The Hilbert boundary-value problem with a~finite index and a~countable set of jump discontinuities in coefficients

R. B. Salimov ; P. L. Shabalin

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 3 (2010), pp. 36-47.

Résumé

We consider the Hilbert boundary-value problem with a finite index for the case, when the coefficients in the boundary condition have two infinite sequences of discontinuity points of the first kind. We obtain a formula for the general solution and study the solvability issues.

Détail

Keywords: Hilbert boundary-value problem, finite index, Schwarz problem.

Bibliographie

[1] Volterra V., “Sopra alkunecondizioni caratteristische per functioni di variabile complessa”, Ann. Mat. pura appl., 11:1 (1882), 1–55 | DOI

[2] Hilbert D., “Über eine Anwendung der Integralgleichungen auf ein Problem der Functionentheorie”, Verhandl. des III Internat. Math. Kongr., Heidelberg, 1904

[3] Hilbert D., Grundzüge einer allgemainen Theorie der linearen Integralgleichungen, Druck und Verlag von B. G. Tuebner, Leipzig–Berlin, 1912, 282 pp. | MR

[4] Muskhelishvili N. I., Singulyarnye integralnye uravneniya, Nauka, M., 1968, 511 pp. | MR | Zbl

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

[6] Stolyarova E. K., “Obschaya lineinaya kraevaya zadacha dlya uravnenii ellipticheskogo tipa”, Uchen. zap. Kazansk. un-ta, 111, no. 8, 1951, 149–160

[7] Chibrikova L. I., Salekhov L. G., “K resheniyu kraevoi zadachi Gilberta”, Tr. seminara po kraevym zadacham, 8, Izd-vo Kazansk. un-ta, Kazan, 1971, 155–175 | MR | Zbl

[8] Salimov R. B., Seleznev V. V., “K resheniyu kraevoi zadachi Gilberta s razryvnymi koeffitsientami”, Tr. seminara po kraevym zadacham, 16, Izd-vo Kazansk. un-ta, Kazan, 1979, 149–162 | MR | Zbl

[9] Bezrodnykh S. I., Vlasov V. I., “Singulyarnaya zadacha Rimana–Gilberta i ee primenenie k fizike plazmy”, Tezisy mezhdunarodn. konf. “Differentsialnye uravneniya i smezhnye voprosy”, posv. pamyati I. G. Petrovskogo, M., 2007, 36

[10] Sandrygailo I. E., “O kraevoi zadache Gilberta s beskonechnym indeksom dlya poluploskosti”, Izv. AN BSSR. Ser. fiz.-matem. nauk, 1974, no. 6, 16–23 | MR

[11] Salimov R. B., Shabalin P. L., “K resheniyu zadachi Gilberta s beskonechnym indeksom”, Matem. zametki, 73:5 (2003), 724–734 | MR | Zbl

[12] Salimov R. B., Shabalin P. L., Kraevaya zadacha Gilberta teorii analiticheskikh funktsii i ee prilozheniya, Izd-vo Kazansk. matem. o-va, Kazan, 2005, 298 pp.

[13] Alekna P. Yu., “Kraevaya zadacha Gilberta s beskonechnym indeksom logarifmicheskogo poryadka dlya poluploskosti”, Litovsk. matem. sb., 1977, no. 1, 5–12 | MR | Zbl

[14] Govorov N. V., Kraevaya zadacha Rimana s beskonechnym indeksom, Nauka, M., 1986, 239 pp. | MR | Zbl

[15] Tolochko M. E., “O razreshimosti kraevoi zadachi Rimana s beskonechnym indeksom dlya poluploskosti”, Izv. AN BSSR. Ser. fiz.-matem. nauk, 1971, no. 3, 32–38

[16] Tolochko M. E., “Ob odnorodnoi zadache Rimana s beskonechnym indeksom dlya poluploskosti”, Izv. AN BSSR. Ser. fiz.-matem. nauk, 1972, no. 5, 34–41 | Zbl

[17] Monakhov V. N., Semenko E. V., “O korrektnykh postanovkakh kraevykh zadach sopryazheniya s beskonechnym indeksom dlya kvazianaliticheskikh funktsii”, Nekorrektnye zadachi matematicheskoi fiziki i analiza, VTs SO AN SSSR, Nauka, Novosibirsk, 1984, 91–102 | MR

[18] Monakhov V. N., Semenko E. V., Kraevye zadachi i psevdodifferentsialnye operatory na rimanovykh poverkhnostyakh, Fizmatlit, M., 2003, 416 pp.

[19] Zhuravleva M. I., “Odnorodnaya kraevaya zadacha Rimana s beskonechnym indeksom so schetnym mnozhestvom razryvov ee koeffitsienta”, Tr. Tbilissk. matem. in-ta AN GruzSSR, 43, 1973, 53–71 | Zbl

[20] Zhuravleva M. I., “Neodnorodnaya kraevaya zadacha s beskonechnym indeksom i so schetnym mnozhestvom nulei i polyusov koeffitsientov”, DAN SSSR, 214:4 (1974), 755–757 | Zbl

[21] Garnett Dzh., Ogranichennye analiticheskie funktsii, Mir, M., 1984, 469 pp. | MR | Zbl