Fibre homogenisation

Shane Cooper; Marcus Waurick

doi:10.1016/j.jfa.2019.03.004

Fibre homogenisation

Shane Cooper, Marcus Waurick^*

^*Corresponding author for this work

Mathematics And Statistics

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

9 Citations (Scopus)

14 Downloads (Pure)

Abstract

In this article we present a novel method for studying the asymptotic behaviour, with order-sharp error estimates, of the resolvents of parameter-dependent operator families. The method is applied to the study of differential equations with rapidly oscillating coefficients in the context of second-order PDE systems and the Maxwell system. This produces a non-standard homogenisation result that is characterised by ‘fibre-wise’ homogenisation of the related Floquet-Bloch PDEs. These fibre-homogenised resolvents are shown to be asymptotically equivalent to a whole class of operator families, including those obtained by standard homogenisation methods.

Original language	English
Pages (from-to)	3363-3405
Number of pages	43
Journal	Journal of Functional Analysis
Volume	276
Issue number	11
Early online date	21 Mar 2019
DOIs	https://doi.org/10.1016/j.jfa.2019.03.004
Publication status	Published - 1 Jun 2019

Funding

S. Cooper was supported by the EPSRC grant EP/M017281/1 (“Operator asymptotics, a new approach to length-scale interactions in metamaterials”). M. Waurick is grateful for the financial support of the EPSRC grant EP/L018802/1 (“Mathematical foundations of metamaterials: homogenisation, dissipation and operator theory”).

Keywords

fibre homogenisation
Maxwell's equations
resolvent estimates
second-order PDE systems

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10.1016/j.jfa.2019.03.004

Cooper-Waurick-JFA-2019-Fibre-homogenisationAccepted author manuscript, 279 KBLicence: CC BY-NC-ND 4.0

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AB - In this article we present a novel method for studying the asymptotic behaviour, with order-sharp error estimates, of the resolvents of parameter-dependent operator families. The method is applied to the study of differential equations with rapidly oscillating coefficients in the context of second-order PDE systems and the Maxwell system. This produces a non-standard homogenisation result that is characterised by ‘fibre-wise’ homogenisation of the related Floquet-Bloch PDEs. These fibre-homogenised resolvents are shown to be asymptotically equivalent to a whole class of operator families, including those obtained by standard homogenisation methods.

KW - fibre homogenisation

KW - Maxwell's equations

KW - resolvent estimates

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UR - https://www.scopus.com/pages/publications/85063425820

UR - http://dro.dur.ac.uk/27748/

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VL - 276

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EP - 3405

JO - Journal of Functional Analysis

JF - Journal of Functional Analysis

IS - 11

ER -

Fibre homogenisation

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