一个多值Plateau问题
A Multiple-Valued Plateau Problem
摘要：
自F。阿姆格伦在这里，我们证明了一个多值模拟的经典高原问题的存在面积极小映射的磁盘。具体来说，我们发现，$K \in \mathbb N,$ $k_1,...,k_K\in \mathbb N$与总和$Q$和任何集合的$K$不相交Lipschitz邻域收缩核约旦曲线，最佳的多值边界数据与这些多重性扩展到Dirichlet最小化$Q$ -值函数与最小Dirichlet能量之间的所有可能的单调参数化的边界曲线。在一个类似于平面域极小化的Douglas条件的条件下，极小化的保形性可由拓扑方法和某些复分析得到。最后，我们分析了两个特殊的情况下：在对比单值道格拉斯解决方案，我们首先给出了一类的例子，我们的多值高原解决方案有分支点。其次，我们给出了退化行为的例子，说明了多值最大值原理的弱点，并为我们类似的道格拉斯条件提供了动力。
英文摘要：
The existence of Dirichlet minimizing multiple-valued functions for given boundary data has been known since pioneering work of F. Almgren. Here we prove a multiple-valued analogue of the classical Plateau problem of the existence of area-minimizing mappings of the disk. Specifically, we find, for $K \in \mathbb N,$ $k_1,...,k_K\in \mathbb N$ with sum $Q$ and any collection of $K$ disjoint Lipschitz Neighborhood Retract Jordan curves, optimal multiple-valued boundary data with these multiplicities which extends to a Dirichlet minimizing $Q$-valued function with minimal Dirichlet energy among all possible monotone parameterizations of the boundary curves. Under a condition analogous to the Douglas condition for minimizers from planar domains, conformality of the minimizer follows from topological methods and some complex analysis. Finally, we analyze two particular cases: in contrast to single-valued Douglas solutions, we first give a class of examples for which our multiple-valued Plateau solution has branch points. Second, we give examples of a degenerate behavior, illustrating the weakness of the multiple-valued maximum principle and provide motivation for our analogous Douglas condition.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
