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

Kirill Cherednichenko; Alexander Kiselev

doi:10.1007/s00220-016-2698-4

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

Kirill Cherednichenko
, Alexander Kiselev

Department of Mathematical Sciences

National Pedagogical Dragomanov University

Research output: Contribution to journal › Article › peer-review

15 Citations (SciVal)

81 Downloads (Pure)

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 language	English
Pages (from-to)	441-480
Number of pages	40
Journal	Communications in Mathematical Physics
Volume	349
Issue number	2
Early online date	25 Jul 2016
DOIs	https://doi.org/10.1007/s00220-016-2698-4
Publication status	Published - 1 Jan 2017

Keywords

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

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10.1007/s00220-016-2698-4Licence: CC BY

https://arxiv.org/abs/1510.03364

Mathematical Foundations of Metamaterials: Homogenisation, Dissipation and Operator Theory
Cherednichenko, K. (PI)
Engineering and Physical Sciences Research Council
23/07/14 → 22/06/19
Project: Research council

Kirill Cherednichenko
- K.Cherednichenkobath.acuk
- Department of Mathematical Sciences - Professor
- EPSRC Centre for Doctoral Training in Statistical Applied Mathematics (SAMBa)
Person: Research & Teaching

Cite this

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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.",

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AU - Kiselev, Alexander

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N2 - 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.

AB - 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.

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

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JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

IS - 2

ER -

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

Abstract

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Mathematical Foundations of Metamaterials: Homogenisation, Dissipation and Operator Theory

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Kirill Cherednichenko

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