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

A Cesaroni,H Kröner,M Novaga

doi:10.1051/cocv/2021096

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.

Highlights

We consider the evolution of sets t → Et in RN+1 governed by the geometric law∂tp · ν(p) = −ψ(ν(p))Hφ(p, Et), (1.1)where ν(p) is the exterior normal at p ∈ ∂Et, ψ is a positive, continuous, 1-homogeneous function representing the mobility, φ is a norm representing the surface tension, and Hφ(p) is the anisotropic mean curvature of ∂Et at p, associated with φ, see Definition 2.1
Where ν(p) is the exterior normal at p ∈ ∂Et, ψ is a positive, continuous, 1-homogeneous function representing the mobility, φ is a norm representing the surface tension, and Hφ(p) is the anisotropic mean curvature of ∂Et at p, associated with φ, see Definition 2.1. This evolution is an analogue of the classical mean curvature flow, which corresponds to the case φ(x) = ψ(x) = |x| and it is studied as model of crystal growth, see [6, 7, 14, 15]
In this paper we consider the evolution of subgraphs of entire Lipschitz functions, in the case in which either φ is regular (see (2.2)) or ψ is a norm

Summary

Introduction

Where ν(p) is the exterior normal at p ∈ ∂Et, ψ is a positive, continuous, 1-homogeneous function representing the mobility, φ is a norm representing the surface tension, and Hφ(p) is the anisotropic mean curvature of ∂Et at p, associated with φ, see Definition 2.1. As in the isotropic case, the long time attractors of the flow starting from entire Lipschitz graphs are selfsimilar expanding solutions, defined as follows: Definition 1.1. In Theorem 3.4 we show that Lipschitz graphical evolutions to (1.1) asymptotically approach self-similar expanding solutions, in an appropriate rescaled setting, provided the initial graph is a sublinear perturbations of a cone. This result is obtained by comparison with large Wulff shapes, and by constructing appropriate 1-dimensional periodic barriers to the evolution. This approach provides a different proof for the same result in the isotropic setting, which was obtained by integral estimates on the flow, see [9, 11]

Preliminary definitions and results

Self-similar expanding solutions and convergence of the rescaled flow

Stability of self-similar solutions asymptotic to mean convex cones

Stability of hyperplanes