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

摘要

This paper is devoted to the computation of the asymptotic boundary terms inentropy methods applied to a fast diffusion equation with weights associatedwith Caffarelli-Kohn-Nirenberg interpolation inequalities. So far, onlyelliptic equations have been considered and our goal is to justify, at leastpartially, an extension of the carr{\'e} du champ / Bakry-Emery / R{\'e}nyientropy methods to parabolic equations. This makes sense because evolutionequations are at the core of the heuristics of the method even when onlyelliptic equations are considered, but this also raises difficult questions onthe regularity and on the growth of the solutions in presence of weights.Wealso 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 generalizedR{\'e}nyi entropy powers under the action of the evolution equation and theoptimal range of parameters for symmetry breaking issues inCaffarelli-Kohn-Nirenberg inequalities, under the assumption that the weightsdo not introduce singular boundary terms at x=0. These considerations are neweven in the case without weights. For instance, we establish the equivalence ofcarr{\'e} du champ and R{\'e}nyi entropy methods and explain why entropymethods produce optimal constants in entropy - entropy production andGagliardo-Nirenberg inequalities in absence of weights, or optimal symmetryranges when weights are present.