On the Recovery of Solutions of a Generalized Cauchy--Riemann System in a Multidimensional Spatial Domain from Their Values on a Piece of the Boundary of This Domain

È. N. Sattorov ; F. E. Ermamatova

Matematičeskie zametki, Tome 110 (2021) no. 3, pp. 405-423.

Résumé

The paper deals with the problem of recovering solutions of a generalized Cauchy–Riemann system in a multidimensional spatial domain from their values on a piece of the boundary of this domain, i.e., an approximate solution of this problem based on the Carleman–Yarmukhamedov matrix method is constructed.

Détail

Keywords: generalized Cauchy–Riemann system, Cauchy problem, ill-posed problem, regularized solution, approximate solution
Mots-clés : Carleman matrix.

Bibliographie

[1] E. N. Sattorov, F. E. Ermamatova, “Formula Karlemana dlya reshenii obobschennoi sistemy Koshi–Rimana v mnogomernoi prostranstvennoi oblasti”, Sovremennye problemy matematiki i fiziki, SMFN, 65, no. 1, Rossiiskii universitet druzhby narodov, M., 2019, 95–108 | DOI

[2] E. I. Obolashvili, “Obobschennaya sistema Koshi–Rimana v mnogomernom evklidovom prostranstve”, Sb. dokl. Mezhdunarodnoi konferentsii po kompleksnomu analizu i ego primeneniyam k uravneniyam s chastnymi proizvodnymi (Galle, GDR, 18–24 oktyabrya 1976 g.), Galle, 1977, 36–39

[3] E. I. Obolashvili, “Obobschennaya sistema Koshi–Rimana v mnogomernom prostranstve”, Tr. Tbilis. matem. In-ta, 58, Tbilisi, 1978, 168–173

[4] I. Stein, G. Veis, Vvedenie v garmonicheskii analiz na evklidovykh prostranstvakh, Mir, M., 1974 | Zbl

[5] V. S. Vladimirov, I. V. Volovich, “Superanaliz. I. Differentsialnoe ischislenie”, TMF, 59:1 (1984), 3–27 | MR | Zbl

[6] V. S. Vladimirov, I. V. Volovich, “Superanaliz. II. Integralnoe ischislenie”, TMF, 60:2 (1984), 169–198 | MR | Zbl

[7] F. Brackx, K. Delanghe, F. Sommen, Clifford Analysis, Res. Notes in Math., 76, Pitman, Boston, MA, 1982 | MR

[8] Zh. Adamar, Zadacha Koshi dlya lineinykh uravnenii s chastnymi proizvodnymi giperbolicheskogo tipa, Nauka, M., 1978 | MR

[9] M. M. Lavrentev, O nekotorykh nekorrektnykh zadachakh matematicheskoi fiziki, VTs SO AN SSSR, Novosibirsk, 1962 | MR

[10] A. Bers, F. Dzhon, M. Shekhter, Uravneniya s chastnymi proizvodnymi, Mir, M., 1966 | MR | Zbl

[11] Sh. Yarmukhamedov, “O zadache Koshi dlya uravneniya Laplasa”, Dokl. AN SSSR, 235:2 (1977), 281–283 | MR | Zbl

[12] Sh. Yarmukhamedov, “O prodolzhenii resheniya uravneniya Gelmgoltsa”, Dokl. AN, 357:3 (1997), 320–323 | MR

[13] A. N. Tikhonov, “O reshenii nekorrektno postavlennykh zadach i metode regulyarizatsii”, Dokl. AN SSSR, 151:3 (1963), 501–504 | MR | Zbl

[14] V. K. Ivanov, “Zadacha Koshi dlya uravneniya Laplasa v beskonechnoi polose”, Differents. uravneniya, 1:1 (1965), 131–136 | MR | Zbl

[15] M. M. Lavrentev, “O zadache Koshi dlya lineinykh ellipticheskikh uravnenii vtorogo poryadka”, Dokl. AN SSSR, 112:2 (1957), 195–197 | MR | Zbl

[16] S. N. Mergelyan, “Garmonicheskaya approksimatsiya i priblizhennoe reshenie zadachi Koshi dlya uravneniya Laplasa”, UMN, 11:5 (71) (1956), 3–26 | MR | Zbl

[17] L. A. Aizenberg, N. N. Tarkhanov, “Abstraktnaya formula Karlemana”, Dokl. AN SSSR, 298:6 (1988), 1292–1296 | MR | Zbl

[18] O. I. Makhmudov, “Zadacha Koshi dlya sistemy uravnenii teorii uprugosti i termouprugosti v prostranstve”, Izv. vuzov. Matem., 2004, no. 2, 43–53 | MR | Zbl

[19] O. Makhmudov, I. Niyozov, N. Tarkhanov, “The Cauchy problem of couple-stress elasticity”, Complex Analysis and Dynamical Systems III, Contemp. Math., 455, Amer. Math. Soc., Providence, RI, 2008, 297–310 | DOI | MR

[20] E. N. Sattorov, Dzh. A. Mardonov, “Zadacha Koshi dlya sistemy uravnenii Maksvella”, Sib. matem. zhurn., 44:4 (2003), 851–861 | MR | Zbl

[21] E. N. Sattorov, “Regulyarizatsiya resheniya zadachi Koshi dlya obobschennoi sistemy Moisila–Teodoresku”, Differents. uravneniya, 44:8 (2008), 1100–1110 | MR

[22] E. N. Sattorov, “O prodolzhenii reshenii obobschennoi sistemy Koshi–Rimana v prostranstve”, Matem. zametki, 85:5 (2009), 768–781 | DOI | MR | Zbl

[23] E. N. Sattorov, “Regulyarizatsiya resheniya zadachi Koshi dlya sistemy uravnenii Maksvella v beskonechnoi oblasti”, Matem. zametki, 86:3 (2009), 445–455 | DOI | MR | Zbl

[24] E. N. Sattorov, “O vosstanovlenii reshenii obobschennoi sistemy Moisila–Teodoresku v prostranstvennoi oblasti po ikh znacheniyam na kuske granitsy”, Izv. vuzov. Matem., 2011, no. 1, 72–84 | MR

[25] Sh. Yarmukhamedov, “Ob analiticheskom prodolzhenii golomorfnogo vektora po ego granichnym znacheniyam na kuske granitsy”, Izv. AN UzSSR. Ser. fiz.-matem. nauk, 1980, no. 6, 34–40

[26] K. O. Makhmudov, O. I. Makhmudov, N. Tarkhanov, “Equations of Maxwell Type”, J. Math. Anal. Appl., 378:1 (2011), 64–75 | DOI | MR

[27] E. N. Sattorov, Z. E. Ermamatova, “O vosstanovlenii reshenii obobschennoi sistemy Moisila–Teodoresku v prostranstvennoi oblasti po ikh znacheniyam na kuske granitsy”, Izv. vuzov. Matem., 2011, no. 1, 72–84 | MR

[28] N. N. Tarkhanov, “O matritse Karlemana dlya ellipticheskikh sistem”, Dokl. AN SSSR, 284:2 (1985), 294–297 | MR | Zbl

[29] L. A. Aizenberg, Formuly Karlemana v kompleksnom analize. Pervye prilozheniya, Nauka, Novosibirsk, 1990 | MR | Zbl

[30] N. N. Tarkhanov, The Cauchy Problem for Solutions of Elliptic Equations, Math. Top., 7, Akademie Verlag, Berlin, 1995 | MR

[31] I. N. Vekua, Obobschennye analiticheskie funktsii, Fizmatlit, M., 1959 | MR | Zbl

[32] T. Ishankulov, “O vozmozhnosti obobschenno-analiticheskogo prodolzheniya v oblast funktsii, zadannykh na kuske ee granitsy”, Sib. matem. zhurn., 41:6 (2000), 1350–1356 | MR | Zbl

[33] E. I. Obolashvili, “Prostranstvennyi analog obobschennykh analiticheskikh funktsii”, Soobsch. AN GSSR, 73:1 (1974), 20–24

[34] L. F. Nikiforov, V. B. Uvarov, Osnovy teorii spetsialnykh funktsii, Nauka, M., 1974 | MR

[35] F. Trikomi, Lektsii po uravneniyam v chastnykh proizvodnykh, IL, M., 1957 | MR | Zbl