具有平边的α-Gauss曲率流
α-Gauss Curvature flows with flat sides
摘要：
本文研究了$\R^{3}$中二维凸曲面的变形，曲面上一点的速度正比于高斯曲率的正部分的$α$ -次方。首先，对于1/2<α \leq 1 $, we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with $ C^{1，1} $-estimate before the collapsing time if the initial surface is only convex. Moreover, we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while. We also show the interface between the flat side and the strictly convex side of the surface remains smooth on $ 0 < t < T_0 $ under certain necessary regularity and non-degeneracy initial conditions, where $ T_0$是平坦边的消失时间.
英文摘要：
In this paper, we study the deformation of the 2 dimensional convex surfaces in $\R^{3}$ whose speed at a point on the surface is proportional to $α$-power of positive part of Gauss Curvature. First, for 1/2<α\leq 1$, we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with $C^{1,1}$-estimate before the collapsing time if the initial surface is only convex. Moreover, we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while. We also show the interface between the flat side and the strictly convex side of the surface remains smooth on $0 < t < T_0$ under certain necessary regularity and non-degeneracy initial conditions, where $ T_0$ is the vanishing time of the flat side.
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