Title:

Twoscale homogenisation of partially degenerating PDEs with applications to photonic crystals and elasticity

In this thesis we study elliptic PDEs and PDE systems with epcriodic coeffi
cients, for small E, using the theory of twoscale homogenisation. We study a
class of PDEs of partially degenerating type: PDEs with coefficients that are not
uniformly elliptic with respect to E, and become degenerate in the limit E t O.
We review a recently developed theory of homogenisation for a general class of
partially degenerating PDEs via the theory of twoscale 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 E2. By passing to the two
scale limit as E t 0 we find the homogenised limit equations to be a genuinely
twoscale system in terms of both the macroscopic variable x and the micro
scopic variable y. We discover that the corresponding twoscale 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 yincompressibility, the spectral problem is an uncoupled twoscale prob
lem in terms of x and y. This gives a simple representation of the twoscale 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 electro
magnetic waves down a photonic fibre with a periodic cross section. We seek
solutions to Maxwell's equations, propagating down the waveguide with wave
number k E2close to some (critical' value. In this setting, Maxwell's equations
are reformulated as a partially degenerating PDE system with zperiodic coeffi
cients. Using the theory of homogenisation we pass to the limit as E t 0 to find a
nonstandard twoscale homogenised limit and prove that the spectral compact
ness result holds. We finally prove that there exist gaps in the limit spectrum
for two particular examples: a onedimensionally periodic 'multilayer ' photonic
crystal and a twodimensionally periodic twophase photonic crystal with the in
clusion phase consisting of arbitrarily small circles. Therefore, we prove that
these photonic fibres have photonic band gaps for certain k.
