Geodesic
Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account
Funkcionalʹnyj analiz i ego priloženiâ, Tome 54 (2020) no. 3, pp. 94-99.

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

An elliptic fourth-order differential operator $A_\varepsilon$ on $L_2(\mathbb{R}^d;\mathbb{C}^n)$ is studied. Here $\varepsilon >0$ is a small parameter. It is assumed that the operator is given in the factorized form $A_\varepsilon = b(\mathbf{D})^* g(\mathbf{x}/\varepsilon) b(\mathbf{D})$, where $g(\mathbf{x})$ is a Hermitian matrix-valued function periodic with respect to some lattice and $b(\mathbf{D})$ is a matrix second-order differential operator. We make assumptions ensuring that the operator $A_\varepsilon$ is strongly elliptic. The following approximation for the resolvent $(A_\varepsilon + I)^{-1}$ in the operator norm of $L_2(\mathbb{R}^d;\mathbb{C}^n)$ is obtained: $$ (A_{\varepsilon}+I)^{-1}=(A^{0}+I)^{-1}+\varepsilon K_{1}+\varepsilon^{2}K_{2}(\varepsilon)+O(\varepsilon^{3}). $$ Here $A^0$ is the effective operator with constant coefficients and $K_{1}$ and $K_{2}(\varepsilon)$ are certain correctors.
Keywords: periodic differential operators, homogenization, operator error estimates, effective operator, corrector. 
@article{FAA_2020_54_3_a7,
     author = {V. A. Sloushch and T. A. Suslina},
     title = {Homogenization of the {Fourth-Order} {Elliptic} {Operator} with {Periodic} {Coefficients} with {Correctors} {Taken} into {Account}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {94--99},
     publisher = {mathdoc},
     volume = {54},
     number = {3},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2020_54_3_a7/}
}
TY  - JOUR
AU  - V. A. Sloushch
AU  - T. A. Suslina
TI  - Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2020
SP  - 94
EP  - 99
VL  - 54
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2020_54_3_a7/
LA  - ru
ID  - FAA_2020_54_3_a7
ER  - 
%0 Journal Article
%A V. A. Sloushch
%A T. A. Suslina
%T Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2020
%P 94-99
%V 54
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2020_54_3_a7/
%G ru
%F FAA_2020_54_3_a7
V. A. Sloushch; T. A. Suslina. Homogenization of the Fourth-Order Elliptic Operator with Periodic Coefficients with Correctors Taken into Account. Funkcionalʹnyj analiz i ego priloženiâ, Tome 54 (2020) no. 3, pp. 94-99. http://geodesic.mathdoc.fr/item/FAA_2020_54_3_a7/

[1] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, Stud. Math. Appl., 5, North-Holland Publ. Co., Amsterdam–New York, 1978 | MR | Zbl

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

[3] V. V. Zhikov, S. M. Kozlov, O. A. Oleinik, Usrednenie differentsialnykh operatorov, Fizmatlit, M., 1993

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

[5] M. Sh. Birman, T. A. Suslina, Algebra i analiz, 17:5 (2005), 69–90

[6] M. Sh. Birman, T. A. Suslina, Algebra i analiz, 17:6 (2005), 1–104

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

[8] N. A. Veniaminov, Algebra i analiz, 22:5 (2010), 69–103 | MR

[9] A. A. Kukushkin, T. A. Suslina, Algebra i analiz, 28:1 (2016), 89–149 | MR

[10] V. V. Zhikov, Dokl. RAN, 403:3 (2005), 305–308 | MR | Zbl

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

[12] V. V. Zhikov, S. E. Pastukhova, UMN, 71:3 (2016), 27–122 | DOI | MR | Zbl

[13] S. E. Pastukhova, Algebra i analiz, 28:2 (2016), 204–226

[14] S. E. Pastukhova, Appl. Anal., 95:7 (2016), 1449–1466 | DOI | MR | Zbl