Level set method for motion by mean curvature
Level set method for motion by mean curvature
复制标题
平均曲率运动的水平集方法
DOI：
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发表时间：
2016
期刊：
影响因子：
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通讯作者：
W. Minicozzi
中科院分区：
文献类型：
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作者：
T. Colding;W. Minicozzi
广泛的物理现象，如晶体生长和火焰传播的建模，导致跟踪前沿曲率相关的速度移动。当速度为曲率时，这就导出了欧氏空间上一类经典的退化非线性二阶微分方程。人们自然会问：“解的规律性是什么？“先验解只是在弱意义上定义的，但事实证明它们总是二次可微的经典解。这个结果是最优的;它们的二阶导数只有在具有简单几何解释的非常严格的情况下才是连续的。这个证明将分析和几何学结合在一起。如果不深入理解基本几何，就不可能证明精细的分析性质。
Modeling of a wide class of physical phenomena, such as crystal growth and flame propagation, leads to tracking fronts moving with curvature-dependent speed. When the speed is the curvature this leads to one of the classical degenerate nonlinear second order differential equations on Euclidean space. One naturally wonders "what is the regularity of solutions?" A priori solutions are only defined in a weak sense, but it turns out that they are always twice differentiable classical solutions. This result is optimal; their second derivative is continuous only in very rigid situations that have a simple geometric interpretation. The proof weaves together analysis and geometry. Without deeply understanding the underlying geometry, it is impossible to prove fine analytical properties.