Improved interpolation inequalities, relative entropy and fast diffusion equations

TL;DR

This paper develops improved interpolation inequalities that connect Sobolev and logarithmic Sobolev inequalities, providing explicit estimates and leveraging nonlinear evolution equations and entropy methods to quantify the distance to optimal functions.

Contribution

It introduces a novel approach using nonlinear evolution equations and entropy estimates to refine interpolation inequalities with explicit constants and remainder terms.

Findings

Explicit remainder estimates for interpolation inequalities

Improved inequalities with detailed constants

New method based on entropy production along flows

Abstract

We consider a family of Gagliardo-Nirenberg-Sobolev interpolation inequalities which interpolate between Sobolev's inequality and the logarithmic Sobolev inequality, with optimal constants. The difference of the two terms in the interpolation inequalities (written with optimal constant) measures a distance to the manifold of the optimal functions. We give an explicit estimate of the remainder term and establish an improved inequality, with explicit norms and fully detailed constants. Our approach is based on nonlinear evolution equations and improved entropy - entropy production estimates along the associated flow. Optimizing a relative entropy functional with respect to a scaling parameter, or handling properly second moment estimates, turns out to be the central technical issue. This is a new method in the theory of nonlinear evolution equations, which can be interpreted as the best fit of the solution in the asymptotic regime among all asymptotic profiles.