A fractional Michael-Simon Sobolev inequality on convex hypersurfaces

TL;DR

This paper establishes a fractional Sobolev inequality on convex hypersurfaces, linking the Gagliardo semi-norm and fractional mean curvature, and applies it to improve bounds in fractional mean curvature flow.

Contribution

It introduces a fractional version of the Michael-Simon Sobolev inequality specifically for convex hypersurfaces, extending classical results to fractional settings.

Findings

Proves a fractional Michael-Simon Sobolev inequality for convex hypersurfaces.

Provides a new upper bound for the maximal existence time in fractional mean curvature flow.

Connects geometric properties of convex sets with fractional curvature analysis.

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, 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.