Harmonic Analysis and Homogenization of Elliptic Equations in Perforated Domains

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Partial differential equations with rapidly oscillating coefficients are used to describe various processes in materials such as composite materials and porous media. The theory of homogenization, whose goal is to describe 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 approximately described via a homogenized or effective 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, arising in applications. The proposed research focuses on several challenging problems in the modelling of fluid flows and acoustic propagation in porous media and inclusions in composite materials. The new approaches and techniques to be developed will provide theoretical foundation and guidance for numerical simulations of diffusion processes in highly heterogenous media. Funding for this project provides support for training of Ph.D. students. The main focus of this project on quantitative homogenization of partial differential equations is on large-scale regularity properties and convergence rates for second-order elliptic equations and systems in perforated domains. More specifically, the problems investigated as part of the project include (1) uniform regularity estimates for Darcy’s law; (2) large-scale regularity estimates for Brinkman’s law; (3) elliptic equations and systems with periodic and high-contrast coefficients; and (4) boundary value problems in perforated domains. The resolution of these problems will provide a deeper understanding of some fundamental issues in periodic homogenization. This project 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 work of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.