Three boundary value problems of the Cauchy–Riemann equation in the Eclipse domain

Abstract

In this article, by the parqueting-reflection technique, we obtain a covering of the complex plane from reflecting the Eclipse domain. Then, by the Cauchy–Pompeiu formula, we derive the generalized Cauchy–Pompeiu representation formula in the Eclipse domain. In addition, we consider the Schwarz boundary value problem (BVP) for the nonhomogeneous Cauchy–Riemann equation and the explicit expressions of solution and the condition of solvability. Besides, by using the Schwarz BVP, we investigate the inhomogeneous Dirichlet and homogeneous Neumann BVPs.

Keywords:

• Complex plane
• parqueting-reflection technique
• boundary value problems
• Cauchy–Riemann equation
• Cauchy–Pompeiu formula
• Poisson kernel
• Schwarz
• Dirichlet
• Neumann
• Eclipse

AMS Subject Classifications:

• 35F15
• 30E258
• 35F30
• 35C15

Acknowledgments

The authors would like to express their sincere gratitude to the Associate Editor and two referees for their useful comments, remarks and suggestions to improve the quality of the paper.

Disclosure statement

No potential conflict of interest was reported by the author(s).