黏性解與等高線流

Abstract

給定以時間 t geq 0 為參數且在 R^{n+1} 中的超曲面 Gamma(t), n geq 2。如果 Gamma(t)上每點的垂直速率與該點的均曲率吻合，我們即說 Gamma(t) 依均曲率型變，或者說，Gamma(t)是均曲率方程的解。有趣的是，Gamma(t)可能在某個時間 t_0 產生奇異點，也就是說，Gamma(t) 在 t_0 時並非一光滑的超曲面。為了把 Gamma(t) 延拓過 t_0, 論文[2]中利用黏性解的水平集來定義均曲率方程的廣義解，我們稱其廣義解為廣義均曲率流或等高線流。此外，論文[1]中證明了一個有關旋轉對稱等高線流的一個漂亮定理，因此在本論中，我們將對於[2]中所定義的等高線流與[1]中的漂亮定理做一個介紹。

We say a family of smooth hypersurfaces Gamma(t) in R^{n+1}, n geq 2 evolve by mean curvature if its normal velocity is equal to its mean curvature, or say it is the solution to the mean curvature equation. However, Gamma(t) may develop singularities in finite time, that is, at some t_0<infty, Gamma(t_0) is not a smooth hypersurface. To extend the solution pass the singularity, Chen, Giga, Goto, [2] defined a generalized flow by level sets of viscosity solutions. We call this flow the generalized mean curvature flow or the level set flow. Moreover, in 1991, Altschuler, Angenent and Giga [1] proved a beautiful theorem regarding rotationally symmetric level set flows. Thus, in this thesis, we present the level set flow defined in [2], and the beautiful theorem in [1].