Nonlocal diffusion of smooth sets

TL;DR

This paper analyzes how smooth sets evolve under fractional diffusion, showing that their normal velocity approximates mean curvature or fractional mean curvature depending on the parameter s, linking fractional heat diffusion to geometric flows.

Contribution

It establishes the near proportionality of normal velocity to mean or fractional mean curvature for small times, connecting fractional diffusion to geometric curvature flows.

Findings

Normal velocity approximates mean curvature for s in [1/2, 1)

Normal velocity approximates fractional mean curvature for s in (0, 1/2)

Diffusion by fractional heat relates to fractional mean curvature flow

Abstract

We consider normal velocity of smooth sets evolving by the $s-$fractional diffusion. We prove that for small time, the normal velocity of such sets is nearly proportional to the mean curvature of the boundary of the initial set for $s\in [\frac{1}{2}, 1)$ while, for $s\in (0, \frac{1}{2})$, it is nearly proportional to the fractional mean curvature of the initial set. Our results show that the motion by (fractional) mean curvature flow can be approximated by fractional heat diffusion and by a diffusion by means of harmonic extension of smooth sets.