Harmonic Analysis and Homogenization of Elliptic Equations in Perforated Domains

具有快速振荡系数的偏微分方程被用来描述复合材料和多孔介质等材料中的各种过程。以描述微观非均质或非均质材料宏观性质为目标的均匀化理论表明，这种性质随空间变量急剧变化的强非均质材料可以近似地用一种均匀或有效的均质材料来描述。因此，具有快速振荡系数的偏微分方程齐化理论在物理、力学和材料科学中有着重要的应用。这个项目的长期目标是在齐次化理论中为一大类在不同环境下产生的应用中出现的偏微分方程建立最优的定量结果。所提出的研究集中在多孔介质和复合材料中夹杂物中流体流动和声传播的模拟中的几个具有挑战性的问题。这些有待发展的新方法和新技术将为高非均匀介质中扩散过程的数值模拟提供理论基础和指导。该项目的资金为博士生的培训提供支持。本文主要研究穿孔区域上二阶椭圆型方程和方程组的大尺度正则性和收敛速度问题。更具体地说，作为项目的一部分，所研究的问题包括(1)达西定律的一致正则性估计；(2)布林克曼定律的大规模正则性估计；(3)具有周期系数和高对比度系数的椭圆型方程和系统；以及(4)穿孔区域中的边值问题。这些问题的解决将有助于更深入地理解周期性均化中的一些基本问题。该项目位于调和分析和偏微分方程组的交界处。来自谐波分析的现有和新技术预计将在该项目的工作中发挥重要作用。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

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.