Abstract
ABSTRACTUsing 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.
Highlights
Asymptotic analysis of differential equations with rapidly oscillating coefficients has featured prominently among the interests of the applied analysis community during the last half a century
The problem of understanding and quantifying the overall behaviour of heterogeneous media has emerged as a natural step within the general progress of material science, wave propagation and mathematical physics
In this period several frameworks have been developed for the analysis of families of differential operators, functionals and random processes describing multiscale media, all of which have benefitted from the invariably deep insight and mathematical elegance of the work of V
Summary
Asymptotic analysis of differential equations with rapidly oscillating coefficients has featured prominently among the interests of the applied analysis community during the last half a century. The problem of understanding and quantifying the overall behaviour of heterogeneous media has emerged as a natural step within the general progress of material science, wave propagation and mathematical physics In this period several frameworks have been developed for the analysis of families of differential operators, functionals and random processes describing multiscale media, all of which have benefitted from the invariably deep insight and mathematical elegance of the work of V. The present manuscript is the first work containing an analysis of random heterogeneous media with high contrast that results in a “complete” Hausdorff-type convergence statement for the spectra of the corresponding differential operators. As we show in the present work, new wave phenomena should be expected in the stochastic setting (e.g. a non-trivial continuous component of the spectral measure of the homogenised operator for a bounded-domain problem), which makes the related future developments even more exciting.
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