Stability of Functional and Geometric Inequalities and Applications

函数和几何不等式的稳定性及其应用

• 批准号: 1901427
• 负责人: Robin Neumayer
• 金额: $ 8.35万
• 依托单位: Northwestern University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2021-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1901427&HistoricalAwards=false
• 关键词: Stability; Functional; Geometric; Inequalities; Applications

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.

函数和几何不等式的研究在现代数学中发挥着举足轻重的作用。 几何不等式的一个例子是等周不等式，它的历史可以追溯到古希腊人，它指出在固定体积的所有形状中，球的周长最小。 这种不等式的稳定性是描述近平衡状态行为的属性。 近年来，稳定性主题引起了人们的极大兴趣，这在很大程度上是由于其在物理模型中令人兴奋的应用。 该项目分析了某些函数和几何不等式的稳定性特性，并将它们作为一种工具来解决各个数学领域以及物理和材料科学中不同应用的问题。给定一个不等式，所有等式情况都被表征，问题稳定性如下：假设一个函数（或集合）几乎在不等式中达到相等。那么，在适当的意义上，它是否接近于平等的情况？在这个项目中，稳定性估计将在不同的背景下使用，包括研究具有标量曲率下界的黎曼流形极限空间的正则性、谱形优化问题的正则性，以及液滴模型中能量最小化器的属性原子核。该项目在技术和应用方面汇集了几何分析、偏微分方程和变分法的思想。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进行评估，被认为值得支持标准。