Finding the coefficients of a~linear elliptic equation

R. A. Aliev

Sibirskij žurnal industrialʹnoj matematiki, Tome 19 (2016) no. 2, pp. 17-28.

Résumé

We study the inverse problems of finding the coefficients of a linear elliptic equation for various boundary conditions in a prescribed rectangle. Existence, uniqueness, and stability theorems are proved for solutions to the inverse problems for the particular statements studied in the paper. An iterative method is employed to construct a regularization algorithm for solving the inverse problems.

Détail

Keywords: inversion problem
Mots-clés : elliptic equation.

Bibliographie

[1] Lavrentev M. M., Romanov V. G. Shishatskii S. P., Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980 | MR

[2] Belolipetskii V. M., Shokin Yu. I., Matematicheskoe modelirovanie v zadachakh zaschity okruzhayuschei sredy, Nauka, Novosibirsk, 1997

[3] Iskenderov A. D., “Obratnaya zadacha ob opredelenii koeffitsientov ellipticheskogo uravneniya”, Differents. uravneniya, 15:5 (1979), 858–867 | MR | Zbl

[4] Klibanov M. V., “Dvumernaya obratnaya zadacha dlya odnogo ellipticheskogo uravneniya”, Differents. uravneniya, 19:6 (1983), 1072–1974 | MR

[5] Klibanov M. V., “Obratnye zadachi v tselom i karlemanovskie otsenki”, Differents. uravneniya, 20:6 (1984), 1035–1041 | MR | Zbl

[6] Klibanov M. V., “Edinstvennost v tselom obratnykh zadach dlya odnogo klassa differentsialnogo uravneniya”, Differents. uravneniya, 20:11 (1984), 1947–1953 | MR | Zbl

[7] Khaidarov A., “Odin klass obratnykh zadach dlya ellipticheskikh uravnenii”, Dokl. AN SSSR, 277:6 (1984), 1335–1337 | MR

[8] Khaidarov A., “Ob odnoi obratnoi zadache dlya ellipticheskikh uravnenii”, Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, Novosibirsk, 1984, 245–249

[9] Vabischevich P. N., “O edinstvennosti nekotorykh obratnykh zadach dlya ellipticheskikh uravnenii”, Differents. uravneniya, 24:12 (1988), 2125–2129 | MR

[10] Solovev V. V., “Obratnye zadachi dlya ellipticheskikh uravnenii v ploskosti. 1”, Differents. uravneniya, 42:8 (2006), 1106–1114 | MR | Zbl

[11] Solovev V. V., “Obratnye zadachi opredeleniya istochnika i koeffitsienta v ellipticheskom uravnenii v pryamougolnike”, Zhurn. vychisl. matematiki i mat. fiziki, 47:8 (2007), 1365–1377 | MR

[12] Solovev V. V., “Obratnaya zadacha opredeleniya koeffitsienta dlya ellipticheskii uravnenii v tsilindre. I”, Differents. uravneniya, 49:8 (2013), 1026–1035 | Zbl

[13] Solovev V. V., “Obratnaya zadacha opredeleniya koeffitsienta dlya ellipticheskikh uravnenii v tsilindre. II”, Differents. uravneniya, 49:12 (2013), 1607–1615 | Zbl

[14] Yang R., Ou Y., “Inverse coefficient problems for elliptic equations”, ANZIAM J., 49:2 (2008), 271–279 | DOI | MR

[15] Vakhitov I. S., “Obratnaya zadacha identifikatsii starshego koeffitsienta v uravnenii diffuzireaktsii”, Dalnevost. mat. zhurn., 10:2 (2010), 93–105 | MR

[16] Johantson T., “An iterative procedure for solving a Caure problem for second arder elliptic equations”, Math. Nachr., 272:1 (2004), 46–54 | DOI | MR

[17] Pyatkov S. G., “O nekotorykh obratnykh zadachakh dlya ellipticheskikh uravnenii i sistem”, Sib. zhurn. industr. matematiki, 13:4 (2010), 83–96 | MR | Zbl

[18] Aliev R. A., “Ob opredelenii neizvestnykh koeffitsientov pri starshikh proizvodnykh v lineinom ellipticheskom uravnenii”, Vestn. Samar. gos. tekh. un-ta. Ser. Fiz.-mat. nauki, 2014, no. 3, 31–43 | DOI

[19] Aliev R. A., “Obratnaya zadacha opredeleniya koeffitsienta v ellipticheskom uravnenii”, Vestn. Yuzhno-Ural. gos. un-ta. Ser. Matematika. Mekhanika. Fizika, 7:2 (2015), 5–13

[20] Ladyzhenskaya O. G., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[21] Miranda K., Uravneniya s chastnymi proizvodnymi ellipticheskogo tipa, Izd-vo inostr. lit., M., 1957