{"title":"On the Structure of the Kernel of the Schwarz Problem for First-Order Elliptic Systems on the Plane","authors":"V. G. Nikolaev","doi":"10.1134/s0012266124050057","DOIUrl":null,"url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> The Schwarz problem for <span>\\(J\\)</span>-analytic functions in\nan arbitrary ellipse is considered. The matrix <span>\\(J\\)</span> is assumed to be\ntwo-dimensional with distinct eigenvalues lying above the real axis. An example of a nonconstant\nsolution of the homogeneous Schwarz problem in the form of a vector polynomial of degree three is\ngiven. A numerical parameter <span>\\(l\\)</span> of the matrix\n<span>\\(J \\)</span>, expressed via its eigenvectors, is introduced. After\nthat, one relation derived earlier by the present author is analyzed. Based on this analysis, a\nmethod for computing the dimension and structure of the kernel of the Schwarz problem in an\narbitrary ellipse is obtained. Sufficient conditions for the triviality of the kernel expressed via the\nellipse parameters, the eigenvalues of the matrix <span>\\(J\\)</span>, and the parameter\n<span>\\(l \\)</span> are obtained. Examples of one-dimensional and\ntrivial kernels are given.\n</p>","PeriodicalId":0,"journal":{"name":"","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2024-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1134/s0012266124050057","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The Schwarz problem for \(J\)-analytic functions in
an arbitrary ellipse is considered. The matrix \(J\) is assumed to be
two-dimensional with distinct 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. A numerical parameter \(l\) of the matrix
\(J \), expressed via its eigenvectors, is introduced. After
that, one relation derived earlier by the present 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 via the
ellipse parameters, the eigenvalues of the matrix \(J\), and the parameter
\(l \) are obtained. Examples of one-dimensional and
trivial kernels are given.
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