# Article

 Title: Some remarks on two-scale convergence and periodic unfolding (English) Author: Franců, Jan Author: Svanstedt, Nils E M Language: English Journal: Applications of Mathematics ISSN: 0862-7940 (print) ISSN: 1572-9109 (online) Volume: 57 Issue: 4 Year: 2012 Pages: 359-375 Summary lang: English . Category: math . 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. (English) Keyword: two-scale convergence Keyword: unfolding Keyword: homogenization MSC: 35B27 MSC: 35C20 MSC: 35J25 idZBL: Zbl 1265.35017 idMR: MR2984608 DOI: 10.1007/s10492-012-0021-z . Date available: 2012-08-19T21:43:48Z Last updated: 2020-07-02 Stable URL: http://hdl.handle.net/10338.dmlcz/142904 . Reference: [1] Allaire, G.: Homogenization and two-scale convergence.SIAM J. Math. Anal. 23 (1992), 1482-1518. Zbl 0770.35005, MR 1185639, 10.1137/0523084 Reference: [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. Zbl 0698.76106, MR 1052874, 10.1137/0521046 Reference: [3] Bensoussan, A., Lions, J. L., Papanicolaou, G.: Asymptotic Analysis for Periodic Structures.North-Holland Amsterdam (1978). Zbl 0404.35001, MR 0503330 Reference: [4] Bourgeat, A., Mikelić, A., Wright, S.: Stochastic two-scale convergence in the mean and applications.J. Reine Angew. Math. 456 (1994), 19-51. Zbl 0808.60056, MR 1301450 Reference: [5] Casado-Díaz, J.: Two-scale convergence for nonlinear Dirichlet problems in perforated domains.Proc. R. Soc. Edinb., Sect. A 130 (2000), 249-276. MR 1750830, 10.1017/S0308210500000147 Reference: [6] Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization.C. R. Math. Acad. Sci. Paris 335 (2002), 99-104. Zbl 1001.49016, MR 1921004, 10.1016/S1631-073X(02)02429-9 Reference: [7] Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization.SIAM J. Math. Anal. 40 (2008), 1585-1620. Zbl 1167.49013, MR 2466168, 10.1137/080713148 Reference: [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. Zbl 1204.35038, MR 2233174 Reference: [9] Ekeland, I., Temam, R.: Convex analysis and variational problems.North-Holland Amsterdam (1976). Zbl 0322.90046, MR 0463994 Reference: [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. Reference: [11] Franců, J.: Modification of unfolding approach to two-scale convergence.Math. Bohem. 135 (2010), 403-412. Zbl 1224.35020, MR 2681014 Reference: [12] Holmbom, A., Silfver, J., Svanstedt, N., Wellander, N.: On two-scale convergence and related sequential compactness topics.Appl. Math. 51 (2006), 247-262. Zbl 1164.40304, MR 2228665, 10.1007/s10492-006-0014-x Reference: [13] Lukkassen, D., Nguetseng, G., Wall, P.: Two-scale convergence.Int. J. Pure Appl. Math. 2 (2002), 35-86. Zbl 1061.35015, MR 1912819 Reference: [14] Murat, F.: Compacité par compensation.Ann. Sc. Norm. Super. Pisa, Cl. Sci. 5 (1978), 489-507 French. Zbl 0399.46022, MR 0506997 Reference: [15] Nechvátal, L.: Alternative approaches to the two-scale convergence.Appl. Math. 49 (2004), 97-110. Zbl 1099.35012, MR 2043076, 10.1023/B:APOM.0000027218.04167.9b Reference: [16] Nguetseng, G.: 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: [17] Nguetseng, G., Svanstedt, N.: $\Sigma$-convergence.Banach J. Math. Anal. 2 (2011), 101-135 Open electronic access: www.emis.de/journals/BJMA/. Zbl 1229.46035, MR 2738525 Reference: [18] Silfver, J.: On general two-scale convergence and its application to the characterization of G-limits.Appl. Math. 52 (2007), 285-302. Zbl 1164.35318, MR 2324728, 10.1007/s10492-007-0015-4 Reference: [19] Silfver, J.: Homogenization.PhD. Thesis Mid-Sweden University (2007). Reference: [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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