Geodesic
Dirichlet problem for Lavrent'ev--Bitsadze equation with loaded summands
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2018), pp. 42-58.

Voir la notice de l'article provenant de la source Math-Net.Ru

We study the first boundary-value problem for loaded equation of elliptic-hyperbolic type in rectangular domain. We establish a criterion of uniqueness. A solution to the problem is constructed in the form of the sum of a series. In substantiation of existence of a solution to a problem small denominators appear. We obtain the estimates about a separation from zero of denominators with the corresponding asymptotics. They allow to prove existence of a solution in a class of regular solutions.
Keywords: loaded equation of mixed type, Dirichlet's problem, criterion of uniqueness, series, small denominator, estimate
Mots-clés : existence, uniform convergence. 
@article{IVM_2018_9_a4,
     author = {Yu. K. Sabitova},
     title = {Dirichlet problem for {Lavrent'ev--Bitsadze} equation with loaded summands},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {42--58},
     publisher = {mathdoc},
     number = {9},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2018_9_a4/}
}
TY  - JOUR
AU  - Yu. K. Sabitova
TI  - Dirichlet problem for Lavrent'ev--Bitsadze equation with loaded summands
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2018
SP  - 42
EP  - 58
IS  - 9
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2018_9_a4/
LA  - ru
ID  - IVM_2018_9_a4
ER  - 
%0 Journal Article
%A Yu. K. Sabitova
%T Dirichlet problem for Lavrent'ev--Bitsadze equation with loaded summands
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2018
%P 42-58
%N 9
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2018_9_a4/
%G ru
%F IVM_2018_9_a4
Yu. K. Sabitova. Dirichlet problem for Lavrent'ev--Bitsadze equation with loaded summands. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 9 (2018), pp. 42-58. http://geodesic.mathdoc.fr/item/IVM_2018_9_a4/

[1] Nakhushev A. M., “O zadache Darbu dlya odnogo vyrozhdayuschegosya nagruzhennogo integro-differentsialnogo uravneniya vtorogo poryadka”, Differents. uravneniya, 12:1 (1976), 103–108

[2] Nakhushev A. M., “Nagruzhennye uravneniya i ikh prilozheniya”, Differents. uravneniya, 19:1 (1983), 86–94

[3] Nakhushev A. M., Nagruzhennye uravneniya i ikh primeneniya, Nauka, M., 2012

[4] Dzhenaliev M. T., K teorii lineinykh kraevykh zadach dlya nagruzhennykh differentsialnykh uravnenii, Inst. teoret. i prikl. matem., Almaty, 1995

[5] Sabitov K. B., “Nachalno-granichnaya zadacha dlya nagruzhennogo uravneniya parabolo-giperbolicheskogo tipa”, Dokl. AMAN. Nalchik, 11:1 (2009), 66–73

[6] Sabitov K. B., “Nachalno-granichnaya zadacha dlya parabolo-giperbolicheskogo uravneniya s nagruzhennymi slagaemymi”, Izv. vuzov. Matem., 2015, no. 6, 31–42

[7] Sabitov K. B., Melisheva E. P., “Zadacha Dirikhle dlya nagruzhennogo uravneniya smeshannogo tipa v pryamougolnoi oblasti”, Izv. vuzov. Matem., 2013, no. 7, 62–76 | Zbl

[8] Repin O. A., “Smeshannaya zadacha dlya nagruzhennogo uravneniya Gellerstedta s operatorom M. Saigo v kraevom uslovii”, Vestn. Samarsk. gos. tekh. un-ta. Ser. Fiz.-matem. nauki, 2008, no. 9, 13–18

[9] Sabitova Yu. K., “Kraevaya zadacha s nelokalnym usloviem dlya nagruzhennogo uravneniya smeshannogo tipa”, Nauchn. vedomosti Belgorodsk. gos. nats. issled. un-ta. Ser.: Matem. Fizika, 25(196):37 (2014), 76–90

[10] Sabitova Yu. K., “Kraevaya zadacha s nelokalnym integralnym usloviem dlya uravnenii smeshannogo tipa s vyrozhdeniem na perekhodnoi linii”, Matem. zametki, 98:3 (2015), 393–406 | DOI | Zbl

[11] Sabitov K. B., “Zadacha Dirikhle dlya differentsialnykh uravnenii v chastnykh proizvodnykh vysokikh poryadkov”, Matem. zametki, 97:2 (2015), 262–276 | DOI | Zbl

[12] Zhegalov V. I., “Nelokalnaya zadacha Dirikhle dlya uravneniya smeshannogo tipa”, Neklassicheskie uravneniya matem. fiziki, IM SO AN SSSR, Novosibirsk, 1985, 168–172 | MR

[13] Arnold V. I., “Malye znamenateli i problemy ustoichivosti dvizheniya v klassicheskoi i nebesnoi mekhanike”, UMN, 18:6 (114) (1963), 91–192 | MR | Zbl

[14] Lomov I. S., “Malye znamenateli v analiticheskoi teorii vyrozhdayuschikhsya differentsialnykh uravnenii”, Differents. uravneniya, 29:12 (1993), 2079–2089 | MR | Zbl

[15] Sabitov K. B., Safin E. M., “Obratnaya zadacha dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa”, Matem. zametki, 87:6 (2010), 907–918 | DOI

[16] Khinchin A. Ya., Tsepnye drobi, Nauka, M., 1978