Geodesic
The Dirichlet problem for a class of composite type equations with a discontinuous coefficient of the highest
Dalʹnevostočnyj matematičeskij žurnal, Tome 14 (2014) no. 1, pp. 48-65.

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This paper investigates the solvability of the Dirichlet problem in classes of regular or almost regular solutions of composite type equations with a discontinuous alternating in sign coefficient. The existence and uniqueness theorems are proved by regularization and continuation methods. 
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A. I. Kozhanov; S. V. Potapova. The Dirichlet problem for a class of composite type equations with a discontinuous coefficient of the highest. Dalʹnevostočnyj matematičeskij žurnal, Tome 14 (2014) no. 1, pp. 48-65. http://geodesic.mathdoc.fr/item/DVMG_2014_14_1_a4/

[1] G. V. Demidenko, S. V. Uspenskii, Uravneniya i sistemy, ne razreshennye otnositelno starshei proizvodnoi, Nauchnaya kniga, Novosibirsk, 1998 | MR | Zbl

[2] A. G. Sveshnikov, A. B. Alshin, N. D. Korpusov, Yu. D. Pyatner, Lineinye i nelineinye uravneniya sobolevskogo tipa, Fizmatlit, M., 2007

[3] A. I. Kozhanov, “Composite Type Equations and Inverse Problems”, Utrecht, the Netherlands, VSP, 1999 | MR

[4] S. V. Potapova, “Boundary value problems for pseudohyperbolic equations with a variable time direction”, TWMS Journal of Pure and Applied Mathematics, 3:1 (2012), 75–91 | MR | Zbl

[5] O. A. Ladyzhenskaya, “O reshenii obschei zadachi difraktsii”, DAN SSSR, 93 (1954), 433–436

[6] O. A. Oleinik, “Ob odnom metode resheniya obschei zadachi difraktsii”, DAN SSSR, 1960, 1054–1057 | MR

[7] A. G. Sveshnikov, A. B. Alshin, N. D. Korpusov, Yu. D. Pyatner, Lineinye i nelineinye uravneniya sobolevskogo tipa, Fizmatlit, M., 2007

[8] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967. | MR

[9] S. A. Tersenov, Vvedenie v teoriyu uravnenii parabolicheskogo tipa s menyayuschimsya napravleniem vremeni, Sib. otd–nie AN SSSR. In–t matematiki, Novosibirsk, 1982 | MR

[10] I. E. Egorov, S. G. Pyatkov, S. V. Popov, Neklassicheskie differentsialno–operatornye uravneniya, Nauka, Novosibirsk, 2000 | MR

[11] M. M. Smirnov, Uravneniya smeshannogo tipa, Nauka, M., 1970 | MR

[12] T. D. Dzhuraev, Kraevye zadachi dlya uravneniya smeshannogo i smeshanno-sostavnogo tipov, FAN, Tashkent, 1986 | MR | Zbl

[13] E. I. Moiseev, Uravneniya smeshannogo tipa so spektralnym parametrom, MGU, Moskva, 1988 | MR

[14] A. P. Soldatov, “Zadacha tipa Dirikhle dlya uravneniya Lavrenteva-Bitsadze. I. Teoremy edinstvennosti. II. Teoremy suschestvovaniya”, Dokl. RAN, 332, 333:6, 1 (1993), 696–698, 16–18 | Zbl

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

[16] M. M. Khachev, Pervaya kraevaya zadacha dlya lineinogo uravneniya smeshannogo tipa, Elbrus, Nalchik, 1998

[17] O. I. Marichev, A. A. Kilbas, O. A. Repin, Kraevye zadachi dlya uravnenii s chastnymi proizvodnymi s razryvnymi koeffitsientami, Samarskii gosudarstvennyi ekonomicheskii universitet, Samara, 2008 | Zbl

[18] O. A. Ladyzhenskaya, L. Stupyalis, “Ob uravneniyakh smeshannogo tipa”, Vestnik LGU, 1967, no. 19, 38–46

[19] O. A. Ladyzhenskaya, L. Stupyalis, “Kraevye zadachi dlya uravnenii smeshannogo tipa”, Trudy MIAN SSSR, 116:19 (1971), 101–136 | Zbl

[20] L. Stupyalis, “Kraevye zadachi dlya elliptiko-giperbolicheskikh uravnenii”, Trudy MIAN SSSR, 125 (1973), 211–229

[21] V. A. Ilin, P. V. Luferenko, “Smeshannye zadachi, opisyvayuschie prodolnye kolebaniya sterzhnya, sostoyaschego iz dvukh uchastkov, imeyuschie raznye plotnosti i raznye uprugosti, no odinakovye impedansy”, Dokl. RAN, 428:1 (2009), 12–15 | MR | Zbl

[22] V. A. Ilin, P. V. Luferenko, “Obobschennye resheniya smeshannykh zadach dlya razryvnogo smeshannogo uravneniya pri uslovii ravenstva impedansov”, Dokl. RAN, 429:3 (2009), 317–321 | MR | Zbl

[23] V. A. Ilin, “Optimizatsiya proizvodimogo smescheniem granichnogo upravleniya kolebaniyami sterzhnya, sostoyaschego iz dvukh raznorodnykh uchastkov”, Differentsialnye uravneniya, 47:7 (2011), 978–986 | MR | Zbl

[24] A. M. Rogozhnikov, “Issledovanie smeshannoi zadachi, opisyvayuschei protsess kolebanii sterzhnya, sostoyaschego iz neskolkikh uchastkov, pri uslovii sovpadeniya vremeni prokhozhdeniya volny po kazhdomu iz etikh uchastkov”, Dokl. RAN, 441:4 (2011), 449–451 | MR | Zbl

[25] A. A. Kuleshov, “Smeshannye zadachi dlya uravneniya prodolnykh kolebanii neodnorodnogo sterzhnya so svobodnym libo zakreplennym pravym kontsom, sostoyaschego iz dvukh uchastkov raznoi plotnosti i uprugosti”, Dokl. RAN, 442:4 (2012), 451–454 | MR | Zbl

[26] A. M. Rogozhnikov, “Issledovanie smeshannoi zadachi, opisyvayuschei protsess kolebanii sterzhnya, sostoyaschego iz neskolkikh uchastkov s proizvolnymi dlinami”, Dokl. RAN, 444:5 (2012), 488–491 | MR | Zbl

[27] E. I. Moiseev, E. I. Likhomanenko, “Ob odnoi nelokalnoi zadache dlya uravneniya Lavrenteva-Bitsadze”, Dokl. RAN, 446:3 (2012), 256–258 | Zbl

[28] V. A. Trenogin, Funktsionalnyi analiz, Nauka, M., 1980 | MR | Zbl

[29] S. Ya. Yakubov, Lineinye differentsialno-operatornye uravneniya i ikh prilozheniya, Elm., Baku, 1985

[30] O. A. Ladyzhenskaya, “Prostoe dokazatelstvo razreshimosti osnovnykh kraevykh zadach i zadachi o sobstvennykh znacheniyakh dlya lineinykh ellipticheskikh uravnenii”, Vestnik LGU, 1955, no. 11, 23–29 | Zbl