Abstract
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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
G. Allaire: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482–1518.
T. Arbogast, J. Douglas, U. Hornung: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21 (1990), 823–836.
A. Bensoussan, J. L. Lions, G. Papanicolaou: Asymptotic Analysis for Periodic Structures. North-Holland, Amsterdam, 1978.
A. Bourgeat, A. Mikelić, S. Wright: Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math. 456 (1994), 19–51.
J. Casado-Díaz: Two-scale convergence for nonlinear Dirichlet problems in perforated domains. Proc. R. Soc. Edinb., Sect. A 130 (2000), 249–276.
D. Cioranescu, A. Damlamian, G. Griso: Periodic unfolding and homogenization. C.R.Math. Acad. Sci. Paris 335 (2002), 99–104.
D. Cioranescu, A. Damlamian, G. Griso: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40 (2008), 1585–1620.
A. Damlamian: An elementary introduction to periodic unfolding. In: Proceedings of the Narvik Conference 2004, GAKUTO International Series, Math. Sci. Appl. 24. Gakkotosho, Tokyo, 2006, pp. 119–136.
I. Ekeland, R. Temam: Convex analysis and variational problems. North-Holland, Amsterdam, 1976.
J. Franců: On two-scale convergence. In: Proceeding of the 6th MathematicalWorkshop, Faculty of Civil Engineering, Brno University of Technology, Brno, October 18, 2007, CD, 7 pages.
J. Franců: Modification of unfolding approach to two-scale convergence. Math. Bohem. 135 (2010), 403–412.
A. Holmbom, J. Silfver, N. Svanstedt, N. Wellander: On two-scale convergence and related sequential compactness topics. Appl. Math. 51 (2006), 247–262.
D. Lukkassen, G. Nguetseng, P. Wall: Two-scale convergence. Int. J. Pure Appl. Math. 2 (2002), 35–86.
F. Murat: Compacité par compensation. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 5 (1978), 489–507. (In French.)
L. Nechvátal: Alternative approaches to the two-scale convergence. Appl. Math. 49 (2004), 97–110.
G. Nguetseng: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), 608–623.
G. Nguetseng, N. Svanstedt: Σ-convergence. Banach J. Math. Anal. 2 (2011), 101–135. Open electronic access: www.emis.de/journals/BJMA/.
J. Silfver: On general two-scale convergence and its application to the characterization of G-limits. Appl. Math. 52 (2007), 285–302.
J. Silfver: Homogenization. PhD. Thesis. Mid-Sweden University, 2007.
V.V. Zhikov, E.V. Krivenko: Homogenization of singularly perturbed elliptic operators. Matem. Zametki 33 (1983), 571–582. (Engl. transl.: Math. Notes 33 (1983), 294–300).
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was supported by Grant No 201/08/0874 of the Grant Agency of Czech Republic.
The editor learnt with great sorrow that Nils E M Svanstedt, Professor of Mathematics at Chalmers University of Technology and the University of Gothenburg, unexpectedly passed away at the age of 53 on April 28, 2012.
Rights and permissions
About this article
Cite this article
Franců, J., Svanstedt, N.E.M. Some remarks on two-scale convergence and periodic unfolding. Appl Math 57, 359–375 (2012). https://doi.org/10.1007/s10492-012-0021-z
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10492-012-0021-z