Harmonic Analysis and Periodic Homogenization

The PI proposes to continue his ongoing research program on quantitative homogenization of partial differential equations. The long-term goal is to establish sharp quantitative results in the homogenization theory for a large class of partial differential equations in various settings, arising in physics, mechanics, and materials sciences. The main focus of this project will be on large-scale geometric and regularity properties and convergence rates for second-order elliptic and parabolic equations. The problems to be investigated include (1) large-scale geometric properties of elliptic equations with periodic coefficients; (2) quantitative homogenization of parabolic systems with time-dependent periodic coefficients; (3) quantitative homogenization of quasi-linear elliptic equations; (4) homogenization of $L^p$ boundary value problems in $C^1$ domains; (5) large-scale regularity estimates in perforated domains.