Parabolic equations for curves on surfaces Part I. Curves with $p$-integrable curvature
Parabolic equations for curves on surfaces Part I. Curves with $p$-integrable curvature
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DOI：
10.2307/1971426
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发表时间：
1990-11
影响因子：
4.9
通讯作者：
S. Angenent
中科院分区：
文献类型：
--
作者：
S. Angenent
这是由两部分组成的论文的第一部分，在该论文中，我们开发了表面曲线的抛物线方程理论，该理论可应用于所谓的流平均曲率问题的曲线缩短，以及许多二维相变模型。我们引入了一类方程，对于具有 p 可积曲率的初始数据，其初始值问题是可解的，并且如果曲线在有限时间内变得奇异，我们还给出了曲率的 p 范数必须爆炸的速率的估计。第二部分的主题将详细讨论解变得奇异的方式以及“通过奇点继续解”的方法。
This is the first of a two-part paper in which we develop a theory of parabolic equations for curves on surfaces which can be applied to the so-called curve shortening of flow-by-mean-curvature problem, as well as to a number of models for phase transitions in two dimensions. We introduce a class of equations for which the initial value problem is solvable for initial data with p-integrable curvature, and we also give estimates for the rate at which the p-norms of the curvature must blow up, if the curve becomes singular in finite time. A detailed discussion of the way in which solutions can become singular and a method for "continuing the solution through a singularity" will be the subject of the second part.