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High-contrast random composites: homogenisation framework and new spectral phenomena

Mikhail Cherdantsev, Kirill Cherednichenko, Igor Velčić

Abstract

We develop a framework for multiscale analysis of elliptic operators with high-contrast random coefficients. For a general class of such operators, we provide a detailed spectral analysis of the corresponding homogenised limit operator. Under some lenient assumptions on the configuration of the random inclusions, we fully characterise the limit of the spectra of the high-contrast operators in question, which unlike in the periodic setting is shown to be different to the spectrum of the homogenised operator. Introducing a new notion of the relevant limiting spectrum, we describe the connection between these two sets.

High-contrast random composites: homogenisation framework and new spectral phenomena

Abstract

We develop a framework for multiscale analysis of elliptic operators with high-contrast random coefficients. For a general class of such operators, we provide a detailed spectral analysis of the corresponding homogenised limit operator. Under some lenient assumptions on the configuration of the random inclusions, we fully characterise the limit of the spectra of the high-contrast operators in question, which unlike in the periodic setting is shown to be different to the spectrum of the homogenised operator. Introducing a new notion of the relevant limiting spectrum, we describe the connection between these two sets.

Paper Structure

This paper contains 36 sections, 54 theorems, 400 equations.

Table of Contents

  1. Introduction
  2. Notation index
  3. Preliminaries and problem setting
  4. Probability framework
  5. The main assumption and the extension property
  6. Problem setting and an overview of the existing results
  7. Homogenised operator and its spectrum
  8. Properties of Zhikov's $\beta$-function
  9. Resolution of identity for $-\Delta_\mathcal{O}$
  10. Point spectrum of the homogenised operator
  11. Example: a finite number of inclusion shapes
  12. Example: a continuum family of scaled copies of one shape
  13. Convergence of the spectra
  14. Main results
  15. Properties of $\beta_\infty(\lambda,\omega)$
  16. ...and 21 more sections

Key Result

Theorem 3.4

Consider a probability space $(\Omega,\mathcal{F}, P)$ and an ergodic dynamical system $\{T_x\}_{x \in \mathbf{R}^d}$ on $\Omega$. Suppose that $f \in L^1(\Omega)$ and ${S}\subset \mathbf{R}^d$ is a bounded open set. Then for $P$- a.e. $\omega \in \Omega$ one has Furthermore, for all $f \in L^p(\Omega)$, $1 \leq p \leq \infty,$ and a.e. $\omega \in \Omega$, the function $f (x, \omega)= f(T_x \ome

Theorems & Definitions (132)

  • Definition 3.1
  • Definition 3.2
  • Remark 3.3
  • Theorem 3.4: "Ergodic Theorem"
  • Definition 3.5
  • Remark 3.7
  • Theorem 3.8
  • proof
  • Remark 3.9
  • Theorem 3.10
  • ...and 122 more