发表机构
The University of Tokyo（东京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究斜线上具有一般接触角的平面开曲线的保面积曲率流，证明了全局存在唯一性及指数收敛到等面积圆弧。
AI 中文摘要
我们研究平面开曲线的保面积曲率流，其中曲线的端点位于斜线上，并以一般接触角与这些线相交。在以往关于高阶几何梯度流的移动边界问题的研究中，通常施加直角条件以确保演化曲线自然满足边界条件。然而，在一般接触角条件下，演化必须由自由边界问题来描述。在关于接触角$\theta_\pm$、斜线之间的夹角$\theta$以及初始数据$\gamma_0$的适当假设下，我们证明了在重参数化意义下的全局时间存在性和唯一性。我们进一步证明，在每一时刻选取常速参数化后，解以指数速度收敛到与初始曲线具有相同有向面积的圆弧。我们的分析基于将演化表述为自由边界问题，并推导出一致先验估计。
英文摘要
We study an area-preserving curvature flow for planar open curves whose end points lie on skew lines and meet the lines at general contact angles. In previous studies of moving boundary problems for higher-order geometric gradient flows, the right-angle condition has often been imposed to ensure that the evolving curves naturally satisfy the boundary conditions. However, under general contact-angle conditions, the evolution must be described by a free boundary problem. Under suitable assumptions on the contact angles $θ_\pm$, the angle $θ$ between the skew lines, and the initial data $γ_0$, we prove global-in-time existence and uniqueness up to reparametrization. We further prove that, after choosing a constant-speed parametrization at each time, the solution converges exponentially to a circular arc with the same signed area as the initial curve. Our analysis is based on a formulation of the evolution as a free boundary problem together with the derivation of uniform a priori estimates.
Comments43 pages, 2 figures. Minor formatting and cross-reference corrections; mathematical content unchanged
