Solutions of the dirichlet problem for elliptic systems in a disk

Abstract

We study 2×2 second-order elliptic systems, which can be written as a single equation with complex coefficients. In an arbitrary bounded region with smooth boundary, we obtain necessary and sufficient conditions on the trace relation of a solution, which we apply in the case of a disk. We prove existence and uniqueness theorems for a solution in a Sobolevskii space of an equation which is not properly elliptic. In particular, we prove that the properties of the problem determine the angle between the bicharacteristics. If it is π-rational, then there is no uniqueness, but if it is π-irrational, then the smoothness of the solution of the Dirichlet problem depends on the order of its approximation by π-rational numbers; but if it is nonreal, then the problem has the usual properties for the elliptic case.