*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

**Abstract:**

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.