alpha-Gauss Curvature flows with flat sides

作者：Kim Lami^*; Lee Ki ahm; Rhee Eunjai

来源：Journal of Differential Equations, 2013, 254(3): 1172-1192.

DOI：10.1016/j.jde.2012.10.012

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 alpha-power of positive part of Gauss Curvature. First, for 1/2 < alpha <= 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,C-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.

• 出版日期2013-2-1