High-contrast random systems of PDEs: homogenisation and spectral theory
Matteo Capoferri, Mikhail Cherdantsev, Igor Velčić
TL;DR
This work develops a stochastic two-scale homogenisation and spectral theory for high-contrast elliptic systems with random coefficients, where inclusions scale as $\varepsilon^2$ in a matrix of order 1. It shows that the limit operator $\mathcal{A}^{\mathrm{hom}}$ captures micro-resonances via $\sigma(\mathcal{A}_0)$ and macroscopic resonances via the Zhikov $\boldsymbol{\beta}(\lambda)$, yielding a spectrum described by $\sigma(\mathcal{A}^{\mathrm{hom}})=\sigma(\mathcal{A}_0) \cup \{\lambda: \boldsymbol{\beta}(\lambda) \ge 0\}$. The paper proves an outer bound $\lim_{\varepsilon\to0}\sigma(\mathcal{A}^\varepsilon) \subseteq \mathcal{G}$ with $\mathcal{G}=\sigma(\mathcal{A}_0) \cup \overline{\{\lambda: \beta_\infty(\lambda) \ge 0\}}$, and, under finite-range dependence, an inner bound showing $\lim_{\varepsilon\to0}\sigma(\mathcal{A}^\varepsilon)=\mathcal{G}$, explaining why Hausdorff convergence does not hold in general. The work is reinforced by explicit examples highlighting the roles of micro- and macro-symmetries and random scaling, illustrating richer spectral behavior than in periodic or scalar settings and guiding future analyses of random high-contrast media with partial degeneracies.
Abstract
We develop a qualitative homogenisation and spectral theory for elliptic systems of partial differential equations in divergence form with highly contrasting (i.e., non uniformly elliptic) random coefficients. The focus of the paper is on the behaviour of the spectrum as the heterogeneity parameter tends to zero; in particular, we show that in general one doesn't have Hausdorff convergence of spectra. The theoretical analysis is complemented by several explicit examples, showcasing the wider range of applications and physical effects of systems with random coefficients, when compared with systems with periodic coefficients or with scalar operators (both random and periodic).