Title:
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A note on the solvability of homogeneous Riemann boundary problem with infinity index (English) |
Author:
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Bory-Reyes, Juan |
Language:
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English |
Journal:
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Communications in Mathematics |
ISSN:
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1804-1388 (print) |
ISSN:
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2336-1298 (online) |
Volume:
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29 |
Issue:
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3 |
Year:
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2021 |
Pages:
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527-534 |
Summary lang:
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English |
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Category:
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math |
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Summary:
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In this note we establish a necessary and sufficient condition for solvability of the homogeneous Riemann boundary problem with infinity index on a rectifiable open curve. The index of the problem we deal with considers the influence of the requirement of the solutions of the problem, the degree of non-smoothness of the curve at the endpoints as well as the behavior of the coefficient at these points. (English) |
Keyword:
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Riemann boundary value problem |
Keyword:
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Phragm\`{e}n Lindel\"of principle. |
MSC:
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30D20 |
MSC:
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30E20 |
MSC:
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30E25 |
idZBL:
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Zbl 07484385 |
idMR:
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MR4355426 |
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Date available:
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2022-01-10T10:15:18Z |
Last updated:
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2022-04-28 |
Stable URL:
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http://hdl.handle.net/10338.dmlcz/149334 |
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Reference:
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[1] Gakhov, F.D.: Boundary value problems. Transl. from the Russian. Reprint of the 1966 translation. (English).1990, Dover Publications, Inc., New York, |
Reference:
|
[2] Lu, J.-K.: Boundary value problems for analytic functions (English).1993, World Scientific, Singapore, Series in Pure Mathematics 16. |
Reference:
|
[3] Peña, D.P., Reyes, J.B.: Riemann boundary value problem on a regular open curve.J. Nat. Geom., 22, 1-2, 2002, 1-18, MR 1906672 |
Reference:
|
[4] Seĭfullaev, R.K.: Solvability of a homogeneous Riemann boundary value problem on an open curve. (Russian).Theory of functions and approximations (Saratov, 1982), 2, 1983, 140-143, Saratov. Gos. Univ., Saratov, 1983, |
Reference:
|
[5] Seĭfullaev, R.K.: A Riemann boundary value problem on a nonsmooth open curve. (Russian).Mat. Sb. (N. S.), 112 (154), 2 (6), 1980, 147-161, |
Reference:
|
[6] Seĭfullaev, R.K.: Relationship of the number of linearly independent solutions of a Riemann boundary value problem to properties of the jump curve. (Russian).Gos. Univ. Uchen. Zap., 6, 1978, 14-20, |
Reference:
|
[7] Kutlu, K.: On characteristic adjoint singular integral equation of Riemann boundary value problem.An. Univ. Timişoara Ser. Mat.-Inform., 41, 1, 2003, 125-129, MR 2245917 |
Reference:
|
[8] Kutlu, K.: On Riemann boundary value problem.An. Univ. Timişoara Ser. Mat.-Inform., 38, 1, 2000, 89-96, |
Reference:
|
[9] Kutlu, K.: On Riemann boundary value problem.J. Indian Acad. Math., 21, 2, 1999, 181-191, |
Reference:
|
[10] Diéguez, B. González, Reyes, J. Bory: The homogeneous Riemann boundary value problem on rectifiable open Jordan curves. (Spanish).Cienc. Mat. (Havana), 9, 2, 1988, 3-9, |
Reference:
|
[11] Reyes, J. Bory, González-Diéguez, B.: On the classical formulation of solvability of the Riemann boundary value problem. (Spanish).Cienc. Mat. (Havana), 10, 1, 1989, 65-73, |
Reference:
|
[12] Reyes, J. Bory: On the Riemann boundary problem on open rectifiable Jordan curves. (Spanish).Cienc. Mat. (Havana), 11, 3, 1990, 211-220, |
Reference:
|
[13] Danilov, E.A.: Dependence of the number of solutions of a homogeneous Riemann problem on the contour and the coefficient module. (Russian).Dokl. Akad. Nauk SSSR, 264, 6, 1982, 1305-1308, |
Reference:
|
[14] Salim, M.S.: Necessary and sufficient conditions for continuity of an integral of Cauchy type up to the boundary along a nonsmooth open curve. (Russian).Scientific reports. Mathematics and physics series, 3, 1979, 75-83, |
Reference:
|
[15] Selim, M.S.: The homogeneous Riemann problem on a nonsmooth open curve. (Russian).Scientific reports. Mathematics and physics series, 5, 1979, 12\IL2\textendash 22, |
Reference:
|
[16] Phragmèn, E.: Sur une extension d'un theoreme classique de la theorie des fonctions.Acta Math., 28, 1, 1904, 351-368, 10.1007/BF02418391 |
Reference:
|
[17] Phragmèn, E., Lindelöf, E.: Sur une extension d'un principe classique de l'analyse et sur quelques proprietes des fonctions monogenes dans le voisinage d'un point singulier.Acta Math., 31, 1, 1908, 381-406, 10.1007/BF02415450 |
Reference:
|
[18] Heins, M.: On the Phragmèn-Lindelöf principle.Trans. Amer. Math. Soc., 60, 1946, 238-244, |
Reference:
|
[19] Ahlfors, L.V.: On Phragmèn-Lindelöf's principle.Trans. Amer. Math. Soc., 41, 1, 1937, 1-8, |
Reference:
|
[20] Govorov, N.V.: Riemann's boundary problem with infinite index.1994, Birkhäuser Verlag, Basel, Edited and with an introduction and an appendix by I. V. Ostrovskii. Translated from the 1986 Russian original by Yu.I. Lyubarskii.. |
Reference:
|
[21] Alekhno, A.G.: On the solvability of the homogeneous Riemann boundary value problem with an infinite index. (Russian).Dokl. Akad. Nauk Belarusi, 41, 2, 1997, 37-44, |
Reference:
|
[22] Alekhno, A.G.: Sufficient conditions for solvability of homogeneous Riemann boundary value problem with infinite index (Russian).Tr. N. I. Lobachevskii Mat. Center, 14, 2002, 71-77, MR 1952217 |
Reference:
|
[23] Alekhno, A.G., Sevruk, A.B.: The homogeneous Riemann boundary value problem with an infinite index of Boutroux refined order.Dokl. Nats. Akad. Nauk Belarusi, 55, 6, 2011, 5-10, MR 2963469 |
Reference:
|
[24] Ostrovski, I.V.: The homogeneous Riemann boundary value problem with an infinite index on a curvilinear contour. I. (Russian).Teor. Funktsi. Funktsional. Anal. i Prilozhen., 56, 1991, 95-105, Translation in J. Math. Sci. 76 (4) (1995), 2517--2524.. |
Reference:
|
[25] Ostrovski, I.V.: The homogeneous Riemann boundary value problem with an infinite index on a curvilinear contour. II. (Russian).Teor. Funktsi. Funktsional. Anal. i Prilozhen., 52, 1992, 3-17, Translation in J. Math. Sci. 77 (1) (1995), 2917--2928.. |
Reference:
|
[26] Plaksa, S.A.: Riemann boundary problem with index plus-infinity on a rectifiable curve (English. Russian original).Ukr. Math. J., 42, 9, 1990, 1070-1077, Translation from Ukr. Mat. Zh. 42 (9) (1990) 1204--1213.. |
Reference:
|
[27] Salimov, R.B.: On a new approach to solving the Riemann boundary value problem with a condition on the ray in the case of an infinite index (Russian).Izv. Sarat. Univ. (N.S.) Ser. Mat. Mekh. Inform., 16, 1, 2016, 20-33, MR 3501501 |
Reference:
|
[28] Salimov, R.B., Suleĭmanov, A.Z.: A new approach to solving a homogeneous Riemann boundary value problem on a ray with an infinite index (Russian).Russian Math. (Iz. VUZ), 61, 5, 2017, 61-65, Translated from Izv. Vyssh. Uchebn. Zaved. Mat. 2017, no. 5, 71--76.. MR 3752716, 10.3103/S1066369X17050085 |
Reference:
|
[29] Tolochko, M.E.: On Solvability of Homogeneous Rieman Boundary Value Problem with Infinite Index for a Half-Plane (Russian).Izv. Akad. Nauk BSSR, Ser. Fiz.-Mat. Nauk, 5, 1972, 34-41, |
Reference:
|
[30] Sandrygailo, I.E.: The Riemann Boundary Value Problem with Infinite Index for a Half-Plane (Russian).Dokl. AN BSSR, 19, 10, 1975, 872-875, |
Reference:
|
[31] Salimov, R.B., Gorskaya, T.Yu.: Solution of homogeneous Riemann boundary value problem with a condition on a real axis and an infinite index of logarithmic order with the new method (Russian).Meždunar. Nauč.-Issled. Žurn., 7, 85, 2019, 6-15, |
Reference:
|
[32] Grudskiĭ, S.M.: Singular integral equations and the Riemann boundary value problem with an infinite index in the space $L_0(\Gamma ,\omega )$ (Russian).Izv. Akad. Nauk SSSR Ser. Mat., 49, 1, 1985, 55-80, |
Reference:
|
[33] Fatykhov, A.Kh., Shabalin, P.L.: Solvability homogeneous Riemann-Hilbert boundary value problem with several points of turbulence.Probl. Anal. Issues Anal., 7, 25, 2018, 31-39, Special Issue.. MR 3866050, 10.15393/j3.art.2018.5530 |
Reference:
|
[34] Privalov, I.I.: Graničnye svoístva analitičeskih funkcií. (Russian) (Boundary properties of analytic functions), 2nd ed.1950, Gosudarstv. Izdat. Tehn.-Teor. Lit., Moscow-Leningrad, |
Reference:
|
[35] Painlevé, P.: Sur les lignes singulières des fonctions analytiques (French).Ann. Fac. Sci. Toulouse Sci. Math. Sci. Phys., 2, 1888, B1-B130, |
Reference:
|
[36] Evgrafov, M.A.: Analytic functions. (Russian) 2nd ed.1968, Nauka (Moscow), |
Reference:
|
[37] Levin, B.Ya.: Distribution of zeros of entire functions.1980, American Mathematical Society, Providence, R.I., Translated from the Russian by R.P. Boas, J.M. Danskin, F.M. Goodspeed, J. Korevaar, A.L. Shields and H.P. Thielman. Revised edition. Translations of Mathematical Monographs, 5.. |
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