Rolling stones with nonconvex sides II: all time regularity of interface and surface
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In this paper we consider the evolution of the rolling stone with a rotationally symmetric nonconvex compact initial surface 0 under the Gauss curvature ow. Let X : Sn[0,1) ! Rn+1 be the embeddings of the sphere in Rn+1 such that (t) = X(Sn, t) is the surface at time t and (0) = 0. As a consequence the parabolic equation describing the motion of the hypersurface becomes degenerate on the interface sep-arating 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 z = f(r, t)and the non-convex part of the surface near the interface as a graph of a function z = φ(r), we show that if at time t = 0, g = 1n fn-1r vanishes linearly at the interface, the g(r; t) will become smooth up to the interface for long time before focusing.
Abstract
In this paper we consider the evolution of the rolling stone with a rotationally symmetric nonconvex compact initial surface 0 under the Gauss curvature ow. Let X : Sn[0,1) ! Rn+1 be the embeddings of the sphere in Rn+1 such that (t) = X(Sn, t) is the surface at time t and (0) = 0. As a consequence the parabolic equation describing the motion of the hypersurface becomes degenerate on the interface sep-arating 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 z = f(r, t)and the non-convex part of the surface near the interface as a graph of a function z = φ(r), we show that if at time t = 0, g = 1n fn-1r vanishes linearly at the interface, the g(r; t) will become smooth up to the interface for long time before focusing.
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