Harmonic Analysis and Partial Differential Equations

在这个提议中，PI将集中研究“局部平坦”集合，它们支持的度量的几何形状，以及与谐波度量的联系。此外，PI将研究最小平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规律性。PI还将研究非线性演化问题，重点关注守恒定律的抛物正则化的liouville定理，以及振荡积分之间的联系，傅里叶变换的限制定理，以及非线性色散方程和系统解的适定性。提案的主要指导原则是发展谐波分析，势理论和几何测量理论的各个方面，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，并赞助一些博士后研究。

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.