Abstract: It is known that the Keldych-Sedov formula gives the representation of solutions of the mixed boundary value problem for analytic functions in the upper half-plane the authors. But for many problems in mechanics and physics, one needs the more general formula of solutions of discontinuous Riemann-Hilbert boundary value problem for analytic functions in the upper half-plane and other special domains. In the present paper, the authors establish the representations of the general discontinuous boundary value problem for analytic functions in the upper half-plane and upper half-disk. Moreover the authors give their applications to some nonlinear elliptic complex equations. The results in present paper have the important applications to the equations of mixed type and nonlinear mechanics.

Key words: Discontinuous Riemann-Hilbert problem, analytic functions, elliptic complex equations

摘要： 凯尔提什-谢多夫公式给出了上半平面内解析函数混合边值问题解的表示式。但在许多力学和物理的问题中，需要解析函数在上半平面和其他特殊区域内间断黎曼-希尔伯特边值问题更一般的表示式。本文建立了解析函数在上半平面和上半圆内一般间断边值问题解的表示式，并给出这些表示式在非线性椭圆型复方程中的应用。这些结果在混合型方程与非线性力学中有着重要的应用。

关键词: 间断黎曼-希尔伯特问题, 解析函数, 椭圆型复方程