Geodesic
On the convergence of bloch eigenfunctions in homogenization problems
Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 3, pp. 47-65.

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

We study the convergence of continuous spectrum eigenfunctions for differential operators of divergence type with $\varepsilon$-periodic coefficients, where $\varepsilon$ is a small parameter. Two cases are considered, the case of classical homogenization, where the coefficient matrix satisfies the ellipticity condition uniformly with respect to $\varepsilon$, and the case of two-scale homogenization, where the coefficient matrix has two phases and is highly contrast with hard-to-soft-phase contrast ratio $1\,{:}\,\varepsilon^2$.
Keywords: homogenization, two-scale convergence, Bloch principle, Bloch eigenfunction, double porosity model.
Mots-clés : convergence of spectra 
@article{FAA_2016_50_3_a3,
     author = {V. V. Zhikov and S. E. Pastukhova},
     title = {On the convergence of bloch eigenfunctions in homogenization problems},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
     pages = {47--65},
     publisher = {mathdoc},
     volume = {50},
     number = {3},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/FAA_2016_50_3_a3/}
}
TY  - JOUR
AU  - V. V. Zhikov
AU  - S. E. Pastukhova
TI  - On the convergence of bloch eigenfunctions in homogenization problems
JO  - Funkcionalʹnyj analiz i ego priloženiâ
PY  - 2016
SP  - 47
EP  - 65
VL  - 50
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/FAA_2016_50_3_a3/
LA  - ru
ID  - FAA_2016_50_3_a3
ER  - 
%0 Journal Article
%A V. V. Zhikov
%A S. E. Pastukhova
%T On the convergence of bloch eigenfunctions in homogenization problems
%J Funkcionalʹnyj analiz i ego priloženiâ
%D 2016
%P 47-65
%V 50
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/FAA_2016_50_3_a3/
%G ru
%F FAA_2016_50_3_a3
V. V. Zhikov; S. E. Pastukhova. On the convergence of bloch eigenfunctions in homogenization problems. Funkcionalʹnyj analiz i ego priloženiâ, Tome 50 (2016) no. 3, pp. 47-65. http://geodesic.mathdoc.fr/item/FAA_2016_50_3_a3/

[1] M. Rid, B. Saimon, Metody sovremennoi matematicheskoi fiziki, v. 4, Mir, M., 1982 | MR

[2] P. Kuchment, Floquet Theory for Partial Differential Equations, Birkhäuser, Basel, 1993 | MR | Zbl

[3] V. V. Zhikov, S. E. Pastukhova, “Bloch principle for elliptic differential operators with periodic coefficients”, Russian J. Math. Phys., 23:2 (2016), 1–21 | DOI | MR

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

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

[6] V. V. Zhikov, “Ob operatornykh otsenkakh v teorii usredneniya”, Dokl. RAN, 403:3 (2005), 305–308 | MR | Zbl

[7] V. V. Zhikov, “Ob odnom rasshirenii i primenenii metoda dvukhmasshtabnoi skhodimosti”, Matem. sb., 191:7 (2000), 31–72 | DOI | MR | Zbl

[8] V. V. Zhikov, “O dvukhmasshtabnoi skhodimosti”, Trudy sem. im. I. G. Petrovskogo, 23, 2003, 149–186

[9] G. Nguetseng, “A general convergence result for a functional related to the theory of homogenization”, SIAM J. Math. Anal., 20:3 (1989), 608–623 | DOI | MR | Zbl

[10] G. Allaire, “Homogenization and two-scale convergence”, SIAM J. Math. Anal., 23:6 (1992), 1482–1518 | DOI | MR | Zbl

[11] V. V. Zhikov, “O lakunakh v spektre divergentnykh ellipticheskikh operatorov s periodicheskimi koeffitsientami”, Algebra i analiz, 16:5 (2004), 34–58 | MR