Previous |  Up |  Next

Article

Title: Alternative approaches to the two-scale convergence (English)
Author: Nechvátal, Luděk
Language: English
Journal: Applications of Mathematics
ISSN: 0862-7940
Volume: 49
Issue: 2
Year: 2004
Pages: 97-110
Summary lang: English
.
Category: math
.
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. (English)
Keyword: two-scale convergence
Keyword: weak convergence
Keyword: homogenization
MSC: 35B27
MSC: 74Q05
MSC: 76M50
idZBL: Zbl 1099.35012
idMR: MR2043076
DOI: 10.1023/B:APOM.0000027218.04167.9b
.
Date available: 2009-09-22T18:17:05Z
Last updated: 2015-05-17
Stable URL: http://hdl.handle.net/10338.dmlcz/134561
.
Reference: [1] G.  Allaire: Homogenization and two-scale convergence.SIAM J.  Math. Anal 23 (1992), 1482–1518. Zbl 0770.35005, MR 1185639, 10.1137/0523084
Reference: [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.
Reference: [3] G.  Allaire, M.  Briane: Multiscale convergence and reiterated homogenization.Proc. Roy. Soc. Edinburgh 126 (1996), 297–342. MR 1386865
Reference: [4] M.  Amar: Two-scale convergence and homogenization on  BV$(\Omega )$.Asymptot. Anal. 16 (1998), 65–84. MR 1600123
Reference: [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. MR 1052874, 10.1137/0521046
Reference: [6] E.  Balder: On compactness results for multi-scale convergence.Proc. Roy. Soc. Edinburgh 129 (1999), 467–476. Zbl 0946.46021, MR 1693641, 10.1017/S0308210500021466
Reference: [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. MR 1856245, 10.1137/S0036141000370260
Reference: [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
Reference: [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
Reference: [10] J.  Casado-Díaz: Two-scale convergence for nonlinear Dirichlet problems in perforated domains.Proc. Roy. Soc. Edinburgh 130 (2000), 249–276. MR 1750830
Reference: [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. MR 1817366, 10.1016/S0764-4442(00)01794-8
Reference: [12] D.  Cioranescu, A.  Damlamian, and G.  Griso: Periodic unfolding and homogenization.C.  R.  Acad. Sci. Paris Sér.  I Math. 335 (2002), 99–104. MR 1921004, 10.1016/S1631-073X(02)02429-9
Reference: [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
Reference: [14] N. D.  Botkin, K. H.  Hoffmann: Homogenization of von Kármán plates excited by piezoelectric patches.Z. Angew. Math. Mech. 80 (2000), 579–590. MR 1778561, 10.1002/1521-4001(200009)80:9<579::AID-ZAMM579>3.0.CO;2-2
Reference: [15] A.  Holmbom: Homogenization of parabolic equations—an alternative approach and some corrector-type results.Appl. Math. 47 (1997), 321–343. Zbl 0898.35008, MR 1467553, 10.1023/A:1023049608047
Reference: [16] M.  Lenczner: Homogénéisation d’un circuit électrique.C.  R.  Acad. Sci. Paris Sér. II  b 324 (1997), 537–542. (French) Zbl 0887.35016
Reference: [17] D.  Lukkassen, G.  Nguetseng, and P.  Wall: Two-scale convergence.Int. J.  Pure Appl. Math. 2 (2002), 35–86. MR 1912819
Reference: [18] G.  Nguetseng: A general convergence result for a functional related to the theory of homogenization.SIAM J.  Math. Anal. 20 (1989), 608–623. Zbl 0688.35007, MR 0990867, 10.1137/0520043
Reference: [19] M.  Valadier: Admissible functions in two-scale convergence.Port. Math. 54 (1997), 147–164. Zbl 0886.35018, MR 1467199
Reference: [20] S.  Wright: Time-dependent Stokes flow through a randomly perforated porous medium.Asymptot. Anal. 23 (2000), 257–272. Zbl 0973.76087, MR 1784454
Reference: [21] V. V.  Zhikov: On an extension and an application of the two-scale convergence method.Sb. Math. 191 (2000), 973–1014. MR 1809928, 10.1070/SM2000v191n07ABEH000491
.

Files

Files Size Format View
AplMat_49-2004-2_2.pdf 1.841Mb application/pdf View/Open
Back to standard record
Partner of
EuDML logo