Harmonic Analysis and Periodic Homogenization

University Of Kentucky Research Foundation, Lexington KY

Investigators

Abstract

Partial differential equations with rapidly oscillating coefficients are used to describe various processes in materials with rapidly oscillating microstructures, such as 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. As a result, the theory of homogenization of partial differential equations with rapidly oscillating coefficients has many important applications in physics, mechanics, and materials science. The long-term goal of this project is to establish optimal quantitative results in the homogenization theory for a large class of partial differential equations in various settings, arising in applications. The proposed research focuses on several challenging problems in the area and will develop new approaches and techniques. The results will provide theoretical foundation and guidance for numerical simulations in strongly inhomogeneous materials. The Principal Investigator proposes to continue his ongoing research program on quantitative homogenization of partial differential equations. 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. More specifically, 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 boundary value problems in non-smooth domains; and (5) large-scale regularity estimates in perforated domains. The proposed research lies at the interface of harmonic analysis and partial differential equations. Existing and new techniques from harmonic analysis are expected to play a significant role in the development. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.