Abstract
The paper provides explicit eigenvalues and eigenfunctions of pluriharmonic functions for the third boundary condition on the unit polydiscs. It is shown that in the case of eigenvalue, for each eigenvalue, there are multiple eigenfunctions. Compatibility and solvability conditions are also studied for the case of inhomogeneous third boundary condition.
To the 80 th birthday of Heinrich Begehr
This work was completed in spring 2018 at Khalifa University.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
M. Akel, H. Begehr, Neumann function for a hyperbolic strip and a class of related plane domains. Math. Nachr. 290(4), 490–506 (2017)
M. Akel, S.R. Mondal, Dirichlet problems in lens and lune. Bull. Malays. Math. Sci. Soc. 41(2), 1029–1043 (2017)
G. Auchmuty, Steklov representations of Green’s functions for Laplacian boundary value problems. Appl. Math. Optim. https://doi.org/10.1007/s00245-016-9370-4. Article electronically published on July 25, 2016
H. Begehr, A. Dzhuraev, An Introduction to Several Complex Variables and Partial Differential Equations. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 88 (Addison Wesley Longman, Harlow, 1997)
H. Begehr, A. Mohammed, Schwarz problem for the Torus related domains. Appl. Anal. 85(9), 1079–1101 (2006)
H. Begehr, S. Burgumbayeva, B. Shupeyeva, Remark on Robin problem for Poisson equation. Complex Variables Elliptic Equ. https://doi.org/10.1080/17476933.2017.1303052. Published online: 19 Mar 2017
C. Bandle, A. Wagner, Domain perturbations for elliptic problems with Robin boundary conditions of opposite sign. St. Petersburg Math. J. 28(2), 153–170 (2017). http://dx.doi.org/10.1090/spmj/1443. Article electronically published on February 15, 2017.
S.G. Krantz, Complex Analysis: The Geometric Viewpoint (Carus Mathematical Monographs) (2004). ISBN-13:978-0883850268
A. Mohammed, The Neumann problem for the inhomogeneous pluriharmonic system in polydiscs, in Complex Methods for Partial Differential and Integral Equations, ed. by H. Begehr, A. Celebi, W. Tutschke (Kluwer Academic Publishers, Dordrecht, 1999), pp. 155–164
A. Mohammed, The classical and the modified Neumann problem for the inhomogeneous pluriholomorphic system in polydiscs. J. Anal. Appl. 19, 539–552 (2000)
A. Mohammed, The classical and the modified Dirichlet problem for the inhomogeneous pluriholomorphic system in polydiscs . Complex Variables Theory Appl. 45, 213–246 (2001)
A. Mohammed, Boundary value problems of complex variables. PhD Thesis, Freie Universität Berlin, 2003. http://www.diss.fu-berlin.de/diss/receive/FUDISS_thesis_000000000885
A. Mohammed, The torus related Riemann problem. J. Math. Anal. Appl. 326(1), 533–555 (2007)
A. Mohammed, The Riemann–Hilbert problem for certain poly domains and its connection to the Riemann problem. J. Math. Anal. Appl. 343(2), 706–723 (2008)
A. Mohammed, Schwarz, Riemann, Riemann-Hilbert problems and their connections in polydomains, in Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations (Birkhäuser, Basel, 2009), pp.143–166
A. Mohammed, A.M. Tuffaha, On boundary control of the Poisson equation with the third boundary condition. J. Math. Anal. Appl. 459(1), https://doi.org/10.1016/j.jmaa.2017.10.059. Published online October 2017
A. Mohammed, M.W. Wong, Solutions of the Riemann–Hilbert–Poincaré problem and the Robin problem for the inhomogeneous Cauch - Riemann equation. Proc. R. Soc. Edinb. Sect. A Math. 139(1), 157–181 (2009)
A. Mohammed, M.W. Wong, Solutions of Robin problems for overdetermined inhomogeneous Cauchy–Riemann systems on the unit polydisc. Complex Anal. Oper. Theory 4(1), 39–53 (2010)
A. Mohammed, M.W. Wong, Eigenvalues of the third boundary condition with variable coefficient for Poisson equation (to be published)
A. Mohammed, D. Siginer, F. Akyildiz, Eigenvalues of holomorphic functions for the third boundary condition. J. Quart. Appl. Math. 73(3), 553–574 (2015)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this chapter
Cite this chapter
Mohammed, A. (2019). Boundary Eigenvalues of Pluriharmonic Functions for the Third Boundary Condition on the Unit Polydiscs. In: Rogosin, S., Çelebi, A. (eds) Analysis as a Life. Trends in Mathematics. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-02650-9_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-02650-9_3
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-02649-3
Online ISBN: 978-3-030-02650-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)