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Homogenization at different linear scales, bounded martingales and the two-scale shuffle limit

Published online by Cambridge University Press:  04 July 2013

Kévin Santugini
Kévin Santugini*
Affiliation:
University Bordeaux, IMB, UMR 5251, 33400 Talence, France. [email protected] CNRS, IMB, UMR 5251, 33400 Talence, France INRIA, 33400 Talence, France
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Abstract

In this paper, we consider two-scale limits obtained with increasing homogenization periods, each period being an entire multiple of the previous one. We establish that, up to a measure preserving rearrangement, these two-scale limits form a martingale which is bounded: the rearranged two-scale limits themselves converge both strongly in L2 and almost everywhere when the period tends to  +∞. This limit, called the Two-Scale Shuffle limit, contains all the information present in all the two-scale limits in the sequence.

Keywords

Homogenizationtwo-scale convergence
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 931 - 946
Copyright
© EDP Sciences, SMAI, 2013

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References

Allaire, G., Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482–1518. Google Scholar
G. Allaire, Alain Damlamian and Ulrich Hornung, Two-scale convergence on periodic surfaces and applications, in Proc. International Conference on Mathematical Modelling of Flow through Porous Media, edited by A. Bourgeat et al. World Scientific Pub., Singapore (1995) 15–25.
Allaire, G. and Conca, C., Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153–208. Google Scholar
Ben Arous, G. and Owhadi, H., Multiscale homogenization with bounded ratios and anomalous slow diffusion. Commun. Pure Appl. Math. 56 (2003) 80–113. Google Scholar
Cioranescu, D. and Saint Jean Paulin, J., Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590–607. Google Scholar
Damlamian, A. and Donato, P., Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition? ESAIM: COCV 8 (2002) 555–585. Google Scholar
O. Kallenberg, Foundations of Modern Probability. Probability and its applications, 2nd edition. Springer (2002).
M. Neuss-Radu, Homogenization techniques. Diplomaarbeit. University of Heidelberg (1992).
Neuss-Radu, M., Some extensions of two-scale convergence. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 899–904. Google Scholar
Nguetseng, G., A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608–623. Google Scholar
W. Rudin, Real and Complex Analysis. 3rd edition. McGraw-Hill, Inc. (1987).
Santugini-Repiquet, K., Homogenization of ferromagnetic multilayers in the presence of surface energies. ESAIM: COCV 13 (2007) 305–330. Google Scholar
Santugini-Repiquet, K., Homogenization of the heat equation in multilayers with interlayer conduction. Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 147–181. Google Scholar