Stability of Functional and Geometric Inequalities and 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.