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Crystalline mean curvature flow of convex sets. (English) Zbl 1148.53049
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.

##### MSC:
 53C44 Geometric evolution equations (mean curvature flow, Ricci flow, etc.) (MSC2010) 58E12 Variational problems concerning minimal surfaces (problems in two independent variables)
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