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

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.