On One Method for Constructing Solutions to the Homogeneous Schwarz Problem

Abstract

The paper considers the homogeneous Schwarz problem for Douglis analytic or J-analytic functions. The 2-2-matrix J has eigenvalues λ,μ, lying above the real axis. The eigenvalues can be either distinct or multiples. In the second section of the paper there are given the problem statement and definitions of J-analytic and λ-holomorphic functions. At the beginning of the third section Lemma 3.1 is proved, establishing some relation between real and holomorphic functions. Then is constructed a special operator basis J. For matrices with multiple eigenvalue this basis coincides with the Jordan basis of the matrix J. Then with the help of this basis and Lemma 3.1 is constructed J-analytic function φ(z) in the form of a quadratic vector polynomial of some special form. If the eigenvalues λ,μ of the matrix J are fixed, then the function φ(z) depends on the elements of the first column of the matrix J as parameters. These parameters are chosen so that the real part of the function φ(z) has the form (P; 0), where P = P(x, y) is a positively defined quadratic form. All J-analytic functions are defined with the accuracy of the additive vector constant Therefore, the function φ(z) – (1, 0) will be the required solution to the homogeneous Schwarz problem in the ellipse Γ : P(x, y) = 1. Then the matrix J is reconstructed by the known elements of the first column and the eigenvalues λ,μ. The obtained result is formalized in the form of Theorem 3.1. At the end of the paper there are given six examples constructed according to the algorithm described above.