Schwarz Problem in Ellipse for Nondiagonalizable Matrices
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Journal of Mathematical Sciences, Vol. 251, No. 6, December, 2020
SCHWARZ PROBLEM IN ELLIPSE FOR NONDIAGONALIZABLE MATRICES V. G. Nikolaev Novgorod State University 41, Bol’shaya St.-Peterburgskaya ul., Velikiy Novgorod 173003, Russia [email protected]
UDC 517.952
We study the Schwarz problem for J-analytic vector-valued functions in an ellipse with a square matrix J admitting a nondiagonal Jordan form. We obtain conditions on the ellipse and matrix J necessary and sufficient for the existence and uniqueness of a solution to the Schwarz problem with an arbitrary boundary function of H¨ older class. Under certain conditions on the matrix J, we show that the homogeneous Schwarz problem in an ellipse has a solution in the form of a vector polynomial of an arbitrary degree. Bibliography: 7 titles. Illustrations: 1 figure. We continue the study [1, 2] of the Schwarz problem for J-analytic functions. In this paper, it is assumed that the 2 × 2-matrix J has nondiagonal Jordan form and the boundary function is H¨older. We study the solvability of the Schwarz problem in H¨ older classes. A simpler case of diagonalizable square matrices J of order n was studied in [1] in detail. In [1], the original problem is divided into simpler auxiliary problems and it is proved that the solutions of the auxiliary problems converge to the solution of the Schwarz problem with an arbitrary boundary function. However, the methods of [1] are not appropriate for matrices with nondiagonal Jordan forms, whereas their modifications lead to complicated and cumbersome calculations. Therefore, we apply another approach in this paper: we use the results of [3] to prove immediately the existence of a solution to the Schwarz problem with an arbitrary boundary function. Nondiagonalizable matrices of order n were studied in [2], where the domain of J-analytic functions was assumed to be bounded by a Lyapunov contour and rather strong conditions were imposed on the Jordan basis of the matrix J. In particular, the existence of a real eigenvector of the matrix J was required there. In the present paper, the domain of J-analytic functions is an ellipse, but no restrictions on the structure of the Jordan basis of the matrix J are required.
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Preliminaries. Statement of the Problem
We assume that all eigenvalues of the matrix J ∈ C× lie in the upper half-plane. We recall the definition of a Douglis analytic vector-valued function (cf. [2]–[6]).
Translated from Problemy Matematicheskogo Analiza 107, 2020, pp. 91-112. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2516-0876
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Definition 1.1. An -vector-valued function ω = ω(z) ∈ C 1 (D) is said to be Douglis analytic or J-analytic with a matrix J in a domain D ⊂ R2 if ∂ω ∂ω −J · = 0, ∂y ∂x
z ∈ D.
(1.1)
We also use the one-dimensional analogue of (1.1) (cf. also [2, 5, 7]). Definition 1.2. Let λ ∈ C, Im λ = 0. A scalar function fλ = fλ (z) ∈ C 1 (D) satisfying the equation ∂fλ ∂fλ −λ· = 0, z ∈ D, (1.2) ∂y ∂x is said to be λ-holomorphic in the domain D ⊂ R2 . Thus, the functions fλ (z) gener
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