Abstract
In this paper, we investigate a Dirichlet type problem, known as Riquier problem, for higher order linear complex differential equations in the unit polydisc of \(\mathbb {C}^2\). After deriving a Green’s function, we present the solution for a model equation with homogeneous boundary conditions. Afterwards we obtain the solution of a linear equation for Riquier boundary value problem on the unit polydisc in \(\mathbb {C}^2\).
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ü. Aksoy, A.O. Çelebi, Dirichlet problems for generalized n-Poisson equation, in Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations, ed. by B.W. Schulze, M.W. Wong. Operator Theory: Advances and Applications, vol. 205 (Birkhäuser, Basel, 2009), pp. 129–141
Ü. Aksoy, A.O. Çelebi, Neumann problem for generalized n-Poisson equation. J. Math. Anal. Appl. 357, 438–446 (2009)
Ü. Aksoy, A.O. Çelebi, Schwarz problem for higher order linear equations in a polydisc. Complex Var. Elliptic Equ. 62, 1558–1569 (2016). https://doi.org/10.1080/17476933.2016.1254627
H. Begehr, Boundary value problems in complex analysis I,II, Boletin de la Asosiación Matemática Venezolana, vol. XII (2005), pp. 65–85, 217–250
H. Begehr, Biharmonic Green functions. Le Mathematiche LXI, 395–405 (2006)
H. Begehr, A particular polyharmonic Dirichlet problem, in Complex Analysis Potential Theory, ed. by T. Aliyev Azeroglu, T. M. Tamrazov (World Scientific Publishing, Singapore, 2007), pp. 84–115
H. Begehr, Six biharmonic Dirichlet problems in complex analysis, in Function Spaces in Complex and Clifford Analysis (National University Publishers, Hanoi, 2008), pp. 243–252
H.G.W. Begehr, A. Dzhuraev, An Introduction to Several Complex Variables and Partial Differential Equations. Pitman Monographs and Surveys in Pure and Applied Mathematics (Longman, Harlow, 1997)
H. Begehr, E. Gaertner, A Dirichlet problem for the inhomogeneous polyharmonic equation in the upper half plane. Georg. Math. J. 14, 33–52 (2007)
H. Begehr, T. Vaitekhovich, Green functions in complex plane domains. Uzbek Math. J. 4, 29–34 (2008)
H. Begehr, T. Vaitekhovich, Iterated Dirichlet problem for the higher order Poisson equation. Le Matematiche LXIII, 139–154 (2008)
H. Begehr, T.N.H. Vu, Z. Zhang, Polyharmonic Dirichlet problems. Proc. Steklov Inst. Math. 255, 13–34 (2006)
H. Begehr, J. Du, Y. Wang, A Dirichlet problem for polyharmonic functions. Ann. Mat. Pure Appl. 187, 435–457 (2008)
A. Krausz, Integraldarstellungen mit Greenschen Funktionen höherer Ordnung in Gebieten und Polygebieten (Integral representations with Green functions of higher order in domains and polydomains). Ph.D. thesis, FU Berlin, 2005. http://www.diss.fu-berlin.de/diss/receive/FUDISSthesis000000001659
A. Krausz, A general higher-order integral representation formula for polydomains. J. Appl. Funct. Anal. 2(3), 197–222 (2007)
A. Kumar, A generalized Riemann boundary problem in two variables. Arch. Math. (Basel) 62, 531–538 (1994)
S.S. Kutateladze, Fundamentals of Functional Analysis (Kluwer Academic Publishers, Dordrecht-Boston-London, 1996)
A. Mohammed, The Riemann-Hilbert problem for certain poly domains and its connection to the Riemann problem. J. Math. Anal. Appl. 343, 706–723 (2008)
A. Mohammed, R. Schwarz, , Riemann-Hilbert problems and their connections in polydomains, in Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations. Operator Theory: Advances and Applications, vol. 205 (Birkhäuser, Basel, 2010), pp. 143–166
M. Nicolescu, Les functions polyharmonique (Hermann, Paris, 1936)
T. Vaitsiakhovich, Boundary value problems for complex partial differential equations in a ring domain, Ph. D. thesis, F.U. Berlin, 2008
I.N. Vekua, Generalized Analytic Functions (Pergamon Press, Oxford, 1962)
Acknowledgements
The author is grateful to the anonymous referees for their careful reading which improved the article, and also to Professor Umit Aksoy for her valuable comments and supports.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2018 Springer International Publishing AG, part of Springer Nature
About this paper
Cite this paper
Çelebi, A.O. (2018). Dirichlet Type Problems in Polydomains. In: Drygaś, P., Rogosin, S. (eds) Modern Problems in Applied Analysis. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-72640-3_5
Download citation
DOI: https://doi.org/10.1007/978-3-319-72640-3_5
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-319-72639-7
Online ISBN: 978-3-319-72640-3
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)