Geodesic
Operator estimates for the averaging of the Riemann-Hilbert problem for the Beltrami equation with a locally periodic coefficient
Daghestan Electronic Mathematical Reports, Tome 16 (2021), pp. 51-61.

Voir la notice de l'article provenant de la source Math-Net.Ru

Local characteristics of mathematical models of strongly inhomogeneous media are usually described by functions of the form $a(\varepsilon^{-1} x)$, $b(x,\varepsilon^{-1} x)$, $c(\varepsilon^{-1} x,\delta^{-1} x)$, $d(\varepsilon^{-1} x,\delta^{-1} x,\gamma^{-1} x)$, etc., where $\varepsilon$, $\delta$, $\gamma,\ldots>0$ — small parameters, while functions $a$, $b$, $c$, $d$, $\ldots$ have an ordered structure (for example, they are periodic in variables $y=\varepsilon^{-1} x$, $z=\delta^{-1} x$, etc.). Consequently, the corresponding mathematical models are differential equations with rapidly oscillating coefficients. This work is devoted to estimates of the averaging error. We study the generalized Beltrami equation with a locally periodic coefficient $\mu(x,\varepsilon^{-1} x)$.
Keywords: Beltrami equation, averaging, asymptotic methods. 
@article{DEMR_2021_16_a3,
     author = {M. M. Sirazhudinov and L. M. Dzhabrailova},
     title = {Operator estimates for the averaging of the {Riemann-Hilbert} problem for the {Beltrami} equation with a locally periodic coefficient},
     journal = {Daghestan Electronic Mathematical Reports},
     pages = {51--61},
     publisher = {mathdoc},
     volume = {16},
     year = {2021},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DEMR_2021_16_a3/}
}
TY  - JOUR
AU  - M. M. Sirazhudinov
AU  - L. M. Dzhabrailova
TI  - Operator estimates for the averaging of the Riemann-Hilbert problem for the Beltrami equation with a locally periodic coefficient
JO  - Daghestan Electronic Mathematical Reports
PY  - 2021
SP  - 51
EP  - 61
VL  - 16
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DEMR_2021_16_a3/
LA  - ru
ID  - DEMR_2021_16_a3
ER  - 
%0 Journal Article
%A M. M. Sirazhudinov
%A L. M. Dzhabrailova
%T Operator estimates for the averaging of the Riemann-Hilbert problem for the Beltrami equation with a locally periodic coefficient
%J Daghestan Electronic Mathematical Reports
%D 2021
%P 51-61
%V 16
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DEMR_2021_16_a3/
%G ru
%F DEMR_2021_16_a3
M. M. Sirazhudinov; L. M. Dzhabrailova. Operator estimates for the averaging of the Riemann-Hilbert problem for the Beltrami equation with a locally periodic coefficient. Daghestan Electronic Mathematical Reports, Tome 16 (2021), pp. 51-61. http://geodesic.mathdoc.fr/item/DEMR_2021_16_a3/

[1] Zhikov V.V., Kozlov S.M., Oleinik O.A., Usrednenie differentsialnykh operatorov, Nauka, M., 1993 | MR

[2] Zhikov V.V., Pastukhova S.E., “Ob operatornykh otsenkakh v teorii usredneniya”, Uspekhi mat. nauk, 71:3 (2016), 27–122 | MR | Zbl

[3] Birman M.Sh., Suslina T.A., “Periodicheskie differentsialnye operatory vtorogo poryadka. Porogovye svoistva i usredneniya”, Algebra i analiz, 15:5 (2003), 1–108

[4] Borisov D.I., “Asimptotiki reshenii ellipticheskikh sistem s bystro ostsilliruyuschimi koeffitsientami”, Algebra i analiz, 20:2 (2008), 19–42

[5] Senik N.N., “Ob usrednenii nesamosopryazhennykh lokalno-periodicheskikh ellipticheskikh operatorov”, Funkts. analiz i ego pril., 51:2 (2017), 92–96 | MR | Zbl

[6] Pastukhova S.E., Tikhomirov R.N., “Operatornye otsenki povtornogo i lokalno-periodicheskogo usredneniya”, Doklady RAN, 415:3 (2007), 304–309 | Zbl

[7] Zhikov V.V., “Ob operatornykh otsenkakh v teorii usredneniya”, Doklady RAN, 403:3 (2007), 305–308

[8] Sirazhudinov M.M., “Asimptoticheskii metod usredneniya obobschennykh operatorov Beltrami”, Matem. sbornik, 208:4 (2017), 87–110 | MR | Zbl

[9] Sirazhudinov M.M., “Operatornye otsenki usredneniya obobschennykh uravnenii Beltrami”, Dagestanskie elektronnye matem. izvestiya, 2017, no. 7, 40–46

[10] Sirazhudinov M.M., Tikhomirova S.V., “Otsenki pogreshnosti usredneniya periodicheskoi zadachi dlya uravneniya Beltrami”, Differentsialnye uravneniya, 56:12 (2020), 1651–1659 | MR | Zbl

[11] Sirazhudinov M.M., Dzhamaludinova S.P., “Otsenki pogreshnosti usredneniya zadachi Rimana–Gilberta dlya uravneniya Beltrami s lokalno-periodicheskim koeffitsientom.”, Vestnik Dagestanskogo gos. universiteta. Seriya1: Estestvennye nauki, 36:4 (2021), 38–53

[12] Sirazhudinov M.M., “O G-skhodimosti i usrednenii obobschennykh operatorov Beltrami”, Matematicheskii sbornik, 199:5 (2008), 124–155 | MR

[13] Sirazhudinov M.M., “O kraevoi zadache Rimana–Gilberta ($L_2$-teoriya)”, Differentsialnye uravneniya, 24:1 (1988), 64–73 | MR | Zbl

[14] Sirazhudinov M.M., “O periodicheskikh resheniyakh odnoi ellipticheskoi sistemy pervogo poryadka”, Matem. zametki, 48:5 (1990), 153–155 | MR | Zbl