부산대학교 도서관

We prove a quantitative estimate on the homogenization length scale in terms of the ellipticity ratio~$\Lambda/\lambda$ of the coefficient field. This upper bound applies to high-contrast elliptic equations demonstrating near-critical behavior. Specifically, we show that, given a suitable decay of correlation, the length scale at which homogenization is observed is at most $\exp(C \log^3(1+\Lambda/\lambda))$. The proof introduces the new concept of coarse-grained ellipticity, which measures the effective ellipticity ratio of the equation -- and thus the strength of the disorder -- after integrating out smaller scales. By a direct analytic argument, we obtain an approximate differential inequality for this coarse-grained ellipticity as a function of the length scale. This approach can be interpreted as a rigorous renormalization group argument and provides a quantitative framework for homogenization that can be iteratively applied across an arbitrary number of length scales.
Comment: 157 pages; previously announced at https://www.scottnarmstrong.com/2024/05/high-contrast-homogenization/