On Riemann boundary value problems for certain linear elliptic systems in the plane

Abstract

The partial differential equations which occur in the theory of elastic plates and shells are among those which may be reduced to a first order elliptic system of the type studied by Douglis [3]. Under certain regularity conditions for the coefficients, a Beltrami transformation exists takmg the general first-order, elliptic systems into a normal-Douglis-form. This form can be further simplified, and more concisely represented by utilizing the algebra CY of hypercomplex numbers. The theory of solutions to these linear systems is known as generakzed hyperunai’yticfimction theory (see Gilbert [S, 6, 71 and Gilbert and HiIe [g, 9])* and bears the same relationship to the Douglis theory of hyperanalytic functions as Vekua’s theory to analytic functions. In [I] Hilbert boundary value problems for generalized hyperanalytic functions were studied. Subsequently semilinear Douglis systems were treated by Wendland [13] and Gilbert [7j. Th e p resent work deals with Riemann boundary value problems for linear systems. It is clear that our results may be extended to nonlinear hypercomplex systems resembling the complex cases investigated by Warowna-Dorau [14] and Wolska-Bochenek 1151. A good survey of the methods encountered in the analytic case may be found in the monographs of Gakhov [4J and Muskhelishvili [lo]. The fundamental kernels for the linear system (see [S]) permit the formulation of the Riemann boundary value problem for generalized hyperanalytic functions as a Cauchy type integral relation. In general, there is no similarity principle