Eigenfunctions Localised on a Defect in High-Contrast Random Media

Matteo Capoferri, Mikhail Cherdantsev, Igor Velčić

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Abstract

We study the properties of eigenvalues and corresponding eigenfunctions generated by a defect in the gaps of the spectrum of a high-contrast random operator. We consider a family of elliptic operators Aε in divergence form whose coefficients possess double porosity type scaling and are perturbed on a fixed-size compact domain. The coefficients of Aε are random variables generated, in an appropriate sense, by an ergodic dynamical system. Working in the gaps of the limiting spectrum of the unperturbed operator Aˆε, we show that the point spectrum of Aϵ converges in the sense of Hausdorff to the point spectrum of the homogenised operator Ahom as ε→0. Furthermore, we prove that the eigenfunctions of Aε decay exponentially at infinity uniformly for sufficiently small ε. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of Ahom.
Original languageEnglish
Pages (from-to)7449-7489
Number of pages41
JournalSIAM Journal on Mathematical Analysis
Volume55
Issue number6
Early online date8 Nov 2023
DOIs
Publication statusPublished - Dec 2023

Keywords

  • defect modes
  • high contrast media
  • localized eigenfunctions
  • random media
  • stochastic homogenization

ASJC Scopus subject areas

  • Computational Mathematics
  • Analysis
  • Applied Mathematics

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