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 Hölder 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.
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V. G. Nikolaev, “Solutions to the Schwarz problem with diagonalizable matrices in ellipse,” J. Math. Sci., New York 244, No 4, 655–670 (2020).
V. G. Nikolaev, “On the Schwarz problem in the case of matrices with nondiagonal Jordan forms,” J. Math. Sci., New York 250, No 1, 83–93 (2020).
A. P. Soldatov, “The Schwarz problem for Douglis analytic functions,” J. Math. Sci., New York 173, No. 2, 221–224 (2011).
A. P. Soldatov, Douglis Analytic Functions [in Russian], Veliky Novgorod (1995).
V. B. Vasil’ev and V. G. Nikolaev, “Schwarz problem for first-order elliptic systems on the plane,” Differ. Equ. 53, No. 10, 1318–1328 (2017).
V. G. Nikolaev, “A criterion for the existence of nontrivial solutions to the homogeneous Schwarz problem,” J. Math. Sci., New York 219, No 2, 220–225 (2016).
V. G. Nikolaev, “A class of orthogonal polynomials on the boundary of an ellipse,” J. Math. Sci., New York 239, No 3, 363–380 (2019).
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Translated from Problemy Matematicheskogo Analiza 107, 2020, pp. 91-112.
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Nikolaev, V.G. Schwarz Problem in Ellipse for Nondiagonalizable Matrices. J Math Sci 251, 876–901 (2020). https://doi.org/10.1007/s10958-020-05134-z
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DOI: https://doi.org/10.1007/s10958-020-05134-z