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.
@article{COCV_2013__19_4_931_0, author = {Santugini, K\'evin}, title = {Homogenization at different linear scales, bounded martingales and the two-scale shuffle limit}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {931--946}, publisher = {EDP-Sciences}, volume = {19}, number = {4}, year = {2013}, doi = {10.1051/cocv/2012039}, mrnumber = {3182675}, zbl = {1284.35051}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2012039/} }
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 SP - 931 EP - 946 VL - 19 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2012039/ DO - 10.1051/cocv/2012039 LA - en ID - COCV_2013__19_4_931_0 ER -
%0 Journal Article %A Santugini, Kévin %T Homogenization at different linear scales, bounded martingales and the two-scale shuffle limit %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 931-946 %V 19 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2012039/ %R 10.1051/cocv/2012039 %G en %F COCV_2013__19_4_931_0
Santugini, Kévin. Homogenization at different linear scales, bounded martingales and the two-scale shuffle limit. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 4, pp. 931-946. doi : 10.1051/cocv/2012039. http://www.numdam.org/articles/10.1051/cocv/2012039/
[1] Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992) 1482-1518. | MR | Zbl
,[2] 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.
, and ,[3] Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77 (1998) 153-208. | MR | Zbl
and ,[4] Multiscale homogenization with bounded ratios and anomalous slow diffusion. Commun. Pure Appl. Math. 56 (2003) 80-113. | MR | Zbl
and ,[5] Homogenization in open sets with holes. J. Math. Anal. Appl. 71 (1979) 590-607. | MR | Zbl
and ,[6] Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition? ESAIM: COCV 8 (2002) 555-585. | Numdam | MR | Zbl
and ,[7] Foundations of Modern Probability. Probability and its applications, 2nd edition. Springer (2002). | MR | Zbl
,[8] Homogenization techniques. Diplomaarbeit. University of Heidelberg (1992).
,[9] Some extensions of two-scale convergence. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 899-904. | MR | Zbl
,[10] A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | MR | Zbl
,[11] Real and Complex Analysis. 3rd edition. McGraw-Hill, Inc. (1987). | MR | Zbl
,[12] Homogenization of ferromagnetic multilayers in the presence of surface energies. ESAIM: COCV 13 (2007) 305-330. | EuDML | Numdam | MR | Zbl
,[13] Homogenization of the heat equation in multilayers with interlayer conduction. Proc. Roy. Soc. Edinburgh Sect. A 137 (2007) 147-181. | MR | Zbl
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