The volume preserving crystalline mean curvature flow of convex sets in RN

Abstract

We prove the existence of a volume preserving crystalline mean curvature flat flow starting from a compact convex set C ⊂ RN 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 in this case the convergence of an approximation algorithm due to Almgren, Taylor and Wang. Next we study the evolution of the volume and anisotropic perimeter, needed for the proof of the convergence to the Wulff shape as t → + ∞. © 2009 Elsevier Masson SAS. All rights reserved.