Crystalline curvature flow of a graph in a variational setting

In this paper, we investigate the properties of a definition of crystalline curvature flow given recently by Fukui and Giga in [6]. This definition coincides with that given by Angenant and Gurtin in [2] and by Taylor et. al. in [12],[13] and [15] for admissible faceted curves. Our investigation is concerned with faceted curves which are not necessarily admissible Wulff curves. We prove that faceted curves remain faceted and attain a stationary state after a finite time. But the evolving curves need not become admissible Wulff curves. We introduce a scheme for the numerical computation of the evolution and visualise our results through our computations. AMS Subject Classification: 35 R 35, 35 K 22, 35 K 65.