Calculus of Variations and Geometric Measure Theory

X. Cabré - M. Cozzi - G. Csato

A fractional Michael-Simon Sobolev inequality on convex hypersurfaces

created by csato on 12 Nov 2021



Inserted: 12 nov 2021

Year: 2020

ArXiv: 2004.13129 PDF


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, but displays on the right-hand side an additional $L^p$ term 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 semi-norm 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.