Calculus of Variations and Geometric Measure Theory

A. Cesaroni - H. Kröner - M. Novaga

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

created by novaga on 13 May 2021
modified on 12 Oct 2021


Published Paper

Inserted: 13 may 2021
Last Updated: 12 oct 2021

Journal: ESAIM: COCV
Volume: 27
Number: Paper n. 97
Pages: 17 pages
Year: 2021

ArXiv: 2105.06359 PDF


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.