Interpolation inequalities, nonlinear flows, boundary terms, optimality and linearization

TL;DR

This paper extends entropy methods to weighted fast diffusion equations, analyzing boundary terms, optimal constants, and symmetry breaking, thereby advancing the understanding of interpolation inequalities and nonlinear flows.

Contribution

It introduces a partial extension of entropy methods to parabolic weighted equations and explores their relation to optimal constants and symmetry breaking in interpolation inequalities.

Findings

Established the equivalence of carré du champ and Rényi entropy methods.

Showed entropy methods yield optimal constants in certain inequalities.

Analyzed symmetry breaking ranges in weighted inequalities.

Abstract

This paper is devoted to the computation of the asymptotic boundary terms in entropy methods applied to a fast diffusion equation with weights associated with Caffarelli-Kohn-Nirenberg interpolation inequalities. So far, only elliptic equations have been considered and our goal is to justify, at least partially, an extension of the carr{\'e} du champ / Bakry-Emery / R{\'e}nyi entropy methods to parabolic equations. This makes sense because evolution equations are at the core of the heuristics of the method even when only elliptic equations are considered, but this also raises difficult questions on the regularity and on the growth of the solutions in presence of weights.We also investigate the relations between the optimal constant in the entropy - entropy production inequality, the optimal constant in the information - information production inequality, the asymptotic growth rate of generalized R{\'e}nyi entropy powers under the action of the evolution equation and the optimal range of parameters for symmetry breaking issues in Caffarelli-Kohn-Nirenberg inequalities, under the assumption that the weights do not introduce singular boundary terms at x=0. These considerations are new even in the case without weights. For instance, we establish the equivalence of carr{\'e} du champ and R{\'e}nyi entropy methods and explain why entropy methods produce optimal constants in entropy - entropy production and Gagliardo-Nirenberg inequalities in absence of weights, or optimal symmetry ranges when weights are present.