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

The volume preserving crystalline mean curvature flow of convex sets in $\mathbf R^N$

created by novaga on 25 Jul 2007
modified on 10 Nov 2018


Published Paper

Inserted: 25 jul 2007
Last Updated: 10 nov 2018

Journal: J. Math. Pures Appl.
Volume: 92
Number: 5
Pages: 499-527
Year: 2009


We prove the existence of a volume preserving crystalline mean curvature flat flow starting from a compact convex subset $C$ of $\mathbf R^N$ and its convergence, modulo a time-dependent translation, to a Wulff shape with the corresponding volume. We also prove that if $C$ satisfies an interior ball condition (the ball being the Wulff shape), then the evolving convex set satisfies a similar condition for some time. To prove these results we establish existence, uniqueness and short-time regularity for the crystalline mean curvature flat flow with a bounded forcing term starting from $C$, showing the convergence of the Almgren-Taylor-Wang's algorithm in this case. Next we study the evolution of the volume and anisotropic perimeter, needed for the proof of the convergence to the Wulff shape as the time tends to infinity.

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