Harmonic Analysis and Quantitative 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。旨在描述微观非均质或非均质材料宏观性质的均匀化理论表明，这种性质随空间变量急剧变化的强非均质材料可以通过所谓的均化(或有效均质)材料来近似描述。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化理论中为一大类不同环境下的偏微分方程建立最优的定量结果，这些偏微分方程大多出现在材料科学中。这项研究集中在该领域的几个具有挑战性的问题上，并将开发新的方法和技术。这些结果将为强非均匀材料的数值模拟提供理论基础和指导。首席研究员将继续他正在进行的偏微分方程组定量齐次化的研究计划。这个项目的主要重点将是关于最优正则性估计(直到边界并且关于非齐次尺度是一致的)和具有有界区域中快速振荡系数的散度形式的二阶椭圆型和抛物型方程的精确收敛速度。更具体地说，要研究的问题包括：(1)椭圆型方程和具有几乎周期系数的系统；(2)具有周期系数的线性弹性系统和Stokes系统；(3)穿孔域上的一致正则性估计；以及(4)抛物型方程和具有含时周期系数的系统的定量齐次化。本文的研究是在调和分析和偏微分方程的交界处进行的。现有的和新的谐波分析技术有望在这一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.