Motivated by recent progress in "non-classical" (high-contrast) homogenization, we present a general approach for treating a broad class of problems involving critically scaled (partial) contrasts.
This leads to explicit limit asymptotic descriptions which (in contrast to the classical homogenization) remain two-scale, although with the microscopic part appropriately constrained.
This is illustrated by particular examples and leads to interesting effects physically (e.g. frequency and/or "directional" localisation, dispersion etc), and mathematically allows treating from a unified perspectives the "classical" homogenization, the high-contrasts one, and the intermediate cases, via developing new versions of e.g.
two-scale convergence, spectral and operator convergence, and of two-scale compactness.
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