Harmonic Analysis and Partial Differential Equations

调和分析和偏微分方程

• 批准号: 0968472
• 负责人: Carlos Kenig
• 金额: $ 40万
• 依托单位: University of Chicago
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2015-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0968472&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Partial; Differential; Equations

The purpose of this project is to advance the principal investigator's research in the development of various aspects of harmonic analysis and partial differential equations. The main foci of the project will be the study of "critical elements" in critical nonlinear and wave equations, the study of universal profiles for blow-up solutions, the study of defocusing energy supercritical wave equations, the development of the connection between issues of uniqueness of solutions to parabolic and dispersive equations with Hardy's uncertainty principle for the Fourier transform, the study of local and global inverse problems, and the development of the theory of homogenization for elliptic equations and systems on Lipschitz domains.Many of the topics under study in this project have their origins in problems coming from physics, engineering, and biotechnology. It is hoped that the proposed research will have a synergistic effect between those fields and the mathematical fields of analysis and geometry. One prominent feature of the proposal is the principal investigator's collaborative research with female mathematicians. The broader impacts of the project reside in the wide dissemination of the results obtained, through publication of articles and monographs and through lectures, courses, and web-sites, in the training of students and postdocs, in the increased participation of underrepresented groups, and in the many connections with other fields of science and technology.

这个项目的目的是促进首席研究员在调和分析和偏微分方程各个方面的发展研究。该项目的主要重点将是研究临界非线性方程和波动方程中的“临界元素”，研究爆破解的普遍轮廓，研究散焦能量超临界波动方程，发展抛物型和色散型方程解的唯一性问题与Hardy傅立叶变换测不准原理之间的联系，研究局部和整体反问题，以及发展Lipschitz域上的椭圆型方程和系统的齐次化理论。希望拟议的研究将在这些领域与分析和几何的数学领域之间产生协同效应。该提案的一个突出特点是首席研究员与女性数学家合作进行研究。该项目的更广泛影响在于通过发表文章和专著以及通过讲座、课程和网站广泛传播所取得的成果，培训学生和博士后，扩大代表性不足群体的参与，以及与其他科学和技术领域的许多联系。