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

奖项：DMS-9970549主要研究者：王长友本课题涉及几何变分领域的几个解析问题。它包括三个部分。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，如极小化双调和映射的部分正则性，四维流形上同伦类间双调和映射表示的存在性，四维区域上双调和映射的热演化等。 在第二部分中，我们尝试在三维欧氏空间中发展平稳调和映射流、Ginzburg-Landau系统流、稳定-平稳调和映射的Blow-up分析以及调和映射整体解的能量集中和冒泡现象。 在这里，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。 最后，PI提出研究p-能量的下界及其在p-调和映射流的奇异性动力学中的应用，当p趋于2时，p-调和映射流从二维区域到圆周的奇异性动力学。我们希望在二维中建立与复杂Ginzburg-Landau方程的涡旋动力学的联系。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。调和映射模型的最佳对象与拓扑自然的能量泛函在家庭的对象满足共同的约束。调和映射热流研究这类族中物体的长时间动力学行为。这些研究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性。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一个很自然的问题。本文的结果在微分几何、材料科学（包括液晶）、弹性/塑性和流体力学等领域具有潜在的应用价值。

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.