POLYANALYTIC EQUATIONS IN DOMAINS FORMED BY THE INTERSECTION OF THREE CIRCLES

Year 2025, Volume: 7 Issue: 2, 63 - 72, 30.10.2025

Bahriye Karaca

https://doi.org/10.47087/mjm.1791862

Abstract

In this paper, we study the Schwarz boundary value problem for
inhomogeneous polyanalytic equations in a triangular domain in the complex
plane defined by intersections of three circles. We introduce a family of higher-order Pompeiu-type integral operators that provide explicit solutions to the
problem.

Keywords

triangular domain , schwarz problem , linear equation

References

• I. N. Vekua, Generalized analytic functions, Pergamon Press, Oxford; (1962).
• H. Begehr, Complex analytic methods for partial di erential equations, An introductory text, World Scientific, Singapore; (1994).
• H. Begehr and G.N. Hile, A hierarchy of integral operators, Rocky Mountain J. Math. 27 (1997) 669-706.
• S.H.Kang and,J.Y. Kim, Toeplitz operators on weighted analytic Bergman spaces of the halfplane, Bull.Korean Math. Soc. 37 (2000) 437-450.
• A. Darya, and N. Taghizadeh, Schwarz and Dirichlet problems for\bar{\partial}-equation in a triangular domain, Russian Mathematics. 68 (11) (2024) 9-17.
• Y. Wang, Schwarz-type boundary value problems for the polyanalytic equation in the half unit disc, Complex Variables and Elliptic Equations, 57(9) (2012) 983-993.
• B. Karaca, Boundary value problems for bi-polyanalytic functions on the upper half plane, Complex Variables and Elliptic Equations, 70(8) (2025) 1309-1320.
• B. Karaca, Extensions of the Schwarz Problem for the Beltrami operator in the half unit disc, Journal of Mathematical Sciences, 289 (2025) 683-689.
• B. Karaca, Schwarz problem for model partial di erential equations with one complex variable, Sakarya University Journal of Science, 28 (2) (2024) 410-417.
• Ü. Aksoy, H. Begehr and A. O. Çelebi, AV Bitsadze's observation on bianalytic functions and the Schwarz problem revisited, Complex Variables and Elliptic Equations, 66(4) (2021) 583-585.
• İ. Gençtürk, Robin Boundary-Value Problem for the Beltrami Equation, Ukrainian Mathematical Journal, 75(4) (2023) 510-518.
• İ.Gençtürk, Neumann Boundary Value Problem for the Beltrami Equation in a Ring Domain, Turkish Journal of Mathematics, 47(5) (2023) 1573-1584.
• İ. Gençtürk, Y. R. Linares, A Robin Boundary Value Problem for the Cauchy- Riemann Operator in a Ring Domain, Georgian Mathematical Journal, 29(2) (2022) 205-209.
• B. Karaca, Boundary Value Problems for Inhomogeneous Polyanalytic Equations in a Triangle, Vladikavkaz Mathematical Journal, 25(4) (2025), to appear.