Motion of Nonadmissible Convex Polygons by Crystalline Curvature

Abstract

Behavior of convex solution polygons to a general crystalline motion is investigated. A polygon is called admissible if the set of its normal angles equals that of the Wulff shape. We prove that if the initial polygon is not an admissible polygon, then all edges disappear simultaneously, or edge disappearing occurs at most finitely many instants and eventually a convex solution polygon becomes an admissible convex polygon. In the latter case, the normal angle of disappearing edge does not belong to the set of the normal angles of the Wulff shape. We also show five typical examples of this motion.