ABOUT THE CORE STRUCTURE OF THE SCHWARTZ PROBLEM FOR FIRST-ORDER ELLIPTIC SYSTEMS ON A PLANE-Reference-Cited by-同舟云学术

ABOUT THE CORE STRUCTURE OF THE SCHWARTZ PROBLEM FOR FIRST-ORDER ELLIPTIC SYSTEMS ON A PLANE

Published:2024-12-15 Issue:5 Volume:60 Page:632-642
ISSN:0374-0641
Container-title:Дифференциальные уравнения
language:
Short-container-title:Differencialʹnye uravneniâ

Author:

Nikolaev V. G.¹

Affiliation:

1. Novgorod State University

Abstract

The Schwarz problem for 𝐽-analytic functions in an arbitrary ellipse is considered. The matrix 𝐽 is assumed to be two-dimensional, with different eigenvalues lying above the real axis. An example of a nonconstant solution of the homogeneous Schwarz problem in the form of a vector-polynomial of degree three is given. The numerical parameter 𝑙 of the matrix 𝐽, which is expressed through its eigenvectors, is introduced. After that, one relation derived earlier by the author is analyzed. Based on this analysis, a method for computing the dimension and structure of the kernel of the Schwarz problem in an arbitrary ellipse is obtained. Sufficient conditions for the triviality of the kernel expressed through the ellipse parameters, the eigenvalues of the matrix 𝐽 and the parameter 𝑙 are obtained. Examples of one-dimensional and trivial kernels are given.

Publisher

The Russian Academy of Sciences

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