Norm-resolvent convergence of one-dimensional high-contrast periodic problems to a Kronig-Penney dipole-type model

TL;DR

This paper establishes operator-norm resolvent convergence for one-dimensional high-contrast periodic differential operators, demonstrating their asymptotic equivalence to a Kronig-Penney dipole-type model using boundary triple techniques.

Contribution

It introduces a novel approach to high-contrast homogenisation by employing boundary triple methods to analyze operator convergence in a challenging setting.

Findings

Proves resolvent convergence estimates in high-contrast periodic problems.

Shows asymptotic equivalence to a Kronig-Penney model.

Extends homogenisation techniques to non-uniformly elliptic operators.

Abstract

We prove operator-norm resolvent convergence estimates for one-dimensional periodic differential operators with rapidly oscillating coefficients in the non-uniformly elliptic high-contrast setting, which has been out of reach of the existing homogenisation techniques. Our asymptotic analysis is based on a special representation of the resolvent of the operator in terms of the $M$-matrix of an associated boundary triple ("Krein resolvent formula''). The resulting asymptotic behaviour is shown to be described, up to a unitary equivalent transformation, by a non-standard version of the Kronig-Penney model on $\mathbb R$.