Geodesic
On the Keldysh problem for mixed type equation with two singular lines
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2022), pp. 3-17.

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

In this paper we consider the Keldysh problem in rectangular domain for mixed type equation with two singular coefficients. We prove the criterion of Keldysh problem solution uniqueness on the basis of corresponding spectral problem eigenfunctions system completeness. We construct the Keldysh problem solution as the sum of eigenfunctions series. We obtained separation from zero estimates of small denominators. We prove the existence theorem using this estimates.
Keywords: mixed type equation, Keldysh problem, uniqueness criterion, irrationality measure, Fourier-Bessel series, small denominator
Mots-clés : singular coefficient, existence. 
@article{IVM_2022_2_a0,
     author = {A. A. Abashkin},
     title = {On the {Keldysh} problem for mixed type equation with two singular lines},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {3--17},
     publisher = {mathdoc},
     number = {2},
     year = {2022},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2022_2_a0/}
}
TY  - JOUR
AU  - A. A. Abashkin
TI  - On the Keldysh problem for mixed type equation with two singular lines
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2022
SP  - 3
EP  - 17
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/IVM_2022_2_a0/
LA  - ru
ID  - IVM_2022_2_a0
ER  - 
%0 Journal Article
%A A. A. Abashkin
%T On the Keldysh problem for mixed type equation with two singular lines
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2022
%P 3-17
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/IVM_2022_2_a0/
%G ru
%F IVM_2022_2_a0
A. A. Abashkin. On the Keldysh problem for mixed type equation with two singular lines. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2022), pp. 3-17. http://geodesic.mathdoc.fr/item/IVM_2022_2_a0/

[1] Keldysh M. V., “O nekotorykh sluchayakh vyrozhdeniya uravnenii ellipticheskogo tipa na granitse oblasti”, DAN SSSR, 77:2 (1951), 181–183

[2] Bitsadze A. V., “K probleme uravnenii smeshannogo tipa v mnogomernykh oblastyakh”, DAN SSSR, 110:6 (1956), 901–902 | MR | Zbl

[3] Pulkin S. P., “K voprosu o postanovke zadachi Trikomi v prostranstve”, Uchen. zap. Kuibyshev. gos. ped. in-ta im. V.V. Kuibysheva, 14 (1958), 63–78

[4] Pulkin S. P., “O edinstvennosti resheniya singulyarnoi zadachi Gellersteda”, Izv. vuzov. Matem., 1960, no. 6, 214–225 | MR

[5] Salakhitdinov M. S., Urinov A. K., “Zadachi na sobstvennye znacheniya dlya uravneniya smeshannogo tipa s dvumya singulyarnymi koeffitsientami”, Sib. matem. zhurn., 48:4 (2007), 882–893 | MR | Zbl

[6] Bitsadze A. V., “Nekorrektnost zadachi Dirikhle dlya uravnenii smeshannogo tipa v smeshannykh oblastyakh”, DAN SSSR, 122:2 (1958), 167–170 | Zbl

[7] Sabitov K. B., “Zadacha Dirikhle dlya uravnenii smeshannogo tipa v pryamougolnoi oblasti”, Dokl. RAN, 413:1 (2007), 23–26 | Zbl

[8] Sabitov K. B., Safina R. M., “Pervaya granichnaya zadacha dlya uravneniya smeshannogo tipa s singulyarnym koeffitsientom”, Izv. RAN. Ser. matem., 82:2 (2018), 79–112 | MR | Zbl

[9] Sabitov K. B., Vagapova E. V., “Zadacha Dirikhle dlya uravneniya smeshannogo tipa s dvumya liniyami vyrozhdeniya v pryamougolnoi oblasti”, Differents. uravneniya, 49:1 (2013), 68–78 | MR | Zbl

[10] Urinov A. K., Karimov K. T., “Zadacha Dirikhle dlya trekhmernogo uravneniya smeshannogo tipa s tremya singulyarnymi koeffitsientami”, Vestn. Samarsk. gos. tekhn. un-ta. Ser. Fiz.-matem. nauki, 21:4 (2017), 665–683 | Zbl

[11] Lebedev N. N., Spetsialnye funktsii i ikh prilozheniya, Lan, S.-P.–M.–Krasnodar, 2010

[12] Abashkin A. A., “Nelokalnaya zadacha dlya uravneniya smeshannogo tipa s singulyarnym koeffitsientom v oblasti, giperbolicheskaya chast kotoroi - vertikalnaya polupolosa”, Vestn. Samarsk. gos. tekhn. un-ta. Ser. Fiz.-matem. nauki, 3:36 (2014), 7–20 | Zbl

[13] Vatson G. V., Teoriya besselevykh funktsii, Izd. Inlit, M., 1949

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

[15] Bondareva I. V., Luchin M. Yu., Salikhov V. Kh., “O mere irratsionalnosti $\ln7$”, Matem. zametki, 107:3 (2020), 366–375 | MR | Zbl

[16] Sabitov K. B., Pryamye i obratnye zadachi dlya uravnenii smeshannogo parabolo-giperbolicheskogo tipa, Nauka, M., 2016 | MR

[17] Roth K. F., “Rational approximations to algebraic numbers”, Math. (London), 1955, no. 2, 1–20 | MR | Zbl

[18] Borwein J. M., Borwein P. B., “Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity”, Mathematics of Computation, 50:181 (1988), 352–354 | DOI | MR

[19] Van Assche W., “Little q-Legendre Polynomials and Irrationality of Certain Lambert Series”, The Ramanujan J., 2001, no. 5, 295–310 | MR | Zbl

[20] Bugeaud Y., Distribution modulo one and diophantine approximation, Cambridge Univ. Press, Cambridge, 2012 | MR | Zbl

[21] Shmidt V., Diofantovy priblizheniya, MIR, M., 1983