Quasi-periodic two-scale homogenisation and effective spatial dispersion in high-contrast media

Cooper, S.

Abstract

The convergence of spectra via two-scale convergence for double-porosity models is well known. A crucial assumption in these works is that the stiff component of the body forms a connected set. We show that under a relaxation of this assumption the (periodic) two-scale limit of the operator is insufficient to capture the full asymptotic spectral properties of high-contrast periodic media. Asymptotically, waves of all periods (or quasi-momenta) are shown to persist and an appropriate extension of the notion of two-scale convergence is introduced. As a result, homogenised limit equations with none trivial quasi-momentum dependence are found as resolvent limits of the original operator family. This results in asymptotic spectral behaviour with a rich dependence on quasimomenta.

Citation

Cooper, S. (2018). Quasi-periodic two-scale homogenisation and effective spatial dispersion in high-contrast media. Calculus of Variations and Partial Differential Equations, 57(3), Article 76. https://doi.org/10.1007/s00526-018-1365-3

• Journal Article Type: Article
• Acceptance Date: Apr 23, 2018
• Online Publication Date: Apr 27, 2018
• Publication Date: Apr 1, 2018
• Deposit Date: Apr 23, 2018
• Publicly Available Date: Apr 23, 2018
• Journal: Calculus of Variations and Partial Differential Equations
• Print ISSN: 0944-2669
• Electronic ISSN: 1432-0835
• Publisher: Springer
• Peer Reviewed: Peer Reviewed
• Volume: 57
• Issue: 3
• Article Number: 76
• DOI: https://doi.org/10.1007/s00526-018-1365-3