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.

Keywords:

• Riemann–Hilbert problem
• Cauchy–Riemann equation
• Pompeiu-type integral
• Schwarz-type integral

AMS Subject Classifications:

• 35F15
• 30E25
• 35C15

Acknowledgements

The author thanks the referee for helpful suggestions to improve the readability of the article. The author would be grateful to Professor Dr. H. Begehr for his valuable help especially in revising the article. The author is grateful to Deanship of Scientific Research, King Faisal University for supporting this work with grant No. 150145.

Notes

No potential conflict of interest was reported by the author.

Additional information

Funding

This work was supported by Deanship of Scientific Research, King Faisal University [grant number 150145].