Grants and Contracts Details
Project Summary: Harmonic Analysis and Homogenization of Elliptic Equations in Perforated Domains Zhongwei Shen Overview The PI proposes to continue his ongoing research program on quantitative homogenization of partial di?erential equations (PDEs). The long-term goal is to establish sharp quantitative results in the homogenization theory for a large class of PDEs in various settings, arising in physics, me- chanics, and materials sciences. The main focus of this project will be on large-scale regularity properties and error estimates for second-order elliptic equations and systems in periodically perfo- rated domains. The problems to be investigated include (1) uniform regularity estimates for Darcy’s law; (2) large-scale regularity estimates for Brinkman’s law; (3) elliptic systems with periodic and high-contrast coe?cients; and (4) boundary value problems in perforated Lipschitz domains. Intellectual Merit Partial Di?erential Equations with rapidly oscillating coe?cients arise in the theory of homog- enization, whose goal is to describe the macroscopic properties of microscopically nonhomogeneous media, such as composite and perforated materials. Together with his collaborators, the PI has made signi?cant progress in elliptic homogenization and solved several longstanding open problems. These include sharp boundary regularity estimates, optimal rates of convergence for solutions and eigenvalues, asymptotic expansions of Green and Neumann functions with sharp remainder esti- mates, and analysis of boundary layers for boundary value problems with oscillating boundary data. Signi?cant progress was also made recently on the doubling property and estimates of nodal sets for elliptic equations with periodic coe?cients, and on the uniform boundary controllability of wave equations with rapidly oscillating coe?cients. The proposed research further extends these quantitative results to other important settings, including the steady Stokes and Navier-Stokes equations in perforated domains and elliptic systems with high-contrast coe?cients. The PI also proposes to study boundary value problems in perfo- rated Lipschitz domains. The resolution of the proposed problems as well as the new techniques and approaches to be developed will provide deeper understanding of some fundamental issues in periodic homogenization. The proposed research lies at the interface of harmonic analysis and PDEs. Existing and new techniques from harmonic analysis are expected to play a signi?cant role in the development. Broader Impacts The theory of homogenization of PDEs has important applications in physics, mechanics, and materials science. The quantitative results in the proposed research would provide theoretical guidance and foundation for computational simulation and experimentation of various processes in strongly inhomogeneous media, such as the ?ltration of a viscous ?uid through a porous medium and inclusions in composite materials. The ?ndings from the research will be disseminated in the scienti?c 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 work with Ph.D. students and junior researchers 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. 1
|Effective start/end date||7/1/22 → 6/30/25|
- National Science Foundation: $213,094.00
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