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 $\mathbb{R}$.

33 pages, 2 figures

References in corpus (7)

• Uniform resolvent convergence for strip with fast oscillating boundary
• Homogenization and norm resolvent convergence for elliptic operators in a strip perforated along a curve
• Resolvent estimates for high-contrast elliptic problems with periodic coefficients
• Two-scale homogenization for a general class of high contrast PDE systems with periodic coefficients
• On band gaps in photonic crystal fibers
• Trace formulae for graph Laplacians with applications to recovering matching conditions
• Isospectrality for graph Laplacians under the change of coupling at graph vertices

Cited by in corpus (5)

• Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. I
• Time-Dispersive Behaviour as a Feature of Critical Contrast Media
• Functional model for boundary-value problems
• Operator-norm resolvent asymptotic analysis of continuous media with high-contrast inclusions
• Two-scale series expansions for travelling wave packets in one-dimensional periodic media