On solutions of the Dirichlet problem for elliptic systems on a circle - Bursky, V. P. (Ukrains’kyi Matematychnyi Zhurnal, 1992)
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.