Riemann–Hilbert problem for the Cauchy–Riemann operator in lens and lune

Abstract

This article is devoted to study the solvability of the Riemann–Hilbert problem, with arbitrary index, for the Cauchy–Riemann operator in a lens and two complementary lunes. We generalize the corresponding results obtained by Begehr and Vaitekhovich (Complex Var Elliptic Eqs. 2014;59:76–84). The parqueting-reflection technique is used. The boundary behaviour of related integral operators of Schwarz type and Pompeiu type is discussed. Then the expressions of the solutions are explicitly obtained.