Motion of non-convex polygons by crystalline curvature and almost convexity phenomena
As it is known, the the mean curvature flow is a typical motion for smooth energies. This flow is obtained as the gradient flow for the total interfacial energy when the interfacial energy is a constant. For non-smooth energies there is a special class of interfacial energies which are called crystalline energies. For these energies, the motion of polygonal plane curves by its crystalline curvature was investigated by S. Angement, M. E. Gurtin and J. E. Taylor. Moreover, admissibility of polygonal curves has been introduced. This paper considers motions of admissible polygons in the plane by crystalline curvature. The admissible polygons have the same normal vectors as the Wulf shape and the change of normal angle at each corner between two edges is the similar to that of the Wulf shape. The generalized crystalline curvature flow with an initial admissible polygon is considered and the behavior of the solution polygon in time is discussed. In this paper, the author clarifies the assumptions on functions \(g\), defining the crystalline curvature flow, and the Wulf shape to guarantee that the solution polygon belongs to the admissible class as long as the enclosed area of solution polygons is positive. Moreover, it is shown that a crystalline curvature of each edges becomes non-negative, and the solution polygon becomes star-shaped strictly before the extinction time. After introduction of some necessary definitions and notations, the conditions on g and the interfacial energy are given. Moreover, the assumption on the point symmetry of the Wulf shape with respect to the origin allows one to formulate and prove two theorems. The first theorem states that the edge-disappearing happens in a finite time. The second one characterizes a shape of solution polygon near the extinction time in the case when there remain two inflection edges, i.e. edges with zero transition number. The obtained results guarantee that the numerical procedure, using crystalline motions as the approximating method for motion of the smooth curves does not break down till the extinction time even if some edges disappear during time evolution.
scientific article; zbMATH DE number 5346976
scientific article