Renormalization group and elliptic homogenization in high contrast

• 成果类型: Article
• 署名作者: Armstrong, Scott; Kuusi, Tuomo
• 署名单位: New York University; University of Helsinki
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-025-01370-9
• 发表日期: 2025
• 页码: 895-1086
• 关键词: quantitative homogenization; Stochastic Homogenization; percolation; REGULARITY

摘要

We prove a quantitative estimate for the homogenization length scale in terms of the ellipticity ratio Lambda/lambda\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\Lambda /\lambda $\end{document} of the coefficient field. This upper bound applies to high-contrast elliptic equations exhibiting near-critical behavior. Specifically, we show, assuming a suitable decay of correlations, the length scale at which homogenization occurs is at most exp(Clog2(1+Lambda/lambda))\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\exp (C \log <^>{2}(1+\Lambda /\lambda ))$\end{document}. 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 derive an approximate differential inequality for this coarse-grained ellipticity as a function of the length scale. This approach may be viewed 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.