Dirichlet problem for an elliptic equation with three singular coefficients

A. K. Urinov ; K. T. Karimov

Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Mathematical Analysis, Tome 156 (2018), pp. 30-40.

Résumé

We prove the unique solvability of the first boundary-value problem for an elliptic equation with three singular coefficients in a rectangular parallelepiped. Using the method of energy integrals, we prove the uniqueness of a solution to the problem. We prove the existence of a solution by the spectral Fourier method based on the separation of variables. A solution to the problem is constructed in the form of a double Fourier–Bessel series. The justification of the uniform convergence of this series is based on asymptotic methods. We obtain an estimate, which allows one to prove the convergence of the series and its derivatives up to the second order and the existence theorem for the class of regular solutions of the equation considered.

Détail

Keywords: Dirichlet problem, spectral method, uniqueness of solution
Mots-clés : elliptic equation, existence of solution.

Bibliographie

[1] Bitsadze A. V., Nekotorye klassy uravnenii v chastnykh proizvodnykh, Nauka, M., 1981

[2] Vatson G. N., Teoriya besselevykh funktsii, v. 1, IL, M., 1949

[3] Karimov K. T., “Zadacha Dirikhle dlya trekhmernogo ellipticheskogo uravneniya s dvumya singulyarnymi koeffitsientami”, Uzb. mat. zh., 2017, no. 1, 96–105

[4] Kuznetsov D. S., Spetsialnye funktsii, Vysshaya shkola, M., 1962

[5] Nazipov I. T., “Reshenie space'ennoi zadachi Trikomi dlya singulyarnogo uravneniya smeshannogo tipa metodom integralnykh uravnenii”, Izv. vuzov. Mat., 2011, no. 3, 69–85 | MR | Zbl

[6] Olver F., Vvedenie v asimptoticheskie metody i spetsialnye funktsii, Mir, M., 1986

[7] Salakhitdinov M. S., Mirsaburov M., Nelokalnye zadachi dlya uravnenii smeshannogo tipa s singulyarnymi koeffitsientami, Tashkent, 2005

[8] Salakhitdinov M. S., Urinov A. K., K spektralnoi teorii uravnenii smeshannogo tipa, Tashkent, 2010

[9] Serbina L. I., “Ob odnoi probleme dlya linearizovannogo uravneniya Bussineska s nelokalnym usloviem Samarskogo”, Differ. uravn., 38:8 (2002), 1113–1119 | MR | Zbl

[10] Smirnov M. M., Vyrozhdayuschiesya ellipticheskie i giperbolicheskie uravneniya, Nauka, M., 1966

[11] Urinov A. K., Karimov K. T., “Zadacha Dirikhle—Neimana dlya trekhmernogo ellipticheskogo uravneniya s dvumya singulyarnymi koeffitsientami”, Vestn. Nats. un-ta Uzbekistana. Ser. Mat. Mekh. Fiz. Inform., 2/1 (2017), 195–206

[12] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya. T. 2, Nauka, M., 1969

[13] Khiiirbekov T. E., “Reshenie zadachi $T_3^+$ dlya odnogo trekhmernogo uravneniya v proizvolnoi oblasti”, Volzh. mat. sb., 1971, no. 8, 215–220

[14] Khiiirbekov T. E., “Reshenie zadachi $N$ dlya odnogo $n$-mernogo uravneniya v normalnoi oblasti”, Volzh. mat. sb., 1971, no. 8, 221–226

[15] Bers L., Mathematical aspects of subsonic and transonic gas dynamics, New York–London, 1958 | MR | Zbl

[16] Gilbert R., Function theoretic methods in partial differential equations, Academic Press, New York–London, 1969 | MR | Zbl

[17] Karimov E. T., “On a boundary problem with Neumann's condition for 3D singular elliptic equations”, Appl. Math. Lett., 23 (2010), 517–522 | DOI | MR | Zbl

[18] Karimov E. T., Nieto J. J., “The Dirichlet problem for a 3D elliptic equation with two singular coefficients”, Comput. Math. Appl., 62 (2011), 214–224 | DOI | MR | Zbl

[19] Nieto J. J., Karimov E. T., “On an analogue of the Holmgren’s problem for 3d singular elliptic equation”, Asian-Europ. J. Math., 5:2 (2012), 1250021 | DOI | MR | Zbl

[20] Salakhitdinov M. S., Karimov E. T., “Spatial boundary problem with the Dirichlet–Neumann condition for a singular elliptic equation”, Appl. Math. Comput., 219 (2012), 3469–3476 | MR | Zbl