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

项目负责人：王畅游本项目研究内容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）的相关问题，例如最小化双调和映射的部分正则性，四维流形同伦类间双调和映射表示的存在性，以及四维域上双调和映射的热演化。在第二部分中，我们尝试对稳态调和图的流动、金兹堡-朗道体系的流动、稳定-平稳调和图以及三维欧几里德空间调和图全解的能量集中和鼓泡现象进行爆破分析。在这里，我们尝试使用几何测量理论和PDE来理解与这些对象相关的缺陷测量。最后，在p趋向于2的情况下，研究p能量的下界及其在p-谐波映射流从二维域到圆的可能的奇点动力学中的应用。我们希望建立二维复杂金兹堡-朗道方程涡旋动力学的联系。非线性偏微分方程是描述微分几何和物理问题的基本工具。调和映射在满足共同约束的目标族中，根据物理上的自然能量泛函对最优目标进行建模。谐波图的热流研究这类物体的长期动力学行为。该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，并预测这些问题解的奇异行为。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都是很自然的。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有潜在的应用。

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.