Schwarz, Riemann, Riemann-Hilbert Problems and Their Connections in Polydomains

  • Alip Mohammed
Part of the Operator Theory: Advances and Applications book series (OT, volume 205)


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.


Several complex variables Schwarz problem Riemann problem Riemann-Hilbert problem polydomain holomorphic functions 

Mathematics Subject Classification (2000)

Primary 32A25 Secondary 32A26 32A30 32A07 32A40 30E25 35N05 35Q15 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    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.Google Scholar
  2. [2]
    H. Begehr and D. Q. Dai, Spatial Riemann problem for analytic functions of two complex variables, J. Anal. Appl. 18 (1999), 827–837.zbMATHMathSciNetGoogle Scholar
  3. [3]
    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.Google Scholar
  4. [4]
    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.Google Scholar
  5. [5]
    H. Begehr, Complex Analytic Methods for Partial Differential Equations, World Scientific, 1994.Google Scholar
  6. [6]
    H. Begehr and A. Mohammed, The Schwarz problem for analysis functions in torus related domains, Appl. Anal. 85 (2006), 1079–1101.zbMATHCrossRefMathSciNetGoogle Scholar
  7. [7]
    H. Begehr and G. C. Wen, Nonlinear Elliptic Boundary Value Problems and Their Applications, Longman, 1996.Google Scholar
  8. [8]
    J. W. Cohen and O. J. Boxma, Boundary Value Problems in Queueing System Analysis, North-Holland, 1983.Google Scholar
  9. [9]
    D. Q. Dai, Fourier method for an over-determined elliptic system with several complex variables, Acta Math. Sinica 22 (2006), 87–94.zbMATHCrossRefGoogle Scholar
  10. [10]
    P. Deift, Orthogonal Polynomials and Random Matrices: A Riemann-Hilbert Approach, American Mathematical Society, 2000.Google Scholar
  11. [11]
    F. D. Gakhov, Boundary Value Problems, Fizmatgiz, Moscow, 1963 (Russian); English Translation, Pergamon Press, 1966.Google Scholar
  12. [12]
    V. A. Kakichev, Boundary value problems of linear conjugation for functions holomorphic in bicylinderical regions, Soviet Math. Dokl. 9 (1968), 222–226.zbMATHGoogle Scholar
  13. [13]
    V. A. Kakichev, Analysis of conditions of solvability for a class of spatial Riemann problems, Ukrainian J. Math. 31 (1979), 205–210.zbMATHCrossRefMathSciNetGoogle Scholar
  14. [14]
    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.Google Scholar
  15. [15]
    S. G. Krantz, Complex Analysis: the Geometric Viewpoint, The Carus Mathematical Monographs, Mathematical Association of America, 1990.Google Scholar
  16. [16]
    A. Kufner and J. Kadlec, Fourier Series, London, ILIFFE Books, Academia, Prague, 1971.zbMATHGoogle Scholar
  17. [17]
    A. Kumar, A generalized Riemann boundary problem in two variables, Arch. Math. 62 (1994), 531–538.zbMATHCrossRefGoogle Scholar
  18. [18]
    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.Google Scholar
  19. [19]
    C.-K. Lu, Boundary Value Problems for Analytic Functions, World Scientific, 1993.Google Scholar
  20. [20]
    C.-K. Lu, Periodic Riemann boundary value problems and their applications in elasticity, Chinese Math 4 (1964), 372–422.Google Scholar
  21. [21]
    V. G. Maz’ya and S. M. Nikol’skii, Analysis IV, Encyclopedia of Mathematical Sciences 27, Springer-Verlag, 1991.Google Scholar
  22. [22]
    A. Mohammed, The torus related Riemann problem, J. Math. Anal. Appl. 326 (2007), 533–555.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [23]
    A. Mohammed, The Riemann-Hilbert problem for polydomains and its connection to the Riemann problem, J. Math. Anal. Appl. 343 (2008), 706–723.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [24]
    A. Mohammed, Boundary Values of Complex Variables, Ph. D. Thesis, Freie Universität Berlin, 2002.Google Scholar
  25. [25]
    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.Google Scholar
  26. [26]
    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.zbMATHCrossRefMathSciNetGoogle Scholar
  27. [27]
    N. I. Muskhelishvili, Singular Integral Equations, Fizmatgiz, Moscow, 1946 (Russian); English Translation, Noordhoof, Groningen, 1953.Google Scholar
  28. [28]
    V. S. Vladimirov, Problems of linear conjugacy of holomorphic functions of several complex variables, Trans. Amer. Math. Soc. 72 (1969), 203–232.Google Scholar

Copyright information

© Birkhäuser Verlag Basel/Switzerland 2009

Authors and Affiliations

  • Alip Mohammed
    • 1
  1. 1.Department of Mathematics and StatisticsYork UniversityTorontoCanada

Personalised recommendations