Harmonic Analysis and Partial Differential Equations

调和分析和偏微分方程

• 批准号: 9988711
• 负责人: Carlos Kenig
• 金额: $ 47.19万
• 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: 2000
• 资助国家: 美国
• 起止时间: 2000-07-01 至 2006-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9988711&HistoricalAwards=false
• 关键词: Harmonic; Analysis; Partial; Differential; Equations

In this proposal,the PI will concentrate on the study of "locally flat" sets,the geometry of the measures that they support,and the connections with harmonicmeasure.Also,the PI will study boundary value problems under minimal smoothnessconditions,and the existence and regularity of solutions to two-phase free boundar problems.The PI will also study non-linear evolution problems,focusing onLiouville theorems for parabolic regularizations of conservation laws,and on theconnection between oscillatory integrals,restriction theorems for the Fouriertransform,and the well-posedness of solutions to non-linear dispersive equationsand systems. The main guiding principle in the proposal is the development of various aspects of harmonic analysis,potential theory and geometric measure theory,with a view to applications to a number of challenging problems in linear and non-linear partial differential equations,and the implementation of these applications.It is expected that this research will lead to the development of new graduatecourses,to Phd dissertations,and to the sposoring of a number of postdoc's reaserch.

在这个计划中，PI将集中研究“局部平坦”集，它们所支持的测度的几何，以及与调和测度的联系。此外，PI将研究最小光滑条件下的边值问题，以及两相自由边界问题解的存在性和正则性。PI还将研究非线性发展问题，重点是守恒律抛物正则化的Liouville定理，振荡积分之间的联系，傅立叶变换的限制定理，以及非线性色散方程和系统解的适定性。该提案的主要指导原则是发展调和分析、势理论和几何测度理论的各个方面，以期应用于线性和非线性偏微分方程中的一些具有挑战性的问题，并实现这些应用。预计这项研究将导致新的研究生课程的发展，博士论文，以及一些博士后的研究成果。