Boundary Theory

边界的概念存在于数学的许多领域。在所涉及的分析中，单位圆盘的经典泊松公式(以及狄里克莱特问题的可解性)可以说提供了最有指导意义的例子。它在圆盘内部的调和函数和边界圆盘上的函数之间建立了一一对应关系。由于边界圆正好是单位圆盘在环境欧几里得平面上的拓扑边界，这种对应关系可能看起来并不那么引人注目。然而，如果考虑到开放圆盘上的调和函数通过共形不变性与双曲平面上的调和函数相同，而双曲平面上的调和函数原本没有任何边界，就变得不那么微不足道了。

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.