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Bloch wave homogenisation of quasiperiodic media

Part of: Qualitative theory Partial differential equations General theory of linear operators Elliptic equations and systems

Published online by Cambridge University Press:  05 October 2020

SISTA SIVAJI GANESH  and
VIVEK TEWARY Open the ORCID record for VIVEK TEWARY [Opens in a new window]
SISTA SIVAJI GANESH
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India emails: sivaji.ganesh@iitb.ac.in; vivektewary@gmail.com
VIVEK TEWARY
Affiliation:
Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai 400076, India emails: sivaji.ganesh@iitb.ac.in; vivektewary@gmail.com
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Abstract

Quasiperiodic media is a class of almost periodic media which is generated from periodic media through a ‘cut and project’ procedure. Quasiperiodic media displays some extraordinary optical, electronic and conductivity properties which call for the development of methods to analyse their microstructures and effective behaviour. In this paper, we develop the method of Bloch wave homogenisation for quasiperiodic media. Bloch waves are typically defined through a direct integral decomposition of periodic operators. A suitable direct integral decomposition is not available for almost periodic operators. To remedy this, we lift a quasiperiodic operator to a degenerate periodic operator in higher dimensions. Approximate Bloch waves are obtained for a regularised version of the degenerate operator. Homogenised coefficients for quasiperiodic media are obtained from the first Bloch eigenvalue of the regularised operator in the limit of regularisation parameter going to zero. A notion of quasiperiodic Bloch transform is defined and employed to obtain homogenisation limit for an equation with highly oscillating quasiperiodic coefficients.

Keywords

Homogenisationalmost periodicityBloch eigenvaluesquasiperiodicity

MSC classification

Primary: 35B27: Homogenization; equations in media with periodic structure
Secondary: 35J15: Second-order elliptic equations 47A55: Perturbation theory 34C27: Almost and pseudo-almost periodic solutions
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 1 , February 2022 , pp. 58 - 78
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Copyright
© The Author(s), 2020. Published by Cambridge University Press

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