Fractional mean curvature flow of Lipschitz graphs

TL;DR

This paper studies the fractional mean curvature flow of Lipschitz graphs, establishing regularity, long-term behavior, and stability results, including convergence to self-similar solutions and stability of hyperplanes and convex cones.

Contribution

It provides new regularity results and analyzes the asymptotic behavior of the flow for Lipschitz graphs, including stability and convergence to self-similar solutions.

Findings

Flow asymptotically approaches self-similar solutions for sublinear perturbations of cones

Hyperplanes and convex cones are stable under the flow

Regularity results for fractional mean curvature flow of Lipschitz graphs

Abstract

We consider the fractional mean curvature flow of entire Lipschitz graphs. We provide regularity results, and we study the long time asymptotics of the flow. In particular we show that in a suitable rescaled framework, if the initial graph is a sublinear perturbation of a cone, the evolution asymptotically approaches an expanding self-similar solution. We also prove stability of hyperplanes and of convex cones in the unrescaled setting.