàtica)

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