Unified approach to critical-contrast homogenisation with explicit links to time-dispersive media

Abstract

A novel approach to critical-contrast homogenisation is proposed. Norm-resolvent asymptotics are explicitly constructed. An essential feature of our approach is that it relates homogenisation limits to a class of time-dispersive media.2000 Mathematics Subject Classification. Primary 34E13 ; Secondary 34E05, 35P20, 47A20, 81Q35.

Effective Behaviour of Critical-Contrast PDEs: Micro-resonances, Frequency Conversion, and Time Dispersive Properties. I

“…[16,17] and references therein for the way to introduce a "proper" boundary triple of [39,31]. Unlike the analysis of [23], [25] and [26], this latter version of the theory does not suit our needs, owing to the PDE setup we are dealing with here.…”

“…Diagonalisation of the M -operator on the stiff component. Our overall strategy is based on the ideas of [25]. The proof of the main result is, however, obtained in a different way, although the rationale (see also [23]) is essentially the same.…”

“…The proof of the main result is, however, obtained in a different way, although the rationale (see also [23]) is essentially the same. Towards the end of the section we outline a sketch of an alternative line of reasoning, which mimics the machinery of [25]. First, we consider the M -function M stiff (z), and observe that…”

“…In [23,25] (see also an earlier paper [26] dealing with critical-contrast homogenisation on R) we considered a rather simple model of a high-contrast graph periodic along one axis. A unified treatment of critical-contrast homogenisation was proposed and carried out in three distinct cases: (i) where neither the soft nor the stiff component of the medium is connected; (ii) where the stiff component of the medium is connected; (iii) where the soft component of the medium is connected.…”

“…The corresponding analysis is based on a version of Birman-Kreȋn-Višik theory [8,46,47,71] as proposed by Ryzhov [58]. In adopting the mentioned approach, we deviate from the boundary triples theory utilised previously in our ODE analysis [23,25]. Still, the theory developed in [58] can be seen as a "version" of the latter, in that it attempts to write the second Green's identity in operator-theoretic terms as…”