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

Kirill Cherednichenko, Alexander Kiselev

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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 transformation, by a non-standard version of the Kronig–Penney model on R.
Original languageEnglish
Pages (from-to)441-480
Number of pages40
JournalCommunications in Mathematical Physics
Volume349
Issue number2
Early online date25 Jul 2016
DOIs
Publication statusPublished - 1 Jan 2017

Keywords

  • High-contrast homogenisation, boundary triples, Krein formula, norm-resolvent estimates, quantum graphs, asymptotic analysis

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