Harmonic Analysis and Quantitative Homogenization

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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 described approximately via a so-called homogenized (or effectively 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, most arising in materials science. The research focuses on several challenging problems in the area and will develop new methods and techniques. The results will provide theoretical foundation and guidance for numerical simulations in strongly inhomogeneous materials.The principal investigator will continue his ongoing research program on quantitative homogenization of partial differential equations. The main focus of this project will be on optimal regularity estimates (up to the boundary and uniform with respect to the inhomogeneity scale) and on sharp convergence rates for second-order elliptic and parabolic equations in divergence form with rapidly oscillating coefficients in bounded domains. More specifically, the problems to be investigated include the following: (1) elliptic equations and systems with almost-periodic coefficients; (2) systems of linear elasticity and Stokes systems with periodic coefficients; (3) uniform regularity estimates in perforated domains; and (4) quantitative homogenization of parabolic equations and systems with time-dependent periodic coefficients. The 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.