Cabré, XavierCozzi, MatteoCsató, Gyula2023-02-132023-02-132022-06-240294-1449https://hdl.handle.net/2445/193498The 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.application/pdfeng(c) Elsevier Masson SAS, 2022Desigualtats (Matemàtica)Espais de SobolevConjunts convexosGeometria diferencialInequalities (Mathematics)Sobolev spacesConvex setsDifferential geometryinfo:eu-repo/semantics/article7255502023-02-13info:eu-repo/semantics/openAccess