Harmonic Analysis and Homogenization of Partial Differential Equations

这个项目涉及椭圆型偏微分方程组和周期系数快速振荡的系统，这是在齐次化理论中出现的。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齐次化中的一些基本问题，包括边值问题解的一致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镇定。具有周期系数的椭圆型方程被视为齐次化的模型情形。针对这种情况开发的新技术和方法将有助于研究其他重要环境中的齐次化，如非一致振荡系数、几乎周期系数、穿孔区域和具有高振荡系数的演化算子。所提出的研究位于调和分析和偏微分方程组的交界处。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。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。均匀化理论表明，这种材料可以用一种均化或有效的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、力学和现代技术中有许多重要的应用。该研究将发展新的方法和技术，为强非均匀材料的数值模拟提供理论基础和指导。这项研究的发现将由沈和他的合作者通过在会议、研讨会和研究生课程上的演讲以及在数学期刊和网站上发表来在科学界传播。沈阳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。

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.