Geodesic
An asymptotic method for homogenization for generalized Beltrami operators
Sbornik. Mathematics, Tome 208 (2017) no. 4, pp. 546-567 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Questions on the homogenization of the Riemann-Hilbert problem for the generalized Beltrami equation with rapidly oscillating periodic coefficients (with small period $\varepsilon$) are considered. Estimates of order $\sqrt\varepsilon$ are obtained for the homogenization error in Lebesgue and Sobolev spaces. Asymptotic methods of homogenization are used. Bibliography: 16 titles.
Keywords: Beltrami's equation, homogenization, asymptotic methods. 
@article{SM_2017_208_4_a4,
     author = {M. M. Sirazhudinov},
     title = {An asymptotic method for homogenization for generalized {Beltrami} operators},
     journal = {Sbornik. Mathematics},
     pages = {546--567},
     year = {2017},
     volume = {208},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2017_208_4_a4/}
}
TY  - JOUR
AU  - M. M. Sirazhudinov
TI  - An asymptotic method for homogenization for generalized Beltrami operators
JO  - Sbornik. Mathematics
PY  - 2017
SP  - 546
EP  - 567
VL  - 208
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/SM_2017_208_4_a4/
LA  - en
ID  - SM_2017_208_4_a4
ER  - 
%0 Journal Article
%A M. M. Sirazhudinov
%T An asymptotic method for homogenization for generalized Beltrami operators
%J Sbornik. Mathematics
%D 2017
%P 546-567
%V 208
%N 4
%U http://geodesic.mathdoc.fr/item/SM_2017_208_4_a4/
%G en
%F SM_2017_208_4_a4
M. M. Sirazhudinov. An asymptotic method for homogenization for generalized Beltrami operators. Sbornik. Mathematics, Tome 208 (2017) no. 4, pp. 546-567. http://geodesic.mathdoc.fr/item/SM_2017_208_4_a4/

[1] V. V. Jikov, S. M. Kozlov, O. A. Oleinik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994, xii+570 pp. | DOI | MR | MR | Zbl | Zbl

[2] N. Bakhvalov, G. Panasenko, Homogenisation: averaging processes in periodic media. Mathematical problems in the mechanics of composite materials, Math. Appl. (Soviet Ser.), 36, Kluwer Acad. Publ., Dordrecht, 1989, xxxvi+366 pp. | DOI | MR | MR | Zbl | Zbl

[3] M. M. Sirazhudinov, “$G$-convergence and homogenization of generalized Beltrami operators”, Sb. Math., 199:5 (2008), 755–786 | DOI | DOI | MR | Zbl

[4] M. M. Sirazhudinov, R. M. Sirazhudinov, “$G$-convergence of systems of generalized Beltrami equations”, Proc. Steklov Inst. Math., 261 (2008), 262–269 | DOI | MR | Zbl

[5] M. M. Sirazhudinov, “$G$-compactness of a class of first-order elliptic systems”, Differ. Equ., 26:2 (1990), 229–235 | MR | Zbl

[6] V. V. Zhikov, M. M. Sirazhudinov, “The averaging of a system of Beltrami equations”, Differ. Equ., 24:1 (1988), 50–56 | MR | Zbl

[7] I. N. Vekua, Generalized analytic functions, Pergamon Press, London–Paris–Frankfurt; Addison-Wesley Publishing Co., Inc., Reading, Mass., 1962, xxix+668 pp. | MR | MR | Zbl | Zbl

[8] M. M. Sirazhudinov, “On the Riemann–Hilbert boundary value problem ($L_2$-theory)”, Differ. Equ., 25:8 (1989), 999–1003 | MR | Zbl

[9] N. S. Bakhvalov, “Osrednennye kharakteristiki tel s periodicheskoi strukturoi”, Dokl. AN SSSR, 218:5 (1974), 1046–1048 | MR

[10] N. S. Bakhvalov, “Averaging of partial differential equations with rapidly oscillating coefficients”, Soviet Math. Dokl., 16 (1975), 351–355 | MR | Zbl

[11] E. Sanchez-Palencia, Non-homogeneous media and vibration theory, Lecture Notes in Phys., 127, Springer-Verlag, Berlin–New York, 1980, ix+398 pp. | DOI | MR | MR | Zbl

[12] B. V. Boyarskii, “Obobschennye resheniya sistemy differentsialnykh uravnenii pervogo poryadka ellipticheskogo tipa s razryvnymi koeffitsientami”, Matem. sb., 43(85):4 (1957), 451–503 | MR | Zbl

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

[14] V. A. Solonnikov, “General boundary value problems for Douglis–Nirenberg elliptic systems. II”, Proc. Steklov Inst. Math., 92 (1968), 269–339 | MR | Zbl

[15] O. A. Ladyzhenskaya, N. N. Ural'tseva, Linear and quasilinear elliptic equations, Academic Press, New York–London, 1968, xviii+495 pp. | MR | MR | Zbl | Zbl

[16] A. A. Samarskii, R. D. Lazarov, V. L. Makarov, Raznostnye skhemy dlya differentsialnykh uravnenii s obobschennymi resheniyami, Vysshaya shkola, M., 1987, 296 pp.