混合解析函数 Schwarz 问题与常系数椭圆方程的边值问题

Schwarz Problem of Mixed AnalyticFunctions and Boundary Value Problemof Elliptic Equations with ConstantCoefficients

本文引用混合解析函数并讨论其 Riemann-Hilbert 型边值问题,先建立了混合解析函数的 lemelj 公式、Cauchy 公式、Cauchy 主值积分(奇异积分)、Privonov 定理. 在此基础上研究混合解析函数的 Schwarz 问题,通过定义分区解析函数找出其解,利用混合解析函数的Schwarz 问题处理常系数椭圆方程的 Poisson 问题,用 Cauchy 型积分给出解的具体表达式。

In this paper, mixed analytic function is introduced and its Riemann-Hilbert boundary value problem is discussed. Firstly, Plemelj formula, Cauchy formula, Cauchy principal integral (singular integral) and Privonov theorem of mixed analytic function are established. On this basis, the Schwarz problem of mixed analytic functions is studied, and its solution is found by defining partition analytic functions. The Schwarz problem of mixed analytic functions is used to deal with Poisson problem of elliptic equations with constant coefficients. The specific expression of the solution is given by Cauchy integral.