Some Problems for Bi-Harmonic Maps, Blow-Up Analysis for Some Variational Problems

双调和映射的一些问题，一些变分问题的放大分析

基本信息

批准号：
9970549
负责人：
Changyou Wang
金额：
$ 5.81万
依托单位：
University of Chicago
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
1999
资助国家：
美国
起止时间：
1999-08-01 至 1999-10-20
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=9970549&HistoricalAwards=false
关键词：
Some Problems Bi Harmonic Maps

项目摘要

Award: DMS-9970549Principal Investigator: Changyou WangWorks on this project concerns several analytic problems arisingfrom the area of geometric variational calculus. It containsthree parts. In the first part, we try to study problems relatedto critical points (or bi-harmonic maps) for Hessian energyfunctionals of maps between manifolds, such as partial regularityfor minimizing bi-harmonic maps, existence of bi-harmonic maprepresentations among homotopy classes from four dimensionalmanifolds, and heat evolutions of bi-harmonic maps from fourdimensional domains. In the second part, we try to developblow-up analysis for flows of stationary harmonic maps, flows ofGinzburg-Landau systems, stable-stationary harmonic maps, andenergy concentrations and bubbling phenomena for entire solutionsto harmonic maps in three dimensional Euclidean space. Here wetry to use geometric measure theory and PDE to understand thedefect measures associated with these objects. In the last part,the PI propose to study lower bound for p-energy and itsapplications to possible dynamics of singularity for p-harmonicmap flows from two dimensional domains to the circle, as p tendsto two. We hope to build connections to vortex dynamics ofcomplex Ginzburg-Landau equations in two dimension.Nonlinear partial differential equations are basic tools todescribe problems arising from both differential geometry andphysics. Harmonic maps model optimal objects with respect tophysically natural energy functionals in families of objectssatisfying common constraints. Heat flows of harmonic maps studythe long-time dynamical behavior of objects in such families. Thestudy will enhance our understanding of these maps, improvemethods to control the singular sets, and predict singularbehavior of solutions to these problems. Bi-harmonic maps arenatural in the study of both fourth order nonlinear PDE andhigher dimensional conformal geometry. Results here will havepotential applications to differential geometry, material scienceincluding liquid crystals, elasticity/plasticity, and fluidmechanics.

奖项：DMS-9970549原理调查员：该项目的Changyou Wangworks涉及几何变异微积分区域引起的几个分析问题。它包含三个部分。在第一部分中，我们试图研究与歧管之间的临界点（或双谐波图）相关的问题，以示为歧管之间地图的Hessian能量功能，例如将双谐光图的存在最小化的部分规律性，在四个Dimensymanifolds和Heational Manyiforts中的同型谐波中的存在，以及来自四个维数型的Heational Manifortions of Four Harmonic Maps的同型谐音。在第二部分中，我们试图为固定谐波图的流量，稳定的谐波图，稳定的谐波图，andenergy浓度和气泡现象进行整个解决方案谐波图的流量进行开发分析。在这里使用几何测量理论和PDE来了解与这些对象相关的措施。在最后一部分中，PI提议研究p能量的下限，并将其应用于p-harmonicmap从二维域流向圆的奇异性动力学，如p tendsto tentsto二。我们希望在二维上建立与Complex Ginzburg-Landau方程的涡流动力学的连接。NonlinelareAr部分微分方程是基本的工具，可以鉴定出由差分几何和物理引起的问题。谐波地图模型模型最佳对象与对象满足共同约束的家族中的自然能量功能。谐波图的热流研究了此类家族中物体的长期动力学行为。 Thestudy将增强我们对这些图的理解，改善控制奇异集的方法，并预测解决方案的奇异行为。在对第四阶非线性PDE和高维形状的研究研究中，双谐波图。此处的结果将对差异几何形状，包括液体晶体，弹性/可塑性和流体力学的材料科学具有潜在的应用。