Schwarz problem for first order elliptic systems in unbounded sectors

Abstract

In this article we deal with a Schwarz-type boundary value problem for both the inhomogeneous Cauchy-Riemann equation and the generalized Beltrami equation on an unbounded sector with angle \( \vartheta = \pi/n, n \in \mathbb{N} \). By the method of plane parqueting-reflection and the Cauchy-Pompeiu formula for the sector, the Schwarz-Poisson integral formula is obtained. We also investigate the boundary behaviour and the \( C^\alpha \)-property of a Schwarz-type as well as of a Pompeiu-type operator. The solution to the Schwarz problem of the Cauchy-Riemann equation is explicitly expressed. Sufficient conditions on the coefficients of the generalized Beltrami equation are obtained under which the corresponding system of integral equations is contractive. This proves the existence of a unique solution to the Schwarz problem of the generalized Beltrami equation.