Boundary Theory

边界理论

• 批准号: RGPIN-2016-06744
• 负责人: Kaimanovich, Vadim
• 金额: $ 1.09万
• 依托单位: University of Ottawa
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2017
• 资助国家: 加拿大
• 起止时间: 2017-01-01 至 2018-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=635204
• 关键词: Boundary; Theory

The idea of boundary is present in many areas of mathematics. In what concerns analysis, the classical Poisson formula (along with the solvability of the Dirichlet problem) for the unit disk arguably provides the most instructive example. It establishes a one-to-one correspondence between harmonic functions on the interior of the disk and functions on the boundary disk. Since the boundary circle is precisely the topological boundary of the unit disk in the ambient Euclidean plane, this correspondence may look not so striking. However, it becomes less trivial if one takes into account that by conformal invariance harmonic functions on the open disk are the same as harmonic functions on the hyperbolic plane, whereas the latter space originally does not come equipped with any boundary.

边界的概念存在于数学的许多领域。在分析中，经典的泊松公式（沿着狄利克雷问题的可解性）为单位圆盘提供了最有启发性的例子。它建立了一个一对一的对应关系，调和函数的内部的磁盘和功能的边界磁盘。由于边界圆恰好是周围欧氏平面中单位圆盘的拓扑边界，所以这种对应关系看起来可能不那么引人注目。然而，如果考虑到通过共形不变性，开圆盘上的调和函数与双曲平面上的调和函数相同，而后者空间最初并不配备任何边界，那么它就变得不那么平凡了。