Poisson公式的推广

The Generalization of Poisson Formula

首先介绍一种更一般的M bius变换及其实数形式,接着引入半径为r的球变形为半径为R的球的映射.在该映射下,证明了一偏微分方程在形式上保持不变,这可看作拓广的Laplace方程不变性的证明.此外,将单位球上Poisson核的4个重要性质推广至半径为r的球上.利用拓广的Laplace方程不变性与Poisson核满足拓广的Laplace方程的特性,证明了半径为r的球上的Poisson积分公式在球内适合于拓广的Laplace方程;利用Poisson核的其它特性,证明积分结果满足一极限条件.从而完全求得半径为r的球上Dirichlet问题的解.

First of all, MObius transformation and the real number form of it were introduced in the n-dimensional space. Then a mapping y of a sphere of radius r to a sphere of radius R was introduced. Under the mapping,a partial derivatives equation keeps the same form. This can be considered as the invariability of a generalized Laplace equation. Moreover,four important properties of generalized Poisson kernel on a sphere of radius r were obtained. With all these characteristic properties,it was proved that a generalized Poisson integral formula in a sphere of radius r satisfies all requirements,which Dirichlet problems in a sphere of radius r provides.