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Keywords:
two-scale convergence; unfolding; homogenization
Summary:
The paper discusses some aspects of the adjoint definition of two-scale convergence based on periodic unfolding. As is known this approach removes problems concerning choice of the appropriate space for admissible test functions. The paper proposes a modified unfolding which conserves integral of the unfolded function and hence simplifies the proofs and its application in homogenization theory. The article provides also a self-contained introduction to two-scale convergence and gives ideas for generalization to non-periodic homogenization.
References:
[1] Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482-1518. DOI 10.1137/0523084 | MR 1185639 | Zbl 0770.35005
[2] Arbogast, T., Douglas, J., Hornung, U.: 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 | Zbl 0698.76106
[3] Bensoussan, A., Lions, J. L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures. North-Holland Amsterdam (1978). MR 0503330 | Zbl 0404.35001
[4] Bourgeat, A., Mikelić, A., Wright, S.: Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math. 456 (1994), 19-51. MR 1301450 | Zbl 0808.60056
[5] Casado-Díaz, J.: Two-scale convergence for nonlinear Dirichlet problems in perforated domains. Proc. R. Soc. Edinb., Sect. A 130 (2000), 249-276. DOI 10.1017/S0308210500000147 | MR 1750830
[6] Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization. C. R. Math. Acad. Sci. Paris 335 (2002), 99-104. DOI 10.1016/S1631-073X(02)02429-9 | MR 1921004 | Zbl 1001.49016
[7] Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40 (2008), 1585-1620. DOI 10.1137/080713148 | MR 2466168 | Zbl 1167.49013
[8] Damlamian, A.: An elementary introduction to periodic unfolding. In: Proceedings of the Narvik Conference 2004, GAKUTO International Series, Math. Sci. Appl. 24 Gakkotosho Tokyo (2006), 119-136. MR 2233174 | Zbl 1204.35038
[9] Ekeland, I., Temam, R.: Convex analysis and variational problems. North-Holland Amsterdam (1976). MR 0463994 | Zbl 0322.90046
[10] Franců, J.: On two-scale convergence. In: Proceeding of the 6th Mathematical Workshop, Faculty of Civil Engineering, Brno University of Technology, Brno, October 18, 2007, CD, 7 pages.
[11] Franců, J.: Modification of unfolding approach to two-scale convergence. Math. Bohem. 135 (2010), 403-412. MR 2681014 | Zbl 1224.35020
[12] Holmbom, A., Silfver, J., Svanstedt, N., Wellander, N.: On two-scale convergence and related sequential compactness topics. Appl. Math. 51 (2006), 247-262. DOI 10.1007/s10492-006-0014-x | MR 2228665 | Zbl 1164.40304
[13] Lukkassen, D., Nguetseng, G., Wall, P.: Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002), 35-86. MR 1912819 | Zbl 1061.35015
[14] Murat, F.: Compacité par compensation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 5 (1978), 489-507 French. MR 0506997 | Zbl 0399.46022
[15] Nechvátal, L.: 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
[16] Nguetseng, G.: 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
[17] Nguetseng, G., Svanstedt, N.: $\Sigma$-convergence. Banach J. Math. Anal. 2 (2011), 101-135 Open electronic access: www.emis.de/journals/BJMA/. MR 2738525 | Zbl 1229.46035
[18] Silfver, J.: On general two-scale convergence and its application to the characterization of G-limits. Appl. Math. 52 (2007), 285-302. DOI 10.1007/s10492-007-0015-4 | MR 2324728 | Zbl 1164.35318
[19] Silfver, J.: Homogenization. PhD. Thesis Mid-Sweden University (2007).
[20] Zhikov, V. V., Krivenko, E. V.: Homogenization of singularly perturbed elliptic operators. Matem. Zametki 33 (1983), 571-582 (Engl. transl.: Math. Notes {\it 33} (1983), 294-300). MR 0704444

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