One form of the scalar two-dimensional Schwarz problem and its applications

The paper deals with the problem of existence and uniqueness of the Schwarz problem solution for 2-vector-functions, being analytic on Douglis, in regions bounded by the Lyapunov contour, and in classes of functions that are Holder continuous. However, the matrix J should have different eigenvalues λ, μ, and at least one eigenvector that is not multiple of the real one. At the beginning of the paper, the inhomogeneous Schwarz problem with a boundary function ψ is transformed. As a result of the performed reduction the Schwarz problem turns into an equivalent boundary problem for an inhomogeneous scalar functional equation. It connects boundary values of λ- and μ-holomorphic functions f, g, defined in the plane region D, with a certain boundary function φ, which is constructed by ψ. This functional equation for different matrices J is distinguished only by a complex coefficient 1, which is calculated using the matrix J. In this case the following circular property is found: the Schwarz problem is solvable or not simultaneously for all matrices, which coefficient module is equal. That’s why without loss of generality 1 can be considered a real number. It’s proved that the studied functional equation for cases l = 0 and |l| = 1 has a unique solution for any right side of φ. The matrices J having complex conjugate eigenvectors and one real eigenvector correspond to these two cases. Therefore, for these matrices the inhomogeneous Schwarz problem in case of any boundary function ψ has the unique solution. We consider absolutely and irrespectively the case when the matrix J has complex conjugate eigenvalues. At the end of the paper it’s shown that in case of |l| = 5 the homogeneous (φ = 0) functional equation has a nontrivial solution.