On a Generalized Riemann-Hilbert Boundary Value Problem for Second Order Elliptic Systems in the Plane

Part of the International Society for Analysis, Applications and Computation book series (ISAA, volume 6)


As a consequence of the unique solvability of the modified Dirichlet problem in a multiply-connected domain it is possible to represent analytic functions in form of Cauchy type integrals with real density satisfying a Hölder condition on the boundary [9]. Such a representation is used in the present paper to investigate the problem
$$\frac{{{\partial ^2}w}}{{\partial {{\bar z}^2}}} + q\left( z \right)\frac{{{\partial ^2}w}}{{\partial z\partial \bar z}} + a\left( z \right)\frac{{\partial w}}{{\partial z}} + c\left( z \right)w = f\left( z \right)inD,$$
$${a_k}\left( t \right)\frac{{\partial u}}{{\partial {x^{2 - k}}\partial {y^{k - 1}}}} + {b_k}\left( t \right)\frac{{\partial v}}{{\partial {x^{2 - k}}\partial {y^{k - 1}}}} = {c_k}\left( t \right),k = 1,2{\text{ }}on{\text{ }}\Gamma {\text{,}}$$
where \(w = u + iv \in W_p^2\left( {\bar D} \right),2 < p{\text{ and }}\Gamma \equiv \partial D\).

The theory of two-dimensional singular integral equations [7] is applied here. In [1, 2] other Riemann-Hilbert problems for second and higher order elliptic systems in the plane are investigated.


Singular Integral Equation Unique Solvability Cauchy Type Generalize Analytic Function Cauchy Type Integral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Akal, M.: Boundary value problems for complex elliptic partial differential equations of higher order. Dissertation, Freie Universität Berlin, Berlin 1996.Google Scholar
  2. [2]
    Akal, M., Begehr, H.: On the Pompeiu operator of higher order and applications. Complex Variables, Theory Appl. 32 (1997), 233–261.MathSciNetMATHCrossRefGoogle Scholar
  3. [3]
    Begehr, H.: Complex analytic methods for partial differential equations. World Scientific, Singapore, 1994.MATHGoogle Scholar
  4. [4]
    Begehr, H., Hile, G.N.: A hierarchy of integral operators. Rocky Mountain J. Math. 27 (1997), 669–706.MathSciNetMATHGoogle Scholar
  5. [5]
    Begehr, H.: Elliptic second order equations. Functional analytic methods in complex analysis and applications to partial differential equations. Ed. W. Tutschke, A. S. Mshimba, World Sci., Singapore, 1995, 115–152.Google Scholar
  6. [6]
    Bojarski, B.: On a boundary value problem for a system of partial differential equations of first order of elliptic type. Dokl. Akad. Nauk. 102 (1955), No. 2, 201–204 (Russian).Google Scholar
  7. [7]
    Dzhuraev, A.: Methods of singular integral equations. Nauka, Moscow, 1987(Russian); English translation, Longman, Harlow, 1992.Google Scholar
  8. [8]
    Gakhov, F.D.: Boundary value problems. Fizmatgiz, Moscow 1963 (Russian); English translation, Pergamon Press, Oxford, 1966.Google Scholar
  9. [9]
    Muskhelishvili, N.I.: Singular integral equations. Fizmatgiz, Moscow, 1946 (Russian); English translation, Noordhoff, Groningen, 1953.Google Scholar
  10. [10]
    Rasulov, K.M.: On the solution of fundamental boundary value problems of Hilbert type for bianalytic functions. Soviet Math. Dokl. 44 (1992), No. 2, 440–445.MathSciNetGoogle Scholar
  11. [11]
    Vekua, I.N.: Generalized analytic functions. Fizmatgiz, Moscow, 1959 (Russian); English translation, Pergamon Press., Oxford, 1962.Google Scholar

Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

  • M. Akal
    • 1
  1. 1.Department of Mathematic Faculty of ScienceSouth Valley UniversityQenaEgypt

Personalised recommendations