On two-scale convergence and related sequential compactness topics. (English) Zbl 1164.40304

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.


40A30 Convergence and divergence of series and sequences of functions
Full Text: DOI EuDML Link


[1] G. Allaire: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482–2518. · Zbl 0770.35005
[2] H. W. Alt: Lineare Funktionalanalysis. Springer-Verlag, Berlin, 1985.
[3] G. Allaire, M. Briane: Multiscale convergence and reiterated homogenization. Proc. Roy. Soc. Edinb. 126 (1996), 297–342. · Zbl 0866.35017
[4] M. Amar: Two-scale convergence and homogenization on BV({\(\Omega\)}). Asymptotic Anal. 16 (1998), 65–84.
[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. · Zbl 0808.60056
[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. · Zbl 0865.46003
[8] D. Cioranescu, A. Damlamian, and G. Griso: Periodic unfolding and homogenization. C. R. Math., Acad. Sci. Paris 335 (2002), 99–104. · Zbl 1001.49016
[9] R. E. Edwards: Functional Analysis. Holt, Rinehart and Winston, New York, 1965.
[10] A. Holmbom, N. Svanstedt, and N. Wellander: Multiscale convergence and reiterated homogenization for parabolic problems. Appl. Math 50 (2005), 131–151. · Zbl 1099.35011
[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. · Zbl 0979.35047
[12] D. Lukkassen, G. Nguetseng, and P. Wall: Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002), 35–86. · Zbl 1061.35015
[13] M. L. Mascarenhas, A.-M. Toader: Scale convergence in homogenization. Numer. Funct. Anal. Optimization 22 (2001), 127–158. · Zbl 0995.49013
[14] L. Nechvátal: Alternative approaches to the two-scale convergence. Appl. Math. 49 (2004), 97–110. · 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. · 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. Pitman, San Francisco, 1979, pp. 136–212.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.