Anisotropic mean curvature flow of Lipschitz graphs and convergence to self-similar solutions

TL;DR

This paper studies the anisotropic mean curvature flow of Lipschitz graphs, establishing existence, uniqueness, and stability of self-similar solutions, and analyzing their long-term behavior and convergence to cones.

Contribution

It introduces new results on the existence, uniqueness, and stability of self-similar solutions for anisotropic mean curvature flow of Lipschitz graphs.

Findings

Existence and uniqueness of expanding self-similar solutions.

Characterization of long-time behavior and convergence to cones.

Stability of solutions under perturbations, including hyperplanes.

Abstract

We consider the anisotropic mean curvature flow of entire Lipschitz graphs. We prove existence and uniqueness of expanding self-similar solutions which are asymptotic to a prescribed cone, and we characterize the long time behavior of solutions, after suitable rescaling, when the initial datum is a sublinear perturbation of a cone. In the case of regular anisotropies, we prove the stability of self-similar solutions asymptotic to strictly mean convex cones, with respect to perturbations vanishing at infinity. We also show the stability of hyperplanes, with a proof which is novel also for the isotropic mean curvature flow.