Numerical study and examples on singularities of solutions to anisotropic crystalline curvature flows of nonconvex polygonal curves

Abstract

We construct explicit solutions of the anisotropic motion of closed polygonal plane curves by a power of crystalline cuvature, in the case where the initial curves are nonconvex and the power is less than one: The solutions develop degenerate pinching singularities of a "whisker"-type and a split-type in finite time, and do not become convex polygons. Moreover, in the splitting case, we conjecture degenerate pinching rate from numerical experiments.