Eigenfunctions localised on a defect in high-contrast random media

Matteo Capoferri, Mikhail Cherdantsev, Igor Velčić

Research output: Working paperPreprint

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
PublisherarXiv
Publication statusPublished - 6 Apr 2021

Keywords

  • math.SP
  • math-ph
  • math.AP
  • math.MP
  • primary 74S25, 74A40, secondary 35B27, 74Q15, 35B40, 60H15, 35P05

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