On a Free Boundary Problem for the Curvature Flow with Driving Force
On a Free Boundary Problem for the Curvature Flow with Driving Force
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DOI：
10.1007/s00205-015-0920-8
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发表时间：
2016-03
影响因子：
2.5
通讯作者：
Jong-Shenq Guo;H. Matano;Masahiko Shimojo;Chang-Hong Wu
中科院分区：
文献类型：
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作者：
Jong-Shenq Guo;H. Matano;Masahiko Shimojo;Chang-Hong Wu
本文研究了上半平面中两个端点沿着水平轴以给定的固定接触角滑动的平面曲线的曲率相关运动的自由边界问题。我们的第一个主要结果涉及的解决方案的分类，每一个解决方案福尔斯落入三个类别之一，即，面积扩张，面积有界和面积收缩类型。然后，我们详细研究在每个类别的解决方案的渐近行为。除其他外，我们表明，解决方案是渐近自相似的区域扩大和区域缩小的情况下，而解决方案收敛到一个固定的解决方案或行波的区域有界的情况下。我们还证明了结果的解的性质。本文的主要工具之一是交数原理，但是为了处理自由边界的解决方案，我们引入了我们所谓的“扩展交数原理”，这在处理端点移动的曲线时非常有用。
We study a free boundary problem associated with the curvature dependent motion of planar curves in the upper half plane whose two endpoints slide along the horizontal axis with prescribed fixed contact angles. Our first main result concerns the classification of solutions; every solution falls into one of the three categories, namely, area expanding, area bounded and area shrinking types. We then study in detail the asymptotic behavior of solutions in each category. Among other things we show that solutions are asymptotically self-similar both in the area expanding and the area shrinking cases, while solutions converge to either a stationary solution or a traveling wave in the area bounded case. We also prove results on the concavity properties of solutions. One of the main tools of this paper is the intersection number principle, however in order to deal with solutions with free boundaries, we introduce what we call “the extended intersection number principle”, which turns out to be exceedingly useful in handling curves with moving endpoints.