On mean curvature flow with forcing
On mean curvature flow with forcing
复制标题
带强迫的平均曲率流
DOI：
10.1080/03605302.2019.1695262
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发表时间：
2015
影响因子：
1.9
通讯作者：
Dohyun Kwon
中科院分区：
文献类型：
--
作者：
Inwon C. Kim;Dohyun Kwon
摘要研究了具有体积依赖强迫的平均曲率流的几何性质和适定性。有了使演化集的体积远离零和无穷大的强迫类，我们证明了随着时间的推移，强版本的星形被保存下来。更准确地说，该流保持ρ反射性质，该性质对应于集合关于最近球的定量Lipschitz性质。基于这一性质，我们证明了问题是适定的，并且它的解从ρ-反射性质开始立即变得光滑。最后，对于一个模型问题，我们将讨论流在Hausdorff拓扑下的指数收敛到唯一平衡点。在分析过程中，我们采用了Feldman-Kim提出的粘性解方法和变分方法相结合的方法。主要的挑战在于缺乏比较原则，这伴随着强制执行惩罚小规模交易的条款。
Abstract This paper investigates geometric properties and well-posedness of a mean curvature flow with volume-dependent forcing. With the class of forcing which bounds the volume of the evolving set away from zero and infinity, we show that a strong version of star-shapedness is preserved over time. More precisely, it is shown that the flow preserves the ρ-reflection property, which corresponds to a quantitative Lipschitz property of the set with respect to the nearest ball. Based on this property we show that the problem is well-posed and its solutions starting with ρ-reflection property become instantly smooth. Lastly, for a model problem, we will discuss the flow’s exponential convergence to the unique equilibrium in Hausdorff topology. For the analysis, we adopt the approach developed by Feldman-Kim to combine viscosity solutions approach and variational method. The main challenge lies in the lack of comparison principle, which accompanies forcing terms that penalize small volume.