ROLLING STONES WITH NONCONVEX SIDES II: ALL TIME REGULARITY OF INTERFACE AND SURFACE Lee, Ki-Ahm; Rhee, Eun-Jai;
In this paper we consider the evolution of the rolling stone with a rotationally symmetric nonconvex compact initial surface under the Gauss curvature flow. Let be the embeddings of the sphere in such that is the surface at time t and . As a consequence the parabolic equation describing the motion of the hypersurface becomes degenerate on the interface separating the nonconvex part from the strictly convex side, since one of the curvature will be zero on the interface. By expressing the strictly convex part of the surface near the interface as a graph of a function and the non-convex part of the surface near the interface as a graph of a function , we show that if at time , vanishes linearly at the interface, the will become smooth up to the interface for long time before focusing.