Dirihlet problem for the generalized Beltrami equation

Pelin Ayşe Gökgöz

doi:10.26650/ijmath.2025.00022

Dirihlet problem for the generalized Beltrami equation

Abstract

In this article, we investigate the Dirichlet problem for the generalized Beltrami equation. Firstly, we introduce the solutions of the Dirichlet problem for the inhomogeneous Cauchy-Riemann equation in the unit disc. Secondly, we state the properties of the integral operators for regular domains. Then, by using Banach fixed point theorem, we obtain the existence of the unique solution of the Dirichlet problem for the generalized Beltrami equation in the unit disc.

Keywords

References

Aksoy, Ü., Çelebi, A.O., 2012, Dirichlet problem for a generalized inhomogeneous polyharmonic equation in an annular domain, Complex Variables and Elliptic Equations, 57 2-4, 229-241. google scholar
Aksoy, Ü., Çelebi, A.O., 2010, Dirichlet problems for generalized /¡-Poisson equation, Oper. Theory Adv. Appl. 205, 129-142. google scholar
Begehr, H., 2005a, Boundary value problems in complex analysis I, Boletin de la Asociacion Matematica Venezolana, 12-1, 65-85. google scholar
Begehr, H., 2005b, Boundary value problems in complex analysis II. Boletin de la Asociacion Matematica Venezolana, 12-2, 217-250. google scholar
Begehr, H., Shupeyeva, B., 2021, Polyanalytic boundary value problems for planar domains with harmonic Green function, Analysis and Mathematical Physics, 11, 137. google scholar
Begehr, H.G.W., 1994, Complex Analytic Methods for Partial Differential Equations: An introductory text, World Scientific, Singapore. google scholar
Begehr, H., Vaitekhovich, T., 2012, Harmonic Dirichlet problem for some equilateral triangle, Complex Variables and Elliptic Equations, 57, 185-196. google scholar
Begehr, H., Gaertner, E., 2007, Dirichlet problem for the inhomogeneous polyharmonic equation in the upper half plane, Georgian Math J. 14, 33-52. google scholar

Begehr, H., Hile, G.N., 1997, A hierarchy of integral operators, Rocky Mountain J. Math. 27, 669-706. google scholar
Begehr, H., Harutyunyan, G., 2009, Neumann problem for the Beltrami operator and for second order operators with Poisson/Bitsadze operator as main part, Complex Variables and Elliptic Equations 54-12, 1129-1150. google scholar
Begehr, H., Obolashvili, E., 1994, Some boundary value problems for a Beltrami equation. Complex Variables and Elliptic Equations, 26, 1-2, 113-122. google scholar
Begehr, H., Vaitekhovich, T., 2007, Complex partial differential equations in a manner of I. N. Vekua. Lecture Notes of TICMI, 8, 15–26. google scholar
Gökgöz, P.A., 2024a, Dirichlet boundary value problem for linear polyanalytic equation in upper half plane, Complex Variables and Elliptic Equations, 70-4, 549–554. google scholar
Gökgöz, P.A., 2024b, Dirichlet Problem for Nonlinear Higher-Order Equations in Upper Half Plane, Complex Analysis and Operator Theory 18, 121. google scholar
Harutyunyan, G., 2007, Boundary value problems for the Beltrami operator, Complex Variables and Elliptic Equations, 52-6, 475-484. google scholar
Tutschke, W., 1983, Partielle Differentialgleichungen - Klassiche, Funktionalanalytische und Komplexe Methoden, Teubner Texte zur Mathematics, Band 27, B.G. Teubner, Leipzig. google scholar
Vaitsiakhovich, T., 2008a, Boundary Value Problems for Complex Partial Differential Equations in a Ring Domain [Ph.D. thesis], Freie Universitat Berlin. google scholar
Vaitekhovich, T.S., 2008b, Boundary value problems to first-order complex partial differential equations, Integral Transforms and Special Functions, 19, 211-233. google scholar
Vaitekhovich, T., 2007, Boundary value problems to second order complex partial differential equations in a ring domain, Siauliai Mathematical Seminar, 2-10, 117-146. google scholar
Vekua, I.N., 1962, Generalized Analytic Functions. Pergamon Press, Oxford. google scholar
Yüksel, U., 2010, A Schwarz problem for the generalized Beltrami equation, Complex Variables and Elliptic Equations, 56-6, 503–511. google scholar
Wang, Y., Du, J., 2015, Harmonic Dirichlet problem in a ring sector. In: Current trends in analysis and its applications: Trends Math. Cham, Switzerland 67-75. google scholar

Details

Primary Language

English

Subjects

Pure Mathematics (Other)

Journal Section

Research Article

Authors

Pelin Ayşe Gökgöz ^*
0000-0003-4381-1535
Türkiye

Publication Date

June 24, 2025

Submission Date

February 18, 2025

Acceptance Date

June 12, 2025

Published in Issue

Year 2025 Volume: 3 Number: 1

DOI

https://doi.org/10.26650/ijmath.2025.00022

IZ

https://izlik.org/JA55JT68ZK

Cite

RIS / Bibtex

APA

Gökgöz, P. A. (2025). Dirihlet problem for the generalized Beltrami equation. Istanbul Journal of Mathematics, 3(1), 12-19. https://doi.org/10.26650/ijmath.2025.00022

AMA

1.Gökgöz PA. Dirihlet problem for the generalized Beltrami equation. Istanbul Journal of Mathematics. 2025;3(1):12-19. doi:10.26650/ijmath.2025.00022

Chicago

Gökgöz, Pelin Ayşe. 2025. “Dirihlet Problem for the Generalized Beltrami Equation”. Istanbul Journal of Mathematics 3 (1): 12-19. https://doi.org/10.26650/ijmath.2025.00022.

EndNote

Gökgöz PA (June 1, 2025) Dirihlet problem for the generalized Beltrami equation. Istanbul Journal of Mathematics 3 1 12–19.

IEEE

[1]P. A. Gökgöz, “Dirihlet problem for the generalized Beltrami equation”, Istanbul Journal of Mathematics, vol. 3, no. 1, pp. 12–19, June 2025, doi: 10.26650/ijmath.2025.00022.

ISNAD

Gökgöz, Pelin Ayşe. “Dirihlet Problem for the Generalized Beltrami Equation”. Istanbul Journal of Mathematics 3/1 (June 1, 2025): 12-19. https://doi.org/10.26650/ijmath.2025.00022.

JAMA

1.Gökgöz PA. Dirihlet problem for the generalized Beltrami equation. Istanbul Journal of Mathematics. 2025;3:12–19.

MLA

Gökgöz, Pelin Ayşe. “Dirihlet Problem for the Generalized Beltrami Equation”. Istanbul Journal of Mathematics, vol. 3, no. 1, June 2025, pp. 12-19, doi:10.26650/ijmath.2025.00022.

Vancouver

1.Pelin Ayşe Gökgöz. Dirihlet problem for the generalized Beltrami equation. Istanbul Journal of Mathematics. 2025 Jun. 1;3(1):12-9. doi:10.26650/ijmath.2025.00022