Harmonic Analysis and Quantitative Homogenization

In Fall 2021, given the ongoing Covid-19 pandemic, I plan to stay mostly at my home institute, the University of Kentucky, with several short-term visits to the University of Chicago and the New York University. In Spring 2022, if the situation allows it, I will visit the Courant Institute for at least two months and several universities in China. Partial differential equations (PDEs) with rapidly oscillating coefficients are used to describe various processes in media with rapidly oscillating microstructures, such as porous media, composite and perforated materials. The theory of homogenization, whose goal is to describe the macroscopic properties of microscopically inhomogeneous or heterogeneous materials, shows that such strongly inhomogeneous material, whose characteristics change sharply with respect to space variables, may be approximately described via a homogenized or effective homogeneous material. The long-term goal of my research program is to establish optimal quantitative results in the homogenization theory for a large class of PDEs in various settings, arising in applications. The proposed research for the sabbatical period focuses on several challenging problems in the area of quantitative homogenization. In particular, I will continue working on my joint project with Fanghua Lin at the Courant Institute. The problems to be studied include (1) large-scale geometric properties of elliptic equations with periodic coefficients, and (2) uniform boundary controllability of wave equations with highly oscillating coefficients. I also plan to continue my joint work with Carlos Kenig at the University of Chicago on optimal regularity estimates, which are uniform with respect to the scale of the microstructure, for elliptic equations and systems with rapidly oscillating coefficients in bounded domains. The emphasis will be on periodically perforated domains. This is a proposal that lies at the interface of harmonic analysis and PDEs. Existing and new techniques from harmonic analysis are expected to play a significant role in the development.