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

奖项：DMS-9970549首席研究员：王畅游这个项目涉及几何变分领域中出现的几个解析问题。它由三部分组成。在第一部分中，我们试图研究流形间映射的Hesse能量泛函的临界点(或双调和映射)的相关问题，如最小化双调和映射的偏正则性，来自四维流形的同伦类之间的双调和映射表示的存在性，以及来自四维区域的双调和映射的热演化。在第二部分中，我们尝试在三维欧氏空间中对定常调和映射流、Ginzburg-Landau系统流、稳定-定常调和映射流以及调和映射流的能量集中和气泡现象进行爆破分析。在这里，我们试图利用几何测量理论和偏微分方程来理解与这些物体相关的缺陷度量。在最后一部分中，PI建议研究p-能量的下界及其在p-调和映射流从二维域到圆的可能的奇性动力学中的应用。我们希望建立与二维复杂Ginzburg-Landau方程涡旋动力学的联系。非线性偏微分方程组是描述微分几何和物理问题的基本工具。调和映射根据满足共同约束的对象族中的自然能量泛函，对最优对象进行建模。调和映射的热流研究这些族中物体的长期动力学行为。这将加深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。双调和映射在研究四阶非线性偏微分方程组和高维共形几何中是不常见的。这一结果将在微分几何、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.