EVOLUTION OF SPIRAL-SHAPED POLYGONAL CURVE BY CRYSTALLINE CURVATURE FLOW WITH A PINNED TIP

Evolution of convex polygonal spiral with fixed center by crystalline eikonal-curvature flow is considered. In this evolution we consider a new facet of the polygonal curve generates from center when a facet associated with the center evolves with enough length, which is equal to the length of the facet in Wulff shape of energy density function. We prove the existence, uniqueness and intersection free of solution to our formulation globally-in-time. In the proof of the existence we also prove that new facets are generated repeatedly in time. The important property for intersection-free result is monotonicity property such that the normal velocity of every facets are positive after the next new facet is generated, so that the center is always behind of the moving facets