多复变函数的一些边值问题

本文主要研究二元复变解析函数与一阶椭圆型复方程组在双圆柱区域上的某些边值问题,包括Dirichlet问题与Riemann-Hilbert问题。文中给出了这些问题适定的变态提法,先证明了相应变态问题解的存在性与唯一性,然后导出原边值问题可解的充要条件。这里,我们使用的方法与别人不同,对于一阶椭圆型复方程组,我们所加的条件较弱,没有看到国内外有其他人获得这样完整的结果。在本文的后一部分,我们还讨论了二元解析函数与一阶椭圆型复方程组在双圆环柱区域上的Dirichlet问题与Riemann—Hilbert问题,给出了这些边值问题可解的充要条件。使用本文中的方法,还可讨论多个复变函数相应边值问题的可解性。

Some Boundary Value Problems for Several Complex Variables

In this paper, we mainly consider some boundary value problems for analytic functions of two complex variabies and systems of first order complex equations in the simply connected bicylinder. The stated boundary value problems include the Dirichlet problem and the Riemann-Hilbert problem. The integral expressions of solutions for the foregoing boundary value problems and their solvability conditions were established . From these, we can see the essential distinction between boundary value problems of two complex variables and those of one complex variable . In addition, the corresponding boundary value problems in some double connected domains for analytic functions and systems of first order complex equations are discussed. Using the similar method, the corresponding boundary value problems for several complex variables can be considered.