Two-Scale Homogenisation of Partially Degenerating PDEs with Applications to Photonic Crystals and Elasticity

Abstract

In this thesis we study elliptic PDEs and PDE systems with ε-periodic coefficients, for small ε, using the theory of two-scale homogenisation. We study a class of PDEs of partially degenerating type: PDEs with coefficients that are not uniformly elliptic with respect to ε, and become degenerate in the limit ε → 0. We review a recently developed theory of homogenisation for a general class of partially degenerating PDEs via the theory of two-scale convergence, and study two such problems from physics. The first problem arises from the study of a linear elastic composite with periodically dispersed inclusions that are isotropic and ‘soft’ in shear: the shear modulus is of order ε2. By passing to the two scale limit as ε → 0 we find the homogenised limit equations to be a genuinely two-scale system in terms of both the macroscopic variable x and the microscopic variable y. We discover that the corresponding two-scale limit solutions must satisfy the incompressibility condition in y and therefore the composite only undergoes microscopic deformations when a ‘microscopically rotational’ force is applied. We analyse the corresponding limit spectral problem and find that, due to the y-incompressibility, the spectral problem is an uncoupled two-scale problem in terms of x and y. This gives a simple representation of the two-scale limit spectrum. We prove the spectral compactness result that states: the spectrum of the original operator converges to the spectrum of the limit operator in the sense of Hausdorff. The second problem we study is the propagation of electromagnetic waves down a photonic fibre with a periodic cross section. We seek solutions to Maxwell’s equations, propagating down the waveguide with wavenumber k ε2-close to some ‘critical’ value. In this setting, Maxwell’s equations are reformulated as a partially degenerating PDE system with ε-periodic coefficients. Using the theory of homogenisation we pass to the limit as ε → 0 to find a non-standard two-scale homogenised limit and prove that the spectral compactness result holds. We finally prove that there exist gaps in the limit spectrum for two particular examples: a one-dimensionally periodic ‘multilayer’ photonic crystal and a two-dimensionally periodic two-phase photonic crystal with the inclusion phase consisting of arbitrarily small circles. Therefore, we prove that these photonic fibres have photonic band gaps for certain k.

Date of Award	17 Jul 2012
Original language	English
Awarding Institution	University of Bath
Supervisor	Valery Smyshlyaev (Supervisor) & Ilia Kamotski (Supervisor)