关于Beltrami方程同胚解的一点注记

A Remark on Homeomorphic Solutions of Beltrami Equations

本文讨论了退化Beltrami方程 (?)w-q(z)(?)_zw=O,z∈D同胚解的存在性及性质,证明了当系数q(z)满足局部一致椭圆型条件时同胚解总存在。特别是证明了,当区域D为Jordan区域,且存在一点z_0∈(?)D的一个邻域U使得(?)(1-|q(z)|)^(-1)dxdy<∞时上述Beltrami方程有一个同胚解把D映为单位圆。这是拟共形映射存在定理的一种推广。

In this paper, the existence of homeomorphic solutions of the Beltrami equationis discussed, where q(z) is a measurable function on the unit disk A such that||q(z)||L∞-(F)<1for every compact set F++A. It is shown that if there is a point △and a number r>0 such that (1-|q(z)|-1dxdy<∞then there exists a homeomorphic solution w =(?)(z) of the Beltrami equation which maps △onto △. If q(z) satisfies the following (1-|q(z)|)-1dxdy<∞and the homeomorphic solution w=(?)(z) satisfies the condition dxdy<∞,Athen w=(?)(z) can be homeomorphically extended to the close disk S.