Singularities and Smoothness in Geometric Partial Differential Equations

几何偏微分方程中的奇异性和光滑性

基本信息

批准号：
1809011
负责人：
Aaron Naber
金额：
$ 22.6万
依托单位：
Northwestern University
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2018
资助国家：
美国
起止时间：
2018-07-01 至 2022-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1809011&HistoricalAwards=false
关键词：
Singularities Smoothness Geometric Partial Differential

项目摘要

The scientific goal of this project is to study various geometrically motivated equations and their applications. One area of investigation is energy-minimizing mappings between curved spaces (known as nonlinear harmonic maps) and their singular sets, i.e., points where the mapping ceases to be smooth. Another area of investigation is the structure of the spaces that arise as limits of smooth spaces with bounded curvature, and a third area is the regularity of solutions to the Yang-Mills equations which arise particle physics. The research project also touches on other mathematical subjects such as Reifenberg analysis, spectral analysis, stochastic analysis and metric-measure spaces. In each area, open problems which are at the forefront of the topic are being pursued. These are topics which have broad applications to many areas of mathematics and physics, and solutions to these problems would open doors to tackling the next generation of problems. Many parts of this project also involve the training of early-career researchers, as several lines of investigation involve collaboration with senior graduate students and postdoctoral scholars.The first part of this overall project is concerned with the energy identity and the W2,1-conjecture for nonlinear harmonic maps. Roughly, the energy identity is a conjectural picture which gives an explicit formula for the blow-up behavior of sequences of nonlinear harmonic maps. The W2,1-conjecture is the easily stated conjecture that stationary harmonic maps have a priori L1 estimates for their hessians. Valtorta and the PI have solved these conjectures for stationary Yang-Mills, but the methods do not work for harmonic maps, and the PI will solve the problems by other means. In another line of investigation, joint with Yu Wang, the PI will answer some open questions by Hardt and Lin on nonlinear harmonic maps. Roughly, the goal is show one can solve for stable harmonic maps in the class of strongly H1-mappings. The second part of the proposal would address issues involving the regularity of spaces with lower and bounded Ricci curvature. Together with Wenshuai Jiang the first project of this part would study the energy identity for limits of manifolds with bounded Ricci curvature. Though similar in spirit, the problem itself is actually very different from the harmonic map case. To begin with, the energy in this context is the L2 curvature form, and it has only been very recently that one even knows this is a bounded measure. Secondly, the predicted form of the defect measure in this context can be computed by the singularity behavior in the limit. Additionally, with Bob Haslhofer, the PI will study connections between bounded Ricci curvature and the analysis on path space. The PI will prove differential Harnack estimates for martingales on spaces with two sided Ricci curvature bounds. The solution to each of these problems involve the development of new techniques and ideas, which themselves one would expect to be even more interesting than the problems.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

该项目的科学目标是研究各种几何动机方程式及其应用。一个调查领域是在弯曲空间（称为非线性谐波图）之间的能量最小化映射及其奇异集合，即映射不再平滑的点。调查的另一个领域是作为曲率有界曲率的平滑空间限制的空间的结构，第三个区域是对Yang-Mills方程的溶液的规律性。该研究项目还涉及其他数学主题，例如Reifenberg分析，光谱分析，随机分析和度量测量空间。在每个领域，正在追求在该主题的最前沿的开放问题。这些主题在许多数学和物理学领域都有广泛的应用，解决这些问题将打开解决下一代问题的大门。该项目的许多部分还涉及对早期研究人员的培训，因为几项调查涉及与高级研究生和博士后学者的合作。该整体项目的第一部分与非线性和谐图的能源身份和W2,1-contionure有关。粗略地，能量身份是一种猜想的图片，为非线性谐波图序列的爆炸行为提供了明确的公式。 W2,1-重点是很容易指出的猜想，即固定的谐波图对其黑森人具有先验的L1估计。瓦尔托塔（Valtorta）和PI已解决了这些固定的阳米矿（Yang-Mills）的猜想，但是这些方法对谐波图不起作用，PI将通过其他方式解决问题。在另一项调查中，与王王的联合，PI将回答Hardt和Lin在非线性谐波图上的一些空旷问题。粗略地，目标是显示一个可以在强烈的H1映射类中求解稳定的谐波图。该提案的第二部分将解决涉及RICCI曲率较低和有限的空间规律性的问题。与Wenshuai Jiang一起，这部分的第一个项目将研究具有有界Ricci曲率的歧管限制的能量身份。尽管精神上相似，但问题本身实际上与谐波地图案例大不相同。首先，在这种情况下的能量是L2曲率形式，直到最近才知道这是一个有限的度量。其次，在这种情况下，缺陷度量的预测形式可以通过极限中的奇异行为来计算。此外，使用Bob Haslhofer，PI将研究有限的RICCI曲率与路径空间的分析之间的联系。 PI将在具有两个方面的RICCI曲率边界的空间上证明Martingales的差异性Harnack估计值。解决这些问题的每个问题的解决方案涉及新技术和思想的发展，这本身就期望比问题更有趣。该奖项反映了NSF的法定任务，并被认为是值得通过基金会的知识分子和更广泛影响的评估审查标准来通过评估来支持的。