Geodesic
Operator Error Estimates in $L_2$ for Homogenization of an Elliptic Dirichlet Problem
Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 3, pp. 91-96 Cet article a éte moissonné depuis la source Math-Net.Ru

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In a bounded domain ${\mathcal O} \subset {\mathbb R}^d$ with $C^{1,1}$ boundary a matrix elliptic second-order operator ${A}_{D,\varepsilon}$ with Dirichlet boundary condition is studied. The coefficients of this operator are periodic and depend on $\mathbf{x}/\varepsilon$, where $\varepsilon >0$ is a small parameter. The sharp-order error estimate $\|{A}_{D,\varepsilon}^{-1} - ({A}_D^0)^{-1} \|_{L_2 \to L_2} \le C \varepsilon$ is obtained. Here ${A}^0_D$ is an effective operator with constant coefficients and Dirichlet boundary condition.
Keywords: periodic differential operators, homogenization, effective operator, operator error estimates. 
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T. A. Suslina. Operator Error Estimates in $L_2$ for Homogenization of an Elliptic Dirichlet Problem. Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 3, pp. 91-96. http://geodesic.mathdoc.fr/item/FAA_2012_46_3_a8/

[1] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic Analysis for Periodic Structures, Stud. Math. Appl., 5, North-Holland Publishing Co., Amsterdam–New York, 1978 | MR

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

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

[4] M. Sh. Birman, T. A. Suslina, Algebra i analiz, 15:5 (2003), 1–108 | MR | Zbl

[5] M. Sh. Birman, T. A. Suslina, Algebra i analiz, 18:6 (2006), 1–130 | MR

[6] V. V. Zhikov, Dokl. RAN, 406:5 (2006), 597–601 | MR | Zbl

[7] V. V. Zhikov, S. E. Pastukhova, Russ. J. Math. Phys., 12:4 (2005), 515–524 | MR | Zbl

[8] G. Griso, Asymptot. Anal., 40:3–4 (2004), 269–286 | MR | Zbl

[9] G. Griso, Anal. Appl., 4:1 (2006), 61–79 | DOI | MR | Zbl

[10] M. A. Pakhnin, T. A. Suslina, arXiv: 1201.2140 | MR

[11] M. A. Pakhnin, T. A. Suslina, Funkts. analiz i ego pril., 46:2, 92–96 | DOI | MR | Zbl

[12] T. A. Suslina, arXiv: 1201.2286

[13] C. E. Kenig, F. Lin, Z. Shen, Arch. Rat. Mech. Anal., 203:3 (2012), 1009–1036 | DOI | MR | Zbl