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

Kévin Santugini — 2013

ESAIM: Control, Optimisation and Calculus of Variations

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 L 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...

Homogenization of ferromagnetic multilayers in the presence of surface energies

Kévin Santugini-Repiquet — 2007

ESAIM: Control, Optimisation and Calculus of Variations

We study the homogenization process of ferromagnetic multilayers in the presence of surface energies: super-exchange, also called interlayer exchange coupling, and surface anisotropy. The two main difficulties are the non-linearity of the Landau-Lifshitz equation and the absence of a good sequence of extension operators for the multilayer geometry. First, we consider the case when surface anisotropy is the dominant term, then the case when the magnitude of the super-exchange interaction is...

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