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.

该项目研究了在变化和部分微分方程计算中产生的几个物理重要模型中的奇异性。最初在力学的背景下开发的变化计算是一个数学研究领域，它研究了最小化能量的形状或功能。例如，在固定的封闭体积下，肥皂气泡会呈圆形，因为它可以最大程度地减少表面张力。在某些情况下，这些天然能量暴露了奇异性 - 解决方案不光滑的地方。该项目研究了这种奇异性的形成和结构。该项目的综合教育组成部分为高中学生提供了数学夏季计划，该学习半设计旨在增加对本地研究的半年度研究半年度研究生课程的访问。项目研究在变异计算的三个领域中的奇异性形成：（平稳）边界问题，培训范围，培养地零件平等的解决方案均可进行副局部差异和能量范围。每个主题都代表了奇异性形成研究中的一个核心问题。什么时候存在奇异性？它们什么时候稳定或通用？什么时候可以精确地描述解决方案在奇异性形成的时空社区中的行为？该项目将开发技术，以区分最小化器中的奇异性形成与稳定溶液中的奇异性形成。该项目的第二个组成部分涉及在不使用单调性的情况下流动奇异性的扰动。该项目的第三个组成部分研究了分析性在泡泡和孤子分辨率独特性中的作用。该项目的教育组成部分旨在通过高中，本科和研究生级别的新且更容易获得的内容来增加对分析和差异方程式感兴趣的学员的供应。这项奖项反映了NSF的法定使命，并通过使用基金会的知识分子优点和更广泛的影响审查标准来诚实地通过评估来诚实地支持。