Harmonic Analysis and Homogenization of Partial Differential Equations

This project concerns partial differential equations with rapidly oscillating periodic co- efficients. The problems to be investigated include (1) Asymptotic behavior of Green’s and Neumann functions, boundary layer phenomenon, and rates of convergence of solutions; (2) Uniform exact boundary controllability; (3) Asymptotic behavior of eigenvalues and uniform estimates of eigenfunctions; (4) Uniform regularity estimates and rates of convergence for systems of elasticity, the Stokes equations, and the Maxwell equations; (5) Uniform regularity estimates in C1 domains and spectral radius on convex domains. Intellectural Merit Partial differential equations with rapidly oscillating coefficients arise in the theory of ho- mogenization, whose goal is to describe the macroscopic properties of microscopically nonho- mogeneous media such as composite and perforated materials. Together with Carlos Kenig and Fanghua Lin, the PI recently has made significant progress in elliptic homogenization and solved several longstanding open problems. These include uniform Rellich estimates, Lipschitz estimates for solutions with Neumann boundary data, and sharp results on the rates of convergence of solutions and eigenvalues. The PI proposes to continue his ongoing research program, much of which is joint with Kenig and Lin, on homogenization of partial differential equations. The proposed research will focus on several challenging problems in the area. Resolution of these problems will provide deep understanding of some fundamental issues in homogenization, such as rates of convergence of solutions and eigenvalues, boundary layer phenomenon, and uniform controllability and stabilization for distributed systems. The proposed research lies at the interface of harmonic analysis and partial differential equations. Broader Impacts The theory of homogenization of partial differential equations with rapidly oscillating coefficients has many important applications in physics, mechanics, and modern technology. The proposed research would provide theoretical foundation for numerical simulations in strongly inhomogeneous media. The findings from the research will be disseminated in the scientific community by the PI and his collaborators through lectures in conferences, seminars and graduate courses and publishing in mathematical journals and websites. The PI has been very active in graduate education and will continue to work with Ph.D students and junior researchers on part of this project. Funds are requested to support the PI’s Ph.D students as research assistants. As chair of the Mathematics Department from 2007 to 2011 and director of the MathExcel program for freshmen from 2004 to 2007, he also played a leading role in undergraduate mathematics education at the University of Kentucky.