J. Korean Math. Soc. 2012; 49(3): 585-604
Printed May 1, 2012
https://doi.org/10.4134/JKMS.2012.49.3.585
Copyright © The Korean Mathematical Society.
Ki-ahm Lee and Eunjai Rhee
Seoul National University, Seoul National University
In this paper we consider the evolution of the rolling stone with a rotationally symmetric nonconvex compact initial surface $\Sigma_0$ under the Gauss curvature flow. Let $X: S^n \times [0, \infty) \to \mathbb R^{n+1}$ be the embeddings of the sphere in $\mathbb R^{n+1}$ such that $\Sigma(t)= X(S^n,t)$ is the surface at time $t$ and $\Sigma(0)=\Sigma_0$. 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 $z=f(r,t)$ and the non-convex part of the surface near the interface as a graph of a function $z=\varphi(r)$, we show that if at time $t=0$, $g=\frac{1}{n} f^{n-1}_r$ vanishes linearly at the interface, the $g(r,t)$ will become smooth up to the interface for long time before focusing.
Keywords: free boundary problems, degenerate fully nonlinear equations, Gauss curvature flow
MSC numbers: 35K20
2012; 49(2): 265-291
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