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

• 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

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 首席研究员：Changyou Wang 该项目涉及几何变分微积分领域中出现的几个解析问题。它包含三个部分。在第一部分中，我们尝试研究流形间映射的Hessian能量泛函的临界点（或双调和映射）相关问题，例如最小化双调和映射的部分正则性、四维流形同伦类之间双调和映射表示的存在性，以及来自四维域的双调和图的热演化。 在第二部分中，我们尝试对稳态调和图流、Ginzburg-Landau 系统流、稳定-稳态调和图以及三维欧几里德空间中调和图整个解的能量集中和冒泡现象进行爆炸分析。 这里我们尝试使用几何测度理论和偏微分方程来理解与这些对象相关的缺陷测度。 在最后一部分中，PI建议研究p能量的下界及其在p调和图从二维域到圆的奇异性可能动力学中的应用，因为p趋于2。我们希望建立与二维复杂Ginzburg-Landau方程的涡动力学的联系。非线性偏微分方程是描述微分几何和物理学所产生的问题的基本工具。调和图根据满足共同约束的对象族中的物理自然能量泛函对最佳对象进行建模。调和图的热流研究了此类族中物体的长期动态行为。这项研究将增强我们对这些映射的理解，改进控制奇异集的方法，并预测这些问题的解决方案的奇异行为。双调和映射在四阶非线性偏微分方程和高维共形几何的研究中是很自然的。这里的结果将在微分几何、材料科学（包括液晶、弹性/塑性和流体力学）方面具有潜在的应用。