Some analytical techniques for partial differential equations on periodic structures and their applications to the study of metamaterials

The work presented in this thesis is a study of homogenisation problems in electromagnetics and elasticity with potential applications to the development of metamaterials. In Chapter 1, I study the leading order frequency approximations of the quasi-static Maxwell equations on the torus. A higher-order asymptotic regime is used to derive a higher-order homogenised equation for the solution of an elliptic second-order partial differential equation. The equivalent variational approach to this problem is studied which leads to an equivalent higher-order homogenised equation. Finally, the derivation of higher-order constitutive laws relating the fields to their inductions is presented. In Chapter 2, I study the governing equations of linearised elasticity where the periodic composite material of interest is made up of a "critically" scaled "stiff" rod framework with the voids in between filled in with a "soft" material which is in high-contrast with the stiff material. Using results from two-scale convergence theory, a well posed homogenised model is presented with features reminiscent of both high-contrast and thin structure homogenised models with the additional feature of a linking relation of Wentzell type. The spectrum of the limiting operator is investigated and the establishment of the convergence of spectra from the initial problem is derived. In the final chapter, I investigate brie y three additional homogenisation problems. In the first problem, I study a periodic dielectric composite and show that there exists a critical scaling between the material parameter of the soft inclusion and the period of the composite. In the second problem, I use of two-scale convergence theory to derive a homogenised model for Maxwell's equations on thin rod structures and in the final problem I study Maxwell's equations in R^3 under a chiral transformation of the coordinates and derive a homogenised model in this special geometry