Stability of Functional and Geometric Inequalities and Applications

泛函不等式和几何不等式的研究在现代数学中起着举足轻重的作用。 几何不等式的一个例子，其历史可以追溯到古希腊人，是等周不等式，它指出球在所有固定体积的形状中具有最小的周长。 这类不等式的稳定性是描述几乎平衡态行为的性质。 近年来，稳定性的主题引起了人们的极大兴趣，这在很大程度上是由于它在物理模型中令人兴奋的应用。 本项目分析了某些函数和几何不等式的稳定性，并将其作为一种工具来解决各种数学领域的问题，并在物理学和材料科学中有着不同的应用。给定一个不等式，其所有的等式情况都被表征，稳定性问题如下：假设一个函数（或集合）几乎在不等式中实现了等式。那么，在适当的意义上，它是否接近于一个平等的情况呢？在这个项目中，稳定性估计将在不同的背景下，包括研究黎曼流形的极限空间的正则性与标量曲率的下限，在频谱形状优化问题的规律性，和能量极小化的性质在液滴模型原子核。该项目汇集了几何分析、偏微分方程和变分法的技术和应用方面的想法。该奖项反映了NSF的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.