Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. I

TL;DR

This paper introduces a novel homogenisation approach for periodic PDEs with critical contrast, explicitly linking homogenisation limits to time-dispersive media through asymptotic analysis of Dirichlet-to-Neumann operators.

Contribution

It develops an explicit asymptotic method for critical-contrast homogenisation, connecting PDE homogenisation to time-dispersive media with norm-resolvent asymptotics.

Findings

Explicit asymptotic formulas for Dirichlet-to-Neumann operators.

Homogenisation limits related to time-dispersive media.

Construction of norm-resolvent asymptotics for oscillating coefficients.

Abstract

A novel approach to critical-contrast homogenisation for periodic PDEs is proposed, via an explicit asymptotic analysis of Dirichlet-to-Neumann operators. Norm-resolvent asymptotics for non-uniformly elliptic problems with highly oscillating coefficients are explicitly constructed. An essential feature of the new technique is that it relates homogenisation limits to a class of time-dispersive media.