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

V. G. Nikolaev

Vestnik Ûžno-Uralʹskogo gosudarstvennogo universiteta. Seriâ, Matematika, mehanika, fizika, Tome 9 (2017) no. 2, pp. 30-35

Résumé

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 $\lambda$, $\mu$, 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 $\psi$ 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 $\lambda$- and $\mu$-holomorphic functions $f$, $g$, defined in the plane region $D$, with a certain boundary function $\varphi$, which is constructed by $\psi$. 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 $\varphi$. 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 $\psi$ 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 ($\varphi = 0$) functional equation has a nontrivial solution.

Détail

Mots-clés : matrix, Lyapunov contour.
Keywords: Jordan canonical form, eigenvalue, eigenvector, holomorphic function