Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
two-scale convergence; compensated compactness; two-scale transform; unfolding
two-scale convergence; compensated compactness; two-scale transform; unfolding
Summary:
A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in $L^{2}(\Omega )$ involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced.
References:
[1] G. Allaire: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482–2518. DOI 10.1137/0523084 | MR 1185639 | Zbl 0770.35005
[2] H. W. Alt: Lineare Funktionalanalysis. Springer-Verlag, Berlin, 1985. Zbl 0577.46001
[3] G. Allaire, M. Briane: Multiscale convergence and reiterated homogenization. Proc. Roy. Soc. Edinb. 126 (1996), 297–342. MR 1386865
[4] M. Amar: Two-scale convergence and homogenization on ${\mathrm BV}(\Omega )$. Asymptotic Anal. 16 (1998), 65–84. MR 1600123
[5] B. Birnir, N. Svanstedt, and N. Wellander: Two-scale compensated compactness. Submitted.
[6] 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
[7] J. Casado-Diaz, I. Gayte: A general compactness result and its application to the two-scale convergence of almost periodic functions. C. R. Acad. Sci. Paris, Série I 323 (1996), 329–334. MR 1408763
[8] D. Cioranescu, A. Damlamian, and G. Griso: Periodic unfolding and homogenization. C. R. Math., Acad. Sci. Paris 335 (2002), 99–104. DOI 10.1016/S1631-073X(02)02429-9 | MR 1921004
[9] R. E. Edwards: Functional Analysis. Holt, Rinehart and Winston, New York, 1965. MR 0221256 | Zbl 0182.16101
[10] A. Holmbom, N. Svanstedt, and N. Wellander: Multiscale convergence and reiterated homogenization for parabolic problems. Appl. Math 50 (2005), 131–151. DOI 10.1007/s10492-005-0009-z | MR 2125155
[11] J.-L. Lions, D. Lukkassen, L.-E. Persson, and P. Wall: Reiterated homogenization of nonlinear monotone operators. Chin. Ann. Math., Ser. B 22 (2001), 1–12. DOI 10.1142/S0252959901000024 | MR 1823125
[12] D. Lukkassen, G. Nguetseng, and P. Wall: Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002), 35–86. MR 1912819
[13] M. L. Mascarenhas, A.-M. Toader: Scale convergence in homogenization. Numer. Funct. Anal. Optimization 22 (2001), 127–158. DOI 10.1081/NFA-100103791 | MR 1841866
[14] L. Nechvátal: Alternative approaches to the two-scale convergence. Appl. Math. 49 (2004), 97–110. DOI 10.1023/B:APOM.0000027218.04167.9b | MR 2043076 | Zbl 1099.35012
[15] 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
[16] L. Tartar: Compensated compactness and applications to partial differential equations. Nonlinear analysis and mechanics: Heriot-Watt Symposium, Vol. IV. Res. Notes Math. 39, (ed.), Pitman, San Francisco, 1979, pp. 136–212. MR 0584398 | Zbl 0437.35004
Search
Partner of
