Eigenfunctions localised on a defect in high-contrast random media

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 ${\mathcal{A}}^{\epsilon }$ 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 ${\hat{\mathcal{A}}}^{\epsilon }$, we show that the point spectrum of ${\mathcal{A}}^{\epsilon }$ converges in the sense of Hausdorff to the point spectrum of the limiting two-scale operator ${\mathcal{A}}^{\mathrm{hom}}$ as $\epsilon \to 0$. Furthermore, we prove that the eigenfunctions of ${\mathcal{A}}^{\epsilon }$ decay exponentially at infinity uniformly for sufficiently small $\epsilon$. This, in turn, yields strong stochastic two-scale convergence of such eigenfunctions to eigenfunctions of ${\mathcal{A}}^{\mathrm{hom}}$.

Published version

References in corpus (1)

• Stochastic homogenization of nonconvex unbounded integral functionals with convex growth

Cited by in corpus (1)

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