On a Certain Functional Equation and Its Application to the Schwarz Problem
The Schwarz problem for <i>J</i>-analytic functions in an ellipse is considered. In this case, the matrix <i>J</i> is assumed to be two-dimensional with different eigenvalues located above the real axis. The Schwarz problem is reduced to an equivalent boundary value problem f...
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MDPI AG
2023-06-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/11/12/2789 |
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| author | Vladimir Nikolaev Vladimir Vasilyev |
| author_facet | Vladimir Nikolaev Vladimir Vasilyev |
| author_sort | Vladimir Nikolaev |
| collection | DOAJ |
| description | The Schwarz problem for <i>J</i>-analytic functions in an ellipse is considered. In this case, the matrix <i>J</i> is assumed to be two-dimensional with different eigenvalues located above the real axis. The Schwarz problem is reduced to an equivalent boundary value problem for the scalar functional equation depending on the real parameter <i>l</i>. This parameter is determined by the Jordan basis of the matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>.</mo></mrow></semantics></math></inline-formula> An analysis of the functional equation was performed. It is shown that for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, the solution of the Schwarz problem with matrix <i>J</i> exists uniquely in the Hölder classes in an arbitrary ellipse. |
| first_indexed | 2024-03-11T02:11:10Z |
| format | Article |
| id | doaj.art-38f5714eb42f406080d55b12dca6a1cf |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-11T02:11:10Z |
| publishDate | 2023-06-01 |
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| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-38f5714eb42f406080d55b12dca6a1cf2023-11-18T11:29:48ZengMDPI AGMathematics2227-73902023-06-011112278910.3390/math11122789On a Certain Functional Equation and Its Application to the Schwarz ProblemVladimir Nikolaev0Vladimir Vasilyev1Yaroslav-the-Wise Novgorod State University, Bolshaya St.-Peterburgskaya ul. 41, 173003 Velikiy Novgorod, RussiaCenter of Applied Mathematics, Belgorod State National Research University, Pobedy St. 85, 308015 Belgorod, RussiaThe Schwarz problem for <i>J</i>-analytic functions in an ellipse is considered. In this case, the matrix <i>J</i> is assumed to be two-dimensional with different eigenvalues located above the real axis. The Schwarz problem is reduced to an equivalent boundary value problem for the scalar functional equation depending on the real parameter <i>l</i>. This parameter is determined by the Jordan basis of the matrix <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>J</mi><mo>.</mo></mrow></semantics></math></inline-formula> An analysis of the functional equation was performed. It is shown that for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>l</mi><mo>∈</mo><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow></semantics></math></inline-formula>, the solution of the Schwarz problem with matrix <i>J</i> exists uniquely in the Hölder classes in an arbitrary ellipse.https://www.mdpi.com/2227-7390/11/12/2789<i>J</i>-analytic functions<i>λ</i>-holomorphic functionsmatrix eigenvalueellipsefunctional equation |
| spellingShingle | Vladimir Nikolaev Vladimir Vasilyev On a Certain Functional Equation and Its Application to the Schwarz Problem Mathematics <i>J</i>-analytic functions <i>λ</i>-holomorphic functions matrix eigenvalue ellipse functional equation |
| title | On a Certain Functional Equation and Its Application to the Schwarz Problem |
| title_full | On a Certain Functional Equation and Its Application to the Schwarz Problem |
| title_fullStr | On a Certain Functional Equation and Its Application to the Schwarz Problem |
| title_full_unstemmed | On a Certain Functional Equation and Its Application to the Schwarz Problem |
| title_short | On a Certain Functional Equation and Its Application to the Schwarz Problem |
| title_sort | on a certain functional equation and its application to the schwarz problem |
| topic | <i>J</i>-analytic functions <i>λ</i>-holomorphic functions matrix eigenvalue ellipse functional equation |
| url | https://www.mdpi.com/2227-7390/11/12/2789 |
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