Geodesic
Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class~$H^1(\mathbb R^d)$
Algebra i analiz, Tome 18 (2006) no. 6, pp. 1-130.

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Investigation of a class of matrix periodic elliptic second-order differential operators $\mathcal A_\varepsilon$ in $\mathbb R^d$ with rapidly oscillating coefficients (depending on $\mathbf x/\varepsilon$) is continued. The homogenization problem in the small period limit is studied. Approximation for the resolvent $(\mathcal A_\varepsilon+I)^{-1}$ in the operator norm from $L_2(\mathbb R^d)$ to $H^1(\mathbb R^d)$ is obtained with an error of order $\varepsilon$. In this approximation, a corrector is taken into account. Moreover, the ($L_2\to L_2$)-approximations of the so-called fluxes are obtained. 
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M. Sh. Birman; T. A. Suslina. Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class~$H^1(\mathbb R^d)$. Algebra i analiz, Tome 18 (2006) no. 6, pp. 1-130. http://geodesic.mathdoc.fr/item/AA_2006_18_6_a0/

[1] N. S. Bakhvalov, G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh, Nauka, M., 1984 | MR | Zbl

[2] M. Sh. Birman, T. A. Suslina, “Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics”, Systems, Approximation, Singular Integral Operators, and Related Topics (Bordeaux, 2000), Oper. Theory: Adv. Appl., 129, Birkhäuser, 2001, 71–107 | MR | Zbl

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

[4] M. Sh. Birman, T. A. Suslina, “Porogovye approksimatsii rezolventy faktorizovannogo samosopryazhennogo semeistva s uchetom korrektora”, Algebra i analiz, 17:5 (2005), 69–90 | MR

[5] M. Sh. Birman, T. A. Suslina, “Usrednenie periodicheskikh ellipticheskikh differentsialnykh operatorov s uchetom korrektora”, Algebra i analiz, 17:6 (2005), 1–104 | MR

[6] M. Giaquinta, Multiple integrals in the calculus of variations and nonlinear elliptic systems, Princeton Univ. Press, 1983 | MR | Zbl

[7] G. Griso, “Error estimate and unfolding for periodic homogenization”, Asymptotic Analysis, 40 (2004), 269–286 | MR | Zbl

[8] G. Griso, “Interior error estimate for periodic homogenization”, C. R. Acad. Sci. Paris, Ser. I, 340 (2005), 251–254 | MR | Zbl

[9] V. V. Zhikov, “O nekotorykh otsenkakh iz teorii usredneniya”, Doklady RAN, 406:5 (2006), 597–601 | MR | Zbl

[10] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Usrednenie differentsialnykh operatorov, Fizmatgiz, M., 1993 | MR | Zbl

[11] V. V. Zhikov, S. E. Pastukhova, “Ob operatornykh otsenkakh dlya nekotorykh zadach teorii usredneniya”, Rus. J. Math. Phys., 12:4 (2005), 501–510 | MR

[12] T. Kato, Teoriya vozmuschenii lineinykh operatorov, Mir, M., 1972 | Zbl

[13] O. A. Ladyzhenskaya, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1964 | MR

[14] S. E. Pastukhova, “O nekotorykh otsenkakh iz usredneniya zadach teorii uprugosti”, Doklady RAN, 406:5 (2006), 604–608 | MR | Zbl

[15] T. A. Suslina, “Ob usrednenii periodicheskoi sistemy Maksvella”, Funkts. analiz i ego pril., 38:3 (2004), 90–94 | MR | Zbl

[16] T. A. Suslina, “Usrednenie statsionarnoi periodicheskoi sistemy Maksvella”, Algebra i analiz, 16:5 (2004), 162–244 | MR

[17] R. G. Shterenberg, “Primer periodicheskogo magnitnogo operatora Shredingera s vyrozhdennym nizhnim kraem spektra”, Algebra i analiz, 16:2 (2004), 177–185 | MR | Zbl

[18] R. G. Shterenberg, “O strukture nizhnego kraya spektra periodicheskogo magnitnogo operatora Shredingera s malym magnitnym potentsialom”, Algebra i analiz, 17:5 (2005), 232–243 | MR