Harmonic Analysis and Homogenization of Partial Differential Equations

偏微分方程的调和分析与齐次化

• 批准号: 1161154
• 负责人: Zhongwei Shen
• 金额: $ 19.34万
• 依托单位: University of Kentucky Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2012
• 资助国家: 美国
• 起止时间: 2012-09-01 至 2016-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1161154&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Homogenization; Partial; Differential

This project concerns elliptic partial differential equations and systems with rapidly oscillating periodic coefficients, which arise in the theory of homogenization. Shen and his collaborators will focus on several challenging problems in the area. Resolution of these problems will provide better understanding of some fundamental issues in homogenization, including uniform sharp regularity estimates and rates of convergence of solutions of boundary value problems, asymptotic behavior of eigenvalues and uniform estimates of eigenfunctions, boundary layer phenomenon, and uniform controllability and stabilization for distributed systems. Elliptic equations with periodic coefficients are considered as a model case in homogenization. New techniques and approaches developed for this case will be useful in studying homogenization in other important settings, such as non-uniformly oscillating coefficients, almost periodic coefficients, perforated domains, and evolution operators with highly oscillating coefficients. 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. 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 shows that such materials may be approximately described via a homogenized or effective homogeneous material. As such the theory of homogenization of partial differential equations with rapidly oscillating coefficients has many important applications in physics, mechanics, and modern technology. The proposed research will develop new methods and techniques that will provide theoretical foundation and guidance for numerical simulations in strongly inhomogeneous materials. The findings from the research will be disseminated in the scientific community by Shen and his collaborators through lectures in conferences, workshops, and graduate courses as well as publishing in mathematical journals and websites. Shen is committed to the training of future generations of mathematicians; graduate students and junior researchers will be involved on the project.

这个项目涉及椭圆型偏微分方程和系统与快速振荡的周期系数，这出现在理论的均匀化。沈和他的合作者将专注于该领域的几个具有挑战性的问题。这些问题的解决将有助于更好地理解均匀化中的一些基本问题，包括边值问题解的一致尖锐正则性估计和收敛速度，特征值的渐近行为和特征函数的一致估计，边界层现象，以及分布式系统的一致可控性和稳定性。周期系数的椭圆方程被认为是均匀化的一个模型。为这种情况下开发的新技术和方法将是有用的，在研究均匀化在其他重要的设置，如非均匀振荡系数，几乎周期系数，穿孔域，并与高振荡系数的演化算子。该研究处于调和分析和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。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。均匀化理论表明，这种材料可以近似地通过均匀化或有效均匀材料来描述。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、力学和现代技术中有许多重要的应用。该研究将为强非均匀材料的数值模拟提供理论基础和指导。研究结果将由沉和他的合作者通过会议，研讨会和研究生课程的讲座以及在数学期刊和网站上发表在科学界传播。沉致力于培养未来几代数学家;研究生和初级研究人员将参与该项目。