Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition
Regularity estimates for solutions to the mean curvature flow with a Neumann boundary condition
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DOI：
10.1007/bf01190825
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发表时间：
1996-06
影响因子：
2.1
通讯作者：
Axel Stahl
中科院分区：
文献类型：
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作者：
Axel Stahl
在这项工作中，我们研究的行为紧凑，光滑，浸入流形的边界下移动的平均曲率流在欧氏空间。因此，我们通过要求浸入的单位法向场与给定的光滑超曲面∑之间的垂直接触角，以纯粹几何的方式规定了诺依曼边界条件。我们推导出一个非常尖锐的局部梯度界仅依赖于浸入和∑的曲率。结合这一点与一个短时存在性结果，我们得到了对任何给定的光滑初值和边界值的唯一解的存在性。这个解存在于任意t>0或极大有限时间区间[0，T]上，使得曲率爆炸ast→T。
In this work we study the behaviour of compact, smooth, immersed manifolds with boundary which move under the mean curvature flow in Euclidian space. We thereby prescribe the Neumann boundary condition in a purely geometric manner by requiring a vertical contact angle between the unit normal fields of the immersions and a given, smooth hypersurface∑. We deduce a very sharp local gradient bound depending only on the curvature of the immersions and∑. Combining this with a short time existence result, we obtain the existence of a unique solution to any given smooth initial and boundary data. This solution either exists for anyt>0 or on a maximal finite time interval [0,T]such that the curvature explodes ast→T.