Harmonic Analysis and Quantitative Homogenization

Overview: 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 science. The main focus of this project will be on uniform regularity estimates and convergence rates for second-order elliptic and parabolic equations in divergence form with rapidly oscillating coefficients. The problems to be investigated include (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. Intellectual Merit : Partial differential equations with rapidly oscillating coefficients arise in the theory of homogenization, whose goal is to describe the macroscopic properties of microscopically inhomogeneous media, such as composite and perforated materials. Together with his collaborators, 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 on quantitative homogenization of partial differential equations in periodic and almost-periodic settings. 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 quantitative homogenization, such as uniform sharp boundary regularity, optimal rates of convergence of solutions, and boundary layer phenomenon. 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 important applications in physics, mechanics, and materials science. The proposed research will provide theoretical foundation and guidance for numerical analysis and computational simulation of various processes in strongly inhomogeneous media, such as composite and perforated materials. 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 by publishing in mathematical journals and websites. The PI has been very active in graduate education and will continue to train Ph.D. students and junior researchers by working with them on part of this project. Funds of this project will provide support for Ph.D. students as research assistants under the PI’s supervision. As chair of the Mathematics Department from 2007 to 2011 and director of the MathExcel Program for freshmen from 2004 to 2007, the PI played a leading role in undergraduate mathematics education at the University of Kentucky. He remains very active in undergraduate education.