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CAREER: Harmonic Analysis and the Stability of Singularities in the Calculus of Variations

职业：变分演算中的调和分析和奇点稳定性

基本信息

批准号：
2143719
负责人：
Max Engelstein
金额：
$ 50万
依托单位：
University of Minnesota-Twin Cities
依托单位国家：
美国
项目类别：
Continuing Grant
财政年份：
2022
资助国家：
美国
起止时间：
2022-07-01 至 2027-06-30
项目状态：
未结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2143719&HistoricalAwards=false
关键词：
CAREER Harmonic Analysis Stability Singularities

项目摘要

This project investigates singularities in several physically important models arising in the calculus of variations and partial differential equations. Developed initially in the context of mechanics, the calculus of variations is a mathematical field of study, which investigates shapes or functions which minimize energy. For example, a soap bubble takes its round shape because it minimizes surface tension given a fixed enclosed volume. In some situations, minimizers to these natural energies exhibit singularities – places where the solution is not smooth. This project investigates the formation and structure of such singularities. The integrated educational component of the project supports a mathematical summer program for high school students, a learning seminar designed to increase access to local research seminars, and a new and interdisciplinary graduate course.The project studies singularity formation in three areas of the calculus of variations: (stationary) free boundary problems, nodal sets of solutions to parabolic partial differential equations, and energy critical evolution on manifolds. Each topic represents a central question in the study of singularity formation. When do singularities exist? When are they stable or generic? When can one precisely describe the behavior of a solution in a space-time neighborhood of the singularity formation? The project will develop techniques to distinguish singularity formation in minimizers from singularity formation in stable solutions. A second component of the project involves the perturbation of singularities in flows without the use of monotonicity. The third component of the project investigates the role of analyticity in the uniqueness of bubbling and soliton resolution. The educational component of the project seeks to increase the supply of trainees interested in analysis and differential equations via new and more accessible content at the high school, undergraduate and graduate levels.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.

这个项目研究在变分法和偏微分方程中产生的几个物理上重要的模型中的奇点。变分法最初是在力学的背景下发展起来的，是一个数学研究领域，它研究使能量最小化的形状或函数。例如，肥皂泡之所以是圆形，是因为它在固定的封闭体积下使表面张力最小化。在某些情况下，这些自然能量的最小化会表现出奇点--解不光滑的地方。这个项目研究这种奇异性的形成和结构。该项目的综合教育部分支持高中生数学暑期项目、旨在增加参与当地研究研讨会机会的学习研讨会以及新的跨学科研究生课程。该项目研究变分法三个领域的奇异性形成：（静态）自由边界问题，抛物型偏微分方程的解的节点集，以及流形上的能量临界演化。每个主题都代表了奇点形成研究中的一个中心问题。奇点何时存在？它们什么时候是稳定的或通用的？什么时候可以精确地描述奇点形成的时空邻域中的解的行为？该项目将开发技术，以区分奇点形成的极小化稳定的解决方案。该项目的第二个组成部分涉及流中奇点的扰动，而不使用单调性。该项目的第三个组成部分调查的作用，解析的唯一性冒泡和孤立子的决议。该项目的教育部分旨在通过高中、本科和研究生阶段新的和更容易获得的内容来增加对分析和微分方程感兴趣的学员的供应。该奖项反映了NSF的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。