On the Structure of the Kernel of the Schwarz Problem for First-Order Elliptic Systems on the Plane

Abstract

The Schwarz problem for \(J\)-analytic functions in an arbitrary ellipse is considered. The matrix \(J\) is assumed to be two-dimensional with distinct eigenvalues lying above the real axis. An example of a nonconstant solution of the homogeneous Schwarz problem in the form of a vector polynomial of degree three is given. A numerical parameter \(l\) of the matrix \(J \), expressed via its eigenvectors, is introduced. After that, one relation derived earlier by the present author is analyzed. Based on this analysis, a method for computing the dimension and structure of the kernel of the Schwarz problem in an arbitrary ellipse is obtained. Sufficient conditions for the triviality of the kernel expressed via the ellipse parameters, the eigenvalues of the matrix \(J\), and the parameter \(l \) are obtained. Examples of one-dimensional and trivial kernels are given.