Functional Inequalities and Improvements: Stability - Part 2

Explore a comprehensive lecture on functional inequalities and their stability properties in this 55-minute talk by María J. Esteban from the Centre National de la Recherche Scientifique in Paris. Delve into the development of functional inequalities and their importance in various mathematical fields, including analysis, spectral analysis, mathematical physics, partial differential equations, and differential geometry. Examine strategies for addressing best constants and extremal functions in inequalities, with a focus on recent advancements in improved inequalities for non-optimizers. Learn about the Caffarelli-Kohn-Nirenberg inequalities, symmetry properties, and the solution to a related conjecture. Investigate magnetic problems, vector-valued functions, and systems. Discover stability properties for critical inequalities and gain insights into open problems and new research directions in this advanced mathematical exploration.

Intro
Outline for today
Caffarelli-Kohn-Nirenberg (CKN) inequalities (1984)
Symmetry (3)
Linear instability of radial minimizers: the Felli-Schneider curve
Solution of the conjecture : A Sobolev type inequality
The flow
Elliptic proof
Corollary: consequences for non-compact manifolds
Magnetic problems
Inequalities involving vector valued functions; systems
go beyond all this and get stability properties for the inequalities
Towards an improved inequality
Stability for critical inequalities
Open problems and new directions