Geometric evolution problems, distance function and viscosity solutions
Geometric evolution problems, distance function and viscosity solutions
复制标题
几何演化问题、距离函数和粘度解决方案
DOI：
10.1007/978-3-642-57186-2_2
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发表时间：
1997
影响因子：
1.4
通讯作者：
L. Ambrosio
中科院分区：
文献类型：
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作者：
L. Ambrosio
平均曲率流是一个几何初值问题。从Rn中的光滑初始曲面Γ 0开始，曲面Γ以等于其平均曲率向量的法向速度随时间演化。通过微分几何的参数化方法，对于凸曲面、图或平面曲线已经获得了许多结果（例如参见Altschleman & Grayson [AG 92]、Ecker & Huisken [EH 89]、Gage &汉密尔顿[GH 86]、Grayson [Gra87]和Huisken [Hui 84]）。然而，当n ≥ 3时，最初光滑的曲面可能会发展出奇点。例如，R3中的“哑铃”区域在有限时间内分裂成两部分（参见图1）。[Gra89a]）或一个足够“胖”的环面在有限的时间内关闭其内部的洞（参见。[SS93]）。还可以看出，R3中的光滑曲线可以在有限时间内自相交。
The mean curvature flow is a geometric initial value problem. Starting from a smooth initial surface Γ0in Rn, the surfacesΓtevolve in time with normal velocity equal to their mean curvature vector. By parametric methods of differential geometry many results have been obtained for convex surfaces, graphs or planar curves (see for instance Altschuler & Grayson [AG92], Ecker & Huisken [EH89], Gage & Hamilton [GH86], Grayson [Gra87], and Huisken [Hui84]). However, for n ≥ 3, initially smooth surfaces may develop singularities. For example, a “dumbbell” region in R3 splits into two pieces in finite time (cf. [Gra89a]) or a “fat” enough torus closes its interior hole in finite time (cf. [SS93]). Also it can be seen that smooth curves in R3may self intersect in finite time.