关于曲率依赖于位置的平面曲线
On planar curves with position-dependent curvature
摘要：
受曲率驱动平面曲线流动的位似解及其许多物理应用的启发，本文对曲率$κ$为给定位置或方向函数的定向曲线进行了系统的研究。该分析是由动力系统的观点。虽然集中在规定的曲率只取决于距离$r$从一个区别点的情况下，基本的动力学概念被认为适用于其他情况下，以及。作为应用，建立了具有实常数$a,b$的$κ= ar^b$的简单封闭解的完全分类.
英文摘要：
Motivated by homothetic solutions to curvature-driven flows of planar curves, as well as their many physical applications, this work carries out a systematic study of oriented curves whose curvature $κ$ is a given function of position or direction. The analysis is informed by a dynamical systems point of view. Though focussed on situations where the prescribed curvature depends only on the distance $r$ from a distinguished point, the basic dynamical concepts are seen to apply in other situations as well. As an application, a complete classification of all simple closed solutions of $κ= ar^b$, with real constants $a,b$, is established.
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