Abstract
This paper presents results on some boundary value problems for holomorphic functions of several complex variables in polydomains. The Cauchy kernel is one of the significant tools for solving the Riemann and the Riemann-Hilbert boundary value problems for holomorphic functions as well as for establishment of the connection between them. For polydomains, the Cauchy kernel is modified in such a way that it corresponds to a certain symmetry of the boundary values of holomorphic functions in polydomains. This symmetry is lost if the classical counterpart of the one-dimensional form of the Cauchy kernel is applied. The general integral representation formulas for the functions, holomorphic in polydomains, the solvability conditions and the solutions of the corresponding Schwarz problems are given explicitly. A necessary and sufficient condition for the boundary values of a holomorphic function for arbitrary polydomains is given and an exact, yet compact way of notation for holomorphic functions in arbitrary polydomains is introduced and applied. The Riemann jump problem and the Riemann-Hilbert problem are solved for holomorphic functions of several complex variables with the unit torus as the jump manifold. The higher-dimensional Plemelj-Sokhotzki formula for holomorphic functions in polydomains is established. The canonical functions of the Riemann problem for polydomains are represented and applied in order to construct solutions for both of the homogeneous and inhomogeneous problems. For all three boundary value problems, well-posed formulations are given which does not demand more solvability conditions than in the one variable case. The connection between the Riemann and the Riemann-Hilbert problem for polydomains is proven. Thus contrary to earlier research the results are similar to the respective ones for just one variable.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
H. Begehr Riemann-Hilbert boundary value problems in ℂn, in Complex Methods for Partial Differential and Integral Equations, Editors: H. Begehr A. Çelebi and W. Tutschke, Kluwer Academic Publishers, 1999, 59–84.
H. Begehr and D. Q. Dai, Spatial Riemann problem for analytic functions of two complex variables, J. Anal. Appl. 18 (1999), 827–837.
H. Begehr and A. Dzhuraev, An introduction to Several Complex Variables and Partial Differential Equations, Pitman Monographs and Surveys in Pure and Applied Mathematics 88, Addison Wesley, 1997.
H. Begehr and A. Dzhuraev, Schwarz problem for Cauchy Riemann system in several complex variables, in Analysis and Topology Editors: C. Andreian Cazacu, et al., World Scientific, 1998, 63–114.
H. Begehr, Complex Analytic Methods for Partial Differential Equations, World Scientific, 1994.
H. Begehr and A. Mohammed, The Schwarz problem for analysis functions in torus related domains, Appl. Anal. 85 (2006), 1079–1101.
H. Begehr and G. C. Wen, Nonlinear Elliptic Boundary Value Problems and Their Applications, Longman, 1996.
J. W. Cohen and O. J. Boxma, Boundary Value Problems in Queueing System Analysis, North-Holland, 1983.
D. Q. Dai, Fourier method for an over-determined elliptic system with several complex variables, Acta Math. Sinica 22 (2006), 87–94.
P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, American Mathematical Society, 2000.
F. D. Gakhov, Boundary Value Problems, Fizmatgiz, Moscow, 1963 (Russian); English Translation, Pergamon Press, 1966.
V. A. Kakichev, Boundary value problems of linear conjugation for functions holomorphic in bicylinderical regions, Soviet Math. Dokl. 9 (1968), 222–226.
V. A. Kakichev, Analysis of conditions of solvability for a class of spatial Riemann problems, Ukrainian J. Math. 31 (1979), 205–210.
V. A. Kakichev, Application of the Fourier Method to the Solution of Boundary Value Problems for Functions Analytic in Disc Bidomains Amer. Math. Soc. Transl. (2) 146 1990.
S. G. Krantz, Complex Analysis: the Geometric Viewpoint, The Carus Mathematical Monographs, Mathematical Association of America, 1990.
A. Kufner and J. Kadlec, Fourier Series, London, ILIFFE Books, Academia, Prague, 1971.
A. Kumar, A generalized Riemann boundary problem in two variables, Arch. Math. 62 (1994), 531–538.
X. Li, An application of the periodic Riemann boundary value problem to a periodic crack problem, in Complex Methods for Partial Differential and Integral Equations, Editors: H. Begehr, A. Çelebi and W. Tutschke, Kluwer Academic Publishers, 1999, 103–112.
C.-K. Lu, Boundary Value Problems for Analytic Functions, World Scientific, 1993.
C.-K. Lu, Periodic Riemann boundary value problems and their applications in elasticity, Chinese Math 4 (1964), 372–422.
V. G. Maz’ya and S. M. Nikol’skii, Analysis IV, Encyclopedia of Mathematical Sciences 27, Springer-Verlag, 1991.
A. Mohammed, The torus related Riemann problem, J. Math. Anal. Appl. 326 (2007), 533–555.
A. Mohammed, The Riemann-Hilbert problem for polydomains and its connection to the Riemann problem, J. Math. Anal. Appl. 343 (2008), 706–723.
A. Mohammed, Boundary Values of Complex Variables, Ph. D. Thesis, Freie Universität Berlin, 2002.
A. Mohammed, The Neumann problem for the inhomogeneous pluriharmonic system in polydiscs, in Complex Methods for Partial Differential and Integral Equations, Editors: H. Begehr, A. Çelebi and W. Tutschke, Kluwer Academic Publishers, 1999, 155–164.
A. Mohammed and M. W. Wong, Solutions of the Riemann-Hilbert-Poincaré and Robin problems for the inhomogeneous Cauchy-Riemann equation, Proc. Royal Soc. Edinburgh Sect. A 139 (2009), 157–181.
N. I. Muskhelishvili, Singular Integral Equations, Fizmatgiz, Moscow, 1946 (Russian); English Translation, Noordhoof, Groningen, 1953.
V. S. Vladimirov, Problems of linear conjugacy of holomorphic functions of several complex variables, Trans. Amer. Math. Soc. 72 (1969), 203–232.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Birkhäuser Verlag Basel/Switzerland
About this paper
Cite this paper
Mohammed, A. (2009). Schwarz, Riemann, Riemann-Hilbert Problems and Their Connections in Polydomains. In: Schulze, BW., Wong, M.W. (eds) Pseudo-Differential Operators: Complex Analysis and Partial Differential Equations. Operator Theory: Advances and Applications, vol 205. Birkhäuser Basel. https://doi.org/10.1007/978-3-0346-0198-6_8
Download citation
DOI: https://doi.org/10.1007/978-3-0346-0198-6_8
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-0346-0197-9
Online ISBN: 978-3-0346-0198-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)