Geodesic
Homogenization of the Elliptic Dirichlet Problem: Error Estimates in the $(L_2\to H^1)$-Norm
Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 2, pp. 92-96.

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

Let $\mathcal{O} \subset \mathbb{R}^d$ be a bounded domain with boundary of class $C^{1,1}$. In $L_2(\mathcal{O};\mathbb{C}^n)$, consider a matrix elliptic second-order differential operator $A_{D,\varepsilon}$ with Dirichlet boundary condition. Here $\varepsilon >\nobreak0$ is a small parameter; the coefficients of $A_{D,\varepsilon}$ are periodic and depend on $\mathbf{x}/\varepsilon$. The operator $A_{D,\varepsilon}^{-1}$ in the norm of operators acting from $L_2(\mathcal{O};\mathbb{C}^n)$ to the Sobolev space $H^1(\mathcal{O};\mathbb{C}^n)$ is approximated with an error of order $\varepsilon^{1/2}$. The approximation is given by the sum of the operator $(A^0_D)^{-1}$ and a first-order corrector. Here $A^0_D$ is an effective operator with constant coefficients and Dirichlet boundary condition.
Keywords: homogenization of periodic differential operators, effective operator, corrector, operator error estimates. 
@article{FAA_2012_46_2_a9,
     author = {M. A. Pakhnin and T. A. Suslina},
     title = {Homogenization of the {Elliptic} {Dirichlet} {Problem:} {Error} {Estimates} in the $(L_2\to H^1)${-Norm}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {92--96},
     publisher = {mathdoc},
     volume = {46},
     number = {2},
     year = {2012},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2012_46_2_a9/}
}
TY  - JOUR
AU  - M. A. Pakhnin
AU  - T. A. Suslina
TI  - Homogenization of the Elliptic Dirichlet Problem: Error Estimates in the $(L_2\to H^1)$-Norm
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2012
SP  - 92
EP  - 96
VL  - 46
IS  - 2
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2012_46_2_a9/
LA  - ru
ID  - FAA_2012_46_2_a9
ER  - 
%0 Journal Article
%A M. A. Pakhnin
%A T. A. Suslina
%T Homogenization of the Elliptic Dirichlet Problem: Error Estimates in the $(L_2\to H^1)$-Norm
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2012
%P 92-96
%V 46
%N 2
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2012_46_2_a9/
%G ru
%F FAA_2012_46_2_a9
M. A. Pakhnin; T. A. Suslina. Homogenization of the Elliptic Dirichlet Problem: Error Estimates in the $(L_2\to H^1)$-Norm. Funkcionalʹnyj analiz i ego priloženiâ, Tome 46 (2012) no. 2, pp. 92-96. http://geodesic.mathdoc.fr/item/FAA_2012_46_2_a9/

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

[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] T. A. Suslina, arXiv: 1201.2286

[11] T. A. Suslina, Funkts. analiz i ego pril. (to appear)

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