Communications in Analysis and Geometry

Volume 27 (2019)

Number 1

On the evolution by fractional mean curvature

Pages: 211 – 249



Mariel Sáez (Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Santiago, Chile)

Enrico Valdinoci (Department of Mathematics and Statistics, University of Western Australia, Crawley, WA, Australia)


In this paper we study smooth solutions to a fractional mean curvature flow equation. We establish a comparison principle and consequences such as uniqueness and finite extinction time for compact solutions. We also establish evolutions equations for fractional geometric objects that in turn yield the preservation of certain quantities, such as the positivity of the fractional mean curvature.

It is a pleasure to thank Eleonora Cinti, Luca Lombardini and Carlo Sinestrari for their very interesting and useful comments on a preliminary version of this manuscript. The first author has been partially supported by Proyecto Fondecyt Regular 1150014. The second author has been supported by the ERC grant 277749 “EPSILON Elliptic Pde’s and Symmetry of Interfaces and Layers for Odd Nonlinearities” and the Australian Research Council Discovery Project DP170104880 “NEW Nonlocal Equations at Work”.

Received 1 January 2017

Published 7 May 2019