Schwarz boundary value problem on Reuleaux triangle

Abstract

We discuss the Schwarz problem on Reuleaux triangle. By the technique of parqueting reflection principle, we translate the Schwarz problem into a Riemann boundary value problem on a system of arcs. We first state that the Riemann problem is solvable, while the solutions are represented by a Cauchy type integral. At last, as the main theorem, we prove that those two problems are equivalent to each other. And then we obtain the explicit integral representation of the solution to a Schwarz problem.

Keywords:

• Reuleaux triangle
• boundary value problem of analytic functions
• Schwarz problem

AMS Subject Classification:

• 30E25

Acknowledgement

Supported by the National Natural Science Foundation of China (Grant Nos. 11802208,11601390).

Disclosure statement

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