On the Schwarz Problem in the Case of Matrices with Nondiagonal Jordan Forms
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
ON THE SCHWARZ PROBLEM IN THE CASE OF MATRICES WITH NONDIAGONAL JORDAN FORMS 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 functions in the case where the Jordan basis Q of the matrix J has real vectors and the Jordan form J1 of J contains Jordan (2 × 2)-cells, whereas the order of real columns of the matrix Q depends on the structure of J1 . We prove the existence and uniqueness of a solution to the Schwarz problem in H¨ older classes. Bibliography: 4 titles. In this paper, we study the Schwarz problem for Douglis analytic (or J-analytic) functions defined by a matrix J having no real eigenvalues. The problem is to find a vector-valued function ω(z) provided that its real part ψ(t) is known on the boundary Γ of a plane domain D. As a rule, it is assumed that Γ is a Lyapunov contour. For general properties of J-analytic functions and the method for reducing the Dirichlet problem for second order elliptic systems to an equivalent Schwarz problem we refer to [1]. The necessary and sufficient solvability condition for the classical Schwarz problem in the H¨ older classes is obtained in [2]. This condition means the orthogonality of the boundary function ψ(t) to all solutions to the homogeneous conjugate Schwarz problem. For the solvability of the Schwarz problem in the H¨older classes it is sufficient that the kernel of the homogeneous conjugate Schwarz problem to be trivial. Sufficient conditions on the matrices Q and J1 guaranteeing the existence and uniqueness of a solution to the Schwarz problem in H¨older classes can be found in [3]. We note that the matrix J is assumed in [3] to be diagonalizable and its Jordan basis Q consists of pairwise complex conjugates. In this paper, we consider the matrices Q and J1 of other structure. The matrix J1 contains Jordan (2 × 2)-cells, whereas the vectors of the Jordan basis Q are divided into two groups. The first group consists of eigenvectors corresponding to a fixed eigenvalue, and the vectors of the second group satisfy a certain condition. The matrix Q is assumed to be real.
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Statement of the Problem
Let all eigenvalues of a matrix J ∈ Cn×n have nonzero complex parts, and let D ⊂ R2 . We denote by ω = ω(z) ∈ C 1 (D) a complex-valued n-vector function and consider the homogeneous Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 75-83. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0083
83
linear system of first order partial differential equations ∂ω ∂ω −J · = 0, ∂y ∂x
z ∈ D.
(1.1)
It is shown in [1] that the system (1.1) is elliptic. Definition 1.1 ([1]–[3]). A complex-valued n-vector function ω = ω(z) ∈ C 1 (D) is Douglis analytic or J-analytic with the matrix J if it is a solution to Equation (1.1) in D ⊂ R2 . The simplest example of J-analytic functions is provided by an n-vector polynomial of the form ω(z) = (xE + yJ)k · c, k = 1, 2, 3 .
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