Stochastic homogenisation of high-contrast media

Kirill Cherednichenko, Mikhail Cherdantsev, Igor Velcic

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Abstract

Using a suitable stochastic version of the compactness argument of [Zhikov VV. On an extension of the method of two-scale convergence and its applications. Sb Math. 2000;191(7–8):973–1014], we develop a probabilistic framework for the analysis of heterogeneous media with high contrast. We show that an appropriately defined multiscale limit of the field in the original medium satisfies a system of equations corresponding to the coupled ‘macroscopic’ and ‘microscopic’ components of the field, giving rise to an analogue of the ‘Zhikov function’, which represents the effective dispersion of the medium. We demonstrate that, under some lenient conditions within the new framework, the spectra of the original problems converge to the spectrum of their homogenisation limit.

Original languageEnglish
Pages (from-to)91-117
Number of pages28
JournalApplicable Analysis
Volume98
Issue number1-2
Early online date6 Aug 2018
DOIs
Publication statusPublished - 2019

Keywords

  • Alexander Pankov
  • High contrast
  • random media
  • stochastic homogenisation

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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