Stability and Symmetry in the Navier Problem for the One-Dimensional Willmore Equation
Stability and Symmetry in the Navier Problem for the One-Dimensional Willmore Equation
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一维威尔莫尔方程纳维问题的稳定性和对称性
DOI：
10.1137/07069033x
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发表时间：
2009
期刊：
影响因子：
--
通讯作者：
H. Grunau
中科院分区：
文献类型：
--
作者：
K. Deckelnick;H. Grunau
我们考虑一维Willmore方程满足Navier边界条件，即边界上的位置和曲率是已知的。在以前的工作中，已经构造了对称数据的显式对称解。在一定的边界曲率范围内，人们恰好有两个对称解，而对于这个范围闭合之外的边界曲率，则没有。解决方案是有序的；一个是“小的”，另一个是“大的”。在本文的第一部分，我们讨论了稳定性问题，证明了小解在允许的边界曲率的整个开放范围内是(线性化的)稳定的，而大解是不稳定的，并且具有Morse指数1。第二个目标是研究小解对于相应的Willmore泛函是否是极小的。结果表明，对于某一子范围的可容许边界曲率，小解是唯一极小值，而对于超出该范围的曲率，则不能达到最小值。..。
We consider the one-dimensional Willmore equation subject to Navier boundary conditions; i.e., the position and the curvature are prescribed on the boundary. In a previous work, explicit symmetric solutions to symmetric data have been constructed. Within a certain range of boundary curvatures one has precisely two symmetric solutions, while for boundary curvatures outside the closure of this range there are none. The solutions are ordered; one is “small,” and the other is “large.” In the first part of this paper we address the stability problem and show that the small solution is (linearized) stable in the whole open range of admissible boundary curvatures, while the large one is unstable and has Morse index 1. A second goal is to investigate whether the small solution is minimal for the corresponding Willmore functional. It turns out that for a certain subrange of admissible boundary curvatures the small solution is the unique minimum, while for curvatures outside that range the minimum is not attained. ...