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A fractional Michael-Simon Sobolev inequality on convex hypersurfaces
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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.
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CABRÉ, Xavier, COZZI, Matteo and CSATÓ, Gyula. A fractional Michael-Simon Sobolev inequality on convex hypersurfaces. Annales de l'Institut Henri Poincare-Analyse non Lineaire. 2022. ISSN 0294-1449. [consulted: 17 of August of 2026]. Available at: https://hdl.handle.net/2445/193498