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Keywords:
two-scale convergence; weak convergence; homogenization
two-scale convergence; weak convergence; homogenization
Summary:
Two-scale convergence is a special weak convergence used in homogenization theory. Besides the original definition by Nguetseng and Allaire two alternative definitions are introduced and compared. They enable us to weaken requirements on the admissibility of test functions $\psi (x,y)$. Properties and examples are added.
References:
[1] G. Allaire: Homogenization and two-scale convergence. SIAM J. Math. Anal 23 (1992), 1482–1518. DOI 10.1137/0523084 | MR 1185639 | Zbl 0770.35005
[2] G. Allaire, A. Damlamian, and V. Hornung: Two-scale convergence on periodic surfaces and applications. In: Proc. International Conference on Mathematical Modelling of Flow through Porous Media, World Scientific Pub., Singapore, 1995, pp. 15–25.
[3] G. Allaire, M. Briane: Multiscale convergence and reiterated homogenization. Proc. Roy. Soc. Edinburgh 126 (1996), 297–342. MR 1386865
[4] M. Amar: Two-scale convergence and homogenization on BV$(\Omega )$. Asymptot. Anal. 16 (1998), 65–84. MR 1600123
[5] T. Arbogast, J. Douglas, and U. Hornung: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21 (1990), 823–836. DOI 10.1137/0521046 | MR 1052874
[6] E. Balder: On compactness results for multi-scale convergence. Proc. Roy. Soc. Edinburgh 129 (1999), 467–476. DOI 10.1017/S0308210500021466 | MR 1693641 | Zbl 0946.46021
[7] G. Bouchitté, I. Fragalà: Homogenization of thin structures by two-scale method with respect to measures. SIAM J. Math. Anal. 32 (2001), 1198–1226. DOI 10.1137/S0036141000370260 | MR 1856245
[8] A. Bourgeat, A. Mikelic, and S. Wright: Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math. 456 (1994), 19–51. MR 1301450
[9] J. Casado-Díaz, I. Gayte: A general compactness result and its application to the two-scale convergence of almost periodic functions. C. R. Acad. Sci. Paris Sér. I Math. 323 (1996), 329–334. MR 1408763
[10] J. Casado-Díaz: Two-scale convergence for nonlinear Dirichlet problems in perforated domains. Proc. Roy. Soc. Edinburgh 130 (2000), 249–276. MR 1750830
[11] J. Casado-Díaz, M. Luna-Laynez, and J. D. Martín: An adaptation of the multi-scale methods for the analysis of very thin reticulated structures. C. R. Acad. Sci. Paris Sér. I Math. 332 (2001), 223–228. DOI 10.1016/S0764-4442(00)01794-8 | MR 1817366
[12] D. Cioranescu, A. Damlamian, and G. Griso: Periodic unfolding and homogenization. C. R. Acad. Sci. Paris Sér. I Math. 335 (2002), 99–104. DOI 10.1016/S1631-073X(02)02429-9 | MR 1921004
[13] G. W. Clark, R. E. Showalter: Two-scale convergence of a model for flow in a partially fissured medium. Electron. J. Differential Equations 2 (1999), 1–20. MR 1670679
[14] N. D. Botkin, K. H. Hoffmann: Homogenization of von Kármán plates excited by piezoelectric patches. Z. Angew. Math. Mech. 80 (2000), 579–590. DOI 10.1002/1521-4001(200009)80:9<579::AID-ZAMM579>3.0.CO;2-2 | MR 1778561
[15] A. Holmbom: Homogenization of parabolic equations—an alternative approach and some corrector-type results. Appl. Math. 47 (1997), 321–343. DOI 10.1023/A:1023049608047 | MR 1467553 | Zbl 0898.35008
[16] M. Lenczner: Homogénéisation d’un circuit électrique. C. R. Acad. Sci. Paris Sér. II b 324 (1997), 537–542. (French) Zbl 0887.35016
[17] D. Lukkassen, G. Nguetseng, and P. Wall: Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002), 35–86. MR 1912819
[18] G. Nguetseng: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), 608–623. DOI 10.1137/0520043 | MR 0990867 | Zbl 0688.35007
[19] M. Valadier: Admissible functions in two-scale convergence. Port. Math. 54 (1997), 147–164. MR 1467199 | Zbl 0886.35018
[20] S. Wright: Time-dependent Stokes flow through a randomly perforated porous medium. Asymptot. Anal. 23 (2000), 257–272. MR 1784454 | Zbl 0973.76087
[21] V. V. Zhikov: On an extension and an application of the two-scale convergence method. Sb. Math. 191 (2000), 973–1014. DOI 10.1070/SM2000v191n07ABEH000491 | MR 1809928
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