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

Kévin Santugini

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 4, page 931-946
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Santugini, Kévin. "Homogenization at different linear scales, bounded martingales and the two-scale shuffle limit." ESAIM: Control, Optimisation and Calculus of Variations 19.4 (2013): 931-946. <http://eudml.org/doc/272787>.

@article{Santugini2013,
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.},
author = {Santugini, Kévin},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {homogenization; two-scale convergence; sequence of two-scale limits},
language = {eng},
number = {4},
pages = {931-946},
publisher = {EDP-Sciences},
title = {Homogenization at different linear scales, bounded martingales and the two-scale shuffle limit},
url = {http://eudml.org/doc/272787},
volume = {19},
year = {2013},
}

TY - JOUR
AU - Santugini, Kévin
TI - Homogenization at different linear scales, bounded martingales and the two-scale shuffle limit
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2013
PB - EDP-Sciences
VL - 19
IS - 4
SP - 931
EP - 946
AB - 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.
LA - eng
KW - homogenization; two-scale convergence; sequence of two-scale limits
UR - http://eudml.org/doc/272787
ER -

References

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  1. [1] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal.23 (1992) 1482–1518. Zbl0770.35005MR1185639
  2. [2] 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. 
  3. [3] G. Allaire and C. Conca, Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl.77 (1998) 153–208. Zbl0901.35005MR1614641
  4. [4] G. Ben Arous and H. Owhadi, Multiscale homogenization with bounded ratios and anomalous slow diffusion. Commun. Pure Appl. Math.56 (2003) 80–113. Zbl1205.76223MR1929443
  5. [5] D. Cioranescu and J. Saint Jean Paulin, Homogenization in open sets with holes. J. Math. Anal. Appl.71 (1979) 590–607. Zbl0427.35073MR548785
  6. [6] A. Damlamian and P. Donato, Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition? ESAIM: COCV 8 (2002) 555–585. Zbl1073.35020MR1932963
  7. [7] O. Kallenberg, Foundations of Modern Probability. Probability and its applications, 2nd edition. Springer (2002). Zbl0996.60001MR1876169
  8. [8] M. Neuss-Radu, Homogenization techniques. Diplomaarbeit. University of Heidelberg (1992). 
  9. [9] M. Neuss-Radu, Some extensions of two-scale convergence. C. R. Acad. Sci. Paris Sér. I Math.322 (1996) 899–904. Zbl0852.76087MR1390613
  10. [10] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal.20 (1989) 608–623. Zbl0688.35007MR990867
  11. [11] W. Rudin, Real and Complex Analysis. 3rd edition. McGraw-Hill, Inc. (1987). Zbl0278.26001MR924157
  12. [12] K. Santugini-Repiquet, Homogenization of ferromagnetic multilayers in the presence of surface energies. ESAIM: COCV 13 (2007) 305–330. Zbl1130.35310MR2306638
  13. [13] K. Santugini-Repiquet, Homogenization of the heat equation in multilayers with interlayer conduction. Proc. Roy. Soc. Edinburgh Sect. A137 (2007) 147–181. Zbl1133.35008MR2359777

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