Functional inequalities: nonlinear flows and entropy methods as a tool for obtaining sharp and constructive results

TL;DR

This paper explores how entropy methods and nonlinear flows can be used to derive sharp, constructive results in functional inequalities, with applications to stability and symmetry in various mathematical inequalities.

Contribution

It introduces recent developments in entropy-entropy production inequalities and demonstrates their use in proving optimality, symmetry, and stability in key functional inequalities.

Findings

Entropy methods relate nonlinear and linear regimes in inequalities.

Optimal constants correspond to decay rates of entropy.

Framework yields new stability and symmetry results.

Abstract

Interpolation inequalities play an essential role in Analysis with fundamental consequences in Mathematical Physics, Nonlinear Partial Differential Equations (PDEs), Markov Processes, etc., and have a wide range of applications in various other areas of Science. Research interests have evolved over the years: while mathematicians were originally focussed on abstract properties (for instance appropriate notions of functional spaces for the existence of weak solutions in PDEs), more qualitative questions (for instance, bifurcation diagrams, multiplicity of the solutions in PDEs and their qualitative behaviour) progressively emerged. The use of entropy methods in nonlinear PDEs is a typical example: in some cases, the optimal constant in the inequality can be interpreted as an optimal rate of decay of an entropy for an associated evolution equation. Much more has been learned by adopting this point of view. This paper aims at illustrating some of these recent aspect of entropy-entropy production inequalities, with applications to stability in Gagliardo-Nirenberg-Sobolev inequalities and symmetry results in Caffarelli-Kohn-Nirenberg inequalities. Entropy methods provide a framework which relates nonlinear regimes with their linearized counterparts. This framework allows to prove optimality results, symmetry results and stability estimates. Some emphasis will be put on the hidden structure which explain such properties. Related open problems will be listed.