Eigenfunctions localised on a defect in high-contrast random media

TL;DR

This paper investigates how defects in high-contrast random media influence eigenvalues and eigenfunctions, showing convergence and exponential decay properties as the scale parameter approaches zero.

Contribution

It establishes the convergence of eigenvalues and eigenfunctions in high-contrast random media with defects, including exponential decay and stochastic two-scale convergence results.

Findings

Eigenvalues converge in Hausdorff sense to the limiting spectrum.

Eigenfunctions decay exponentially at infinity.

Eigenfunctions exhibit strong stochastic two-scale convergence.

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 $\mathcal{A}^\varepsilon$ in divergence form whose coefficients are random, possess double porosity type scaling, and are perturbed on a fixed-size compact domain (a defect). Working in the gaps of the limiting spectrum of the unperturbed operator $\hat{\mathcal{A}}^\varepsilon$, we show that the point spectrum of $\mathcal{A}^\varepsilon$ converges in the sense of Hausdorff to the point spectrum of the limiting two-scale operator $\mathcal{A}^\mathrm{hom}$ as $\varepsilon \to 0$. Furthermore, we prove that the eigenfunctions of $\mathcal{A}^\varepsilon$ decay exponentially at infinity uniformly for sufficiently small $\varepsilon$. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of $\mathcal{A}^\mathrm{hom}$.