Harmonic Analysis and Periodic Homogenization

具有快速振荡系数的偏微分方程组被用来描述具有快速振荡微结构的材料中的各种过程，例如复合材料和穿孔材料。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。拟议的研究侧重于该领域的几个具有挑战性的问题，并将开发新的方法和技术。这一结果将为强非均匀材料的数值模拟提供理论基础和指导。首席研究员建议继续他正在进行的偏微分方程组定量齐次化的研究计划。这个项目的主要焦点将是研究二阶椭圆型和抛物型方程的大规模几何和正则性以及收敛速度。更具体地说，要研究的问题包括：(1)周期系数椭圆型方程的大规模几何性质；(2)周期系数含时抛物型方程的定量齐次化；(3)拟线性椭圆型方程的定量齐次化；(4)非光滑区域上边值问题的齐次化；以及(5)穿孔区域上的大规模正则性估计。所提出的研究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发展中发挥重要作用。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 with rapidly oscillating microstructures, such as composite and perforated materials. The theory of homogenization, whose goal is to describe the 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 area and will develop new approaches and techniques. The results will provide theoretical foundation and guidance for numerical simulations in strongly inhomogeneous materials.The Principal Investigator proposes to continue his ongoing research program on quantitative homogenization of partial differential equations. The main focus of this project will be on large-scale geometric and regularity properties and convergence rates for second-order elliptic and parabolic equations. More specifically, the problems to be investigated include (1) large-scale geometric properties of elliptic equations with periodic coefficients; (2) quantitative homogenization of parabolic systems with time-dependent periodic coefficients; (3) quantitative homogenization of quasi-linear elliptic equations; (4) homogenization of boundary value problems in non-smooth domains; and (5) large-scale regularity estimates in perforated domains. 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.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.