A coarse-graining theory for elliptic operators and homogenization in high contrast

Abstract: We review a coarse-graining theory for divergence-form elliptic operators. The construction centers on a pair of coarse-grained matrices defined on spatial blocks that encode a scale-dependent notion of ellipticity, transmit precise information from small to large scales, and yield coarse-grained counterparts of standard elliptic estimates. Under simplifying assumptions, we give a complete proof of the result of [arXiv:2405.10732] that homogenization is reached within at most $C\log^2(1+\Theta)$ dyadic length scales in the high-contrast regime, where $\Theta$ is the ellipticity contrast. We argue that this scale-local notion of ellipticity is genuinely iterable across arbitrarily many scales, providing a framework for a rigorous renormalization group analysis.