Stability of Functional and Geometric Inequalities and Applications

函数和几何不等式的稳定性及其应用

基本信息

批准号：
2200886
负责人：
Robin Neumayer
金额：
$ 8.35万
依托单位：
Carnegie-Mellon University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2021
资助国家：
美国
起止时间：
2021-10-01 至 2022-09-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=2200886&HistoricalAwards=false
关键词：
Stability Functional Geometric Inequalities Applications

项目摘要

The study of functional and geometric inequalities plays a pivotal role in modern mathematics. An example of a geometric inequality, with a history dating back to the ancient Greeks, is the isoperimetric inequality, which states that a ball has the least perimeter among all shapes of a fixed volume. The stability of such inequalities is a property that describes behavior of almost-equilibrium states. The topic of stability has generated substantial interest in recent years, due in large part to its exciting applications to physical models. This project analyzes the stability properties of certain functional and geometric inequalities, and applies them as a tool to address problems across various mathematical fields and with diverse applications in physics and materials science.Given an inequality for which all equality cases are characterized, the question of stability is the following: suppose a function (or set) almost achieves equality in the inequality. Then is it close, in a suitable sense, to an equality case? In this project, stability estimates will be employed in varied contexts, including the study of regularity properties for limit spaces of Riemannian manifolds with lower bounds on scalar curvature, regularity in spectral shape optimization problems, and properties of energy minimizers in the liquid drop model for atomic nuclei. The project brings together ideas across geometric analysis, partial differential equations and the calculus of variations, in terms of both techniques and applications.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.

功能和几何不平等的研究在现代数学中起着关键作用。等级不等式的几何不等式的一个例子是等等的不平等现象，它指出球在固定体积的所有形状中的周长最小。这种不平等的稳定性是描述几乎平衡状态的行为的特性。近年来，稳定性的主题引起了巨大的兴趣，这在很大程度上是由于其令人兴奋的物理模型应用程序。该项目分析了某些功能和几何不等式的稳定性，并将其应用于解决各个数学领域的问题以及在物理和材料科学中的各种应用中解决问题的工具。赋予所有平等案例的不平等现象，其稳定性的特征是，稳定性的问题在于：假设功能（或集合）在不平等的情况下达到等值。那么，从适当的意义上讲，它是平等案例的近距离吗？在该项目中，将在各种环境中使用稳定性估计，包括研究标量曲率上的较低界限的riemannian歧管的限制性能，光谱形状优化问题的规律性以及液体液滴模型中的能量最小化的特性。该项目在几何分析，部分微分方程和变化的计算方面汇集了思想，从技术和应用方面。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智力优点和更广泛的影响审查标准通过评估来获得支持的。