Geodesic
Solution of the basic boundary value problems for a degenerate elliptic equation by the method of potentials
Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 157 (2015) no. 1, pp. 5-14 Cet article a éte moissonné depuis la source Math-Net.Ru

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Fundamental solutions to a degenerate elliptic equation are found. Using these fundamental solutions, simple and double layer potentials are built. The basic boundary value problems for a degenerate elliptic equation are reduced to the equivalent Fredholm integral equations of the second kind. Their solvability is proved.
Keywords: degenerate elliptic equation, method of potentials, double layer potential, simple layer potential, Fredholm equation, generalized translation operator. 
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R. M. Askhatov; R. N. Abaydullin. Solution of the basic boundary value problems for a degenerate elliptic equation by the method of potentials. Učënye zapiski Kazanskogo universiteta. Seriâ Fiziko-matematičeskie nauki, Uchenye Zapiski Kazanskogo Universiteta. Seriya Fiziko-Matematicheskie Nauki, Tome 157 (2015) no. 1, pp. 5-14. http://geodesic.mathdoc.fr/item/UZKU_2015_157_1_a0/

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

[2] Bitsadze A. V., Uravneniya smeshannogo tipa, Izd-vo AN SSSR, M., 1959, 164 pp.

[3] Oleinik O. A., “Zadacha Koshi i kraevaya zadacha dlya giperbolicheskikh uravnenii vtorogo poryadka, vyrozhdayuschikhsya v oblasti i na ee granitse”, Dokl. AN SSSR, 169:3 (1966), 525–528 | MR

[4] Oleinik O. A., “O giperbolicheskikh uravneniyakh vtorogo poryadka, vyrozhdayuschikhsya vnutri oblasti i na ee granitse”, Usp. matem. nauk, 24:2 (1969), 229–230 | MR | Zbl

[5] Smirnov M. M., Vyrozhdayuschiesya ellipticheskie i giperbolicheskie uravneniya, Nauka, M., 1966, 292 pp. | MR

[6] Tersenov S. A., “K teorii uravnenii ellipticheskogo tipa, vyrozhdayuschikhsya na granitse oblasti”, Sib. matem. zhurn., 6:5 (1965), 1120–1143 | MR | Zbl

[7] Mukhlisov F. G., Nigmedzyanova A. M., “Reshenie kraevykh zadach dlya vyrozhdayuschegosya ellipticheskogo uravneniya vtorogo roda metodom potentsialov”, Izv. vuzov. Matem., 2009, no. 8, 57–70 | MR | Zbl

[8] Tikhonov A. N., Samarskii A. A., Uravneniya matematicheskoi fiziki, Nauka, M., 1972, 736 pp. | MR

[9] Kuznetsov D. S., Spetsialnye funktsii, Vyssh. shk., M., 1962, 248 pp.

[10] Mukhlisov F. G., Potentsialy, porozhdennye operatorom obobschennogo sdviga, i kraevye zadachi dlya odnogo klassa singulyarnykh ellipticheskikh uravnenii, Dis. $\dots$ d-ra fiz.-matem. nauk, Kazan, 1993, 324 pp.

[11] Askhatov R. M., “Reshenie osnovnykh kraevykh zadach dlya odnogo singulyarnogo ellipticheskogo uravneniya metodom potentsialov”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 155, no. 4, 2013, 5–15

[12] Mikhlin S. G., Kurs matematicheskoi fiziki, Nauka, M., 1968, 576 pp. | MR