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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)

Abstract

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.

Keywords

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.

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Copyright information

© Kluwer Academic Publishers 1999

Authors and Affiliations

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

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