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

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 $\mathbb R$.