Abstract

Co-author: Ilia Kamotski (University College London)

There has been lot of recent interest in composite materials whose macroscopic physical properties can be radically different from those of conventional materials, often due to effects of the so-called “micro-resonances”. Mathematically this leads to studying high-contrast homogenization of (periodic or not) problems with a `critically scaled high contrast, where the resulting two-scale asymptotic behaviour appears to display a number of interesting effects. Mathematical analysis of these problems requires development of “two-scale” versions of operator and spectral convergences, of compactness, etc. We will review some background, as well as some more recent generalizations and applications. One is two-scale analysis of general “partially-degenerating” periodic problems, where strong two-scale resolvent convergence appears to hold under a rather generic decomposition assumptions, implying in particular (two-scale) convergence of se migroups with applications to a wide class of micro-resonant dynamic problems. Another is two-scale homogenization with random micro-resonances, which appears to yield macroscopic dynamics effects akin to Anderson localization. Some of the work is joined with Ilia Kamotski.