an:05000053
Zbl 1148.53049
Bellettini, Giovanni; Caselles, Vicent; Chambolle, Antonin; Novaga, Matteo
Crystalline mean curvature flow of convex sets
EN
Arch. Ration. Mech. Anal. 179, No. 1, 109-152 (2006).
00121870
2006
j
53C44 58E12
Summary: We prove a local existence and uniqueness result of crystalline mean curvature flow starting from a compact convex admissible set in \(\mathbb R^n\). 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.