Calculus of Variations and Geometric Measure Theory
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G. Bellettini - V. Caselles - A. Chambolle - M. Novaga

Crystalline mean curvature flow of convex sets

created on 02 Nov 2004
modified by belletti on 23 Dec 2005


Accepted Paper

Inserted: 2 nov 2004
Last Updated: 23 dec 2005

Journal: Arch. Rational Mech. Anal.
Year: 2005


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

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