Calculus of Variations and Geometric Measure Theory

A. Chambolle - M. Morini - M. Novaga - M. Ponsiglione

Existence and uniqueness for anisotropic and crystalline mean curvature flows

created by morini on 09 Feb 2017
modified by novaga on 08 May 2019


Published Paper

Inserted: 9 feb 2017
Last Updated: 8 may 2019

Journal: J. Amer. Math. Soc.
Volume: 32
Number: 3
Pages: 779–824
Year: 2019

ArXiv: 1702.03094 PDF


An existence and uniqueness result, up to fattening, for crystalline mean curvature flows with forcing and arbitrary (convex) mobilities is proven. This is achieved by introducing a new notion of solution to the corresponding level set formulation. Such a solution satisfies the comparison principle and a stability property with respect to the approximation by suitably regularized problems. The results are valid in any dimension and for arbitrary, possibly unbounded, initial closed sets. The approach accounts for the possible presence of a time-dependent bounded forcing term, with spatial Lipschitz continuity. As a byproduct of the analysis, the problem of the convergence of the Almgren-Taylor-Wang minimizing movements scheme to a unique (up to fattening) ''flat flow'' in the case of general, possibly crystalline, anisotropies is settled.