Harmonic Analysis and Quantitative Homogenization

具有快速振荡系数的偏微分方程用于描述具有快速振荡微结构的材料（如复合材料和穿孔材料）中的各种过程。均匀化理论的目标是描述微观上不均匀或非均匀材料的宏观性质，该理论表明，这种特征相对于空间变量急剧变化的强烈非均匀材料可以通过所谓的均匀化（或有效均匀）材料近似描述。因此，具有快速振荡系数的偏微分方程的均匀化理论在物理学、力学和材料科学中有许多重要的应用。这个项目的长期目标是建立最佳的定量结果，在均匀化理论的偏微分方程在各种设置，大多数出现在材料科学。该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。研究结果将为强非均匀材料的数值模拟提供理论基础和指导。主要研究者将继续他正在进行的偏微分方程定量均匀化的研究计划。该项目的主要重点将是最佳正则性估计（直到边界和均匀的不均匀尺度）和在有界域中具有快速振荡系数的发散形式的二阶椭圆和抛物方程的急剧收敛速度。更具体地说，要研究的问题包括：（1）椭圆方程和系统与几乎周期系数;（2）系统的线性弹性和斯托克斯系统与周期系数;（3）一致的正则性估计穿孔域;和（4）定量均匀化抛物方程和系统与时间相关的周期系数。研究处于调和分析与偏微分方程的交叉点。谐波分析的现有技术和新技术预计将在发展中发挥重要作用。

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 described approximately via a so-called homogenized (or effectively 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, most arising in materials science. The research focuses on several challenging problems in the area and will develop new methods and techniques. The results will provide theoretical foundation and guidance for numerical simulations in strongly inhomogeneous materials.The principal investigator will continue his ongoing research program on quantitative homogenization of partial differential equations. The main focus of this project will be on optimal regularity estimates (up to the boundary and uniform with respect to the inhomogeneity scale) and on sharp convergence rates for second-order elliptic and parabolic equations in divergence form with rapidly oscillating coefficients in bounded domains. More specifically, the problems to be investigated include the following: (1) elliptic equations and systems with almost-periodic coefficients; (2) systems of linear elasticity and Stokes systems with periodic coefficients; (3) uniform regularity estimates in perforated domains; and (4) quantitative homogenization of parabolic equations and systems with time-dependent periodic coefficients. The 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.