Harmonic Analysis and Homogenization of Partial Differential Equations

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This project concerns elliptic partial differential equations and systems with rapidly oscillating periodic coefficients, which arise in the theory of homogenization. Shen and his collaborators will focus on several challenging problems in the area. Resolution of these problems will provide better understanding of some fundamental issues in homogenization, including uniform sharp regularity estimates and rates of convergence of solutions of boundary value problems, asymptotic behavior of eigenvalues and uniform estimates of eigenfunctions, boundary layer phenomenon, and uniform controllability and stabilization for distributed systems. Elliptic equations with periodic coefficients are considered as a model case in homogenization. New techniques and approaches developed for this case will be useful in studying homogenization in other important settings, such as non-uniformly oscillating coefficients, almost periodic coefficients, perforated domains, and evolution operators with highly oscillating coefficients. 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. 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 shows that such materials may be approximately described via a homogenized or effective homogeneous material. As such 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 will develop new methods and techniques that will provide theoretical foundation and guidance for numerical simulations in strongly inhomogeneous materials. The findings from the research will be disseminated in the scientific community by Shen and his collaborators through lectures in conferences, workshops, and graduate courses as well as publishing in mathematical journals and websites. Shen is committed to the training of future generations of mathematicians; graduate students and junior researchers will be involved on the project.