Published Paper

Inserted: 2 nov 2004
Last Updated: 10 nov 2018

Journal: Arch. Rational Mech. Anal.
Volume: 179
Number: 1
Pages: 109-152
Year: 2006

Abstract:

We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in $\mathbf{R}^N$. This theorem can handle the facet breaking-bending phenomena, and can be generalized to any anisotropic mean curvature flow. The method provides also a generalized geometric evolution starting from any compact convex set, existing up to the extinction time, satisfying a comparison principle, and defining a continuous semigroup in time. We prove that, when the initial set is convex, our evolution coincides with the flat $\phi$-curvature flow in the sense of Almgren-Taylor-Wang. As a by-product, it turns out that the flat $\phi$-curvature flow starting from a compact convex set is unique.

Keywords: crystalline mean curvature, flat flows, convex bodies