Local uniqueness of solution of a mixed boundary value problem for Riemann surfaces with branch-points

S. R. Nasyrov   ; I. Z. Faizov

Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Kazanskii Gosudarstvennyi Universitet. Uchenye Zapiski. Seriya Fiziko-Matematichaskie Nauki, Tome 148 (2006) no. 2, pp. 97-108

Résumé

A mixed boundary value problem for Riemann surfaces with branch-points, with boundary conditions depending on the parameter $x$ is investigated. Is is supposed that the known part of the boundary of the required surface is a polygon. We obtain an integral representation of solution to the problem; it depends on accessory parameters. Local uniqueness of the solution is proved.

Bibliographie

[1] Avkhadiev F. G., Aksentev L. A., Elizarov A. M., “Dostatochnye usloviya konechnolistnosti analiticheskikh funktsii i ikh prilozheniya”, Itogi nauki i tekhniki. Ser. Matem. analiz, 25, VINITI, M., 1987, 3–121

[2] Avkhadiev F. G., Aksentev L. A., Elizarov A. M., Nasyrov S. R., “Nauchnyi seminar po geometricheskoi teorii funktsii: osnovnye rezultaty dvukh poslednikh desyatiletii”, Tr. Matem. tsentra im. N. I. Lobachevskogo, 14, Kazan. matem. o-vo, Kazan, 2002, 7–38 | MR | Zbl

[3] Monakhov V. N., Kraevye zadachi so svobodnymi granitsami dlya ellipticheskikh sistem uravnenii, Nauka, Novosibirsk, 1977, 424 pp. | MR

[4] Salimov R. B., Shabalin P. L., Kraevaya zadacha Gilberta teorii analiticheskikh funktsii i ee prilozheniya, Kazan. matem. o-vo, Kazan, 2005, 297 pp. | Zbl

[5] Nasyrov S. R., O metode poligonalnoi approksimatsiei v smeshannykh obratnykh kraevykh zadachakh po parametru $x$, Dep. v VINITI 17.05.82, No 2459-82, Kazan. gos. un-t., Kazan, 1982, 48 pp.

[6] Nasyrov S. R., “Smeshannaya obratnaya kraevaya zadacha na rimanovykh poverkhnostyakh”, Izv. vuzov. Matematika, 1990, no. 10, 25–36 | MR | Zbl

[7] Nasyrov S. R., “Metod dvizhuschegosya razreza v smeshannykh obratnykh kraevykh zadachakh”, Konstr. teoriya funktsii i ee prilozheniya, Makhachkala, 1994, 71–73

[8] Tlyusten S. R., “Smeshannaya kraevaya zadacha so svobodnoi granitsei v neodnolistnykh oblastyakh”, Dinamika sploshnoi sredy, 76, Novosibirsk, 1986, 148–156 | MR | Zbl

[9] Tlyusten S. R., “Neodnolistnye otobrazheniya so svobodnoi granitsei”, Dinamika sploshnoi sredy, 86, Novosibirsk, 1988, 141–148 | MR

[10] Tlyusten S. R., “Apriornye otsenki reshenii smeshannoi kraevoi zadachi so svobodnoi granitsei dlya analiticheskikh funktsii”, Dinamika sploshnoi sredy, 92, Novosibirsk, 1989, 108–121 | MR | Zbl

[11] Tlyusten S. R., “Geometricheskie svoistva reshenii smeshannoi kraevoi zadachi so svobodnoi granitsei”, Dinamika sploshnoi sredy, 97, Novosibirsk, 1990, 114–123 | MR | Zbl

[12] Salimov R. B., Strezhneva E. V., “K resheniyu obratnoi smeshannoi kraevoi zadachi”, Tr. seminara po kraev. zadacham, 27, Izd-vo Kazan. un-ta, Kazan, 1992, 95–117 | MR

[13] Salimov R. B., Nasyrova E. V., Shabalin P. L., “Odnolistnaya razreshimost odnoi obratnoi smeshannoi kraevoi zadachi”, Izv. vuzov. Matematika, 1998, no. 4, 78–82 | MR

[14] Strezhneva E. V., “Dostatochnye usloviya odnolistnosti odnoi smeshannoi obratnoi kraevoi zadachi po parametru $x$ dlya odnosvyaznoi oblasti v sluchae poligona”, Teoriya funktsii i priblizhenii. Mezhvuz. sb. nauchn. tr., ch. 3., Izd-vo Saratov. un-ta, Saratov, 1995, 109–115

[15] Salimov R. B., Strezhneva E. V., Reshenie obratnoi kraevoi zadachi dlya dvusvyaznoi oblasti v vidoizmenennoi postanovke, Dep. v VINITI 29.12.90, No 6487-V90, Kazan, 1990, 26 pp.

[16] Galiullina G. R., Nasyrov S. R., “Uravnenie Gakhova dlya vneshnei smeshannoi obratnoi kraevoi zadachi po parametru $x$”, Izv. vuzov. Matematika, 2002, no. 10, 48–55 | MR | Zbl

[17] Lavrentev M. A., Shabat B. V., Metody teorii funktsii kompleksnogo peremennogo, Nauka, M., 1987, 688 pp. | MR

[18] Gakhov F. D., Kraevye zadachi, Nauka, M., 1977, 640 pp. | MR