Journal
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
Volume 30, Issue 5, Pages 917-934Publisher
GAUTHIER-VILLARS/EDITIONS ELSEVIER
DOI: 10.1016/j.anihpc.2012.12.004
Keywords
Gagliardo-Nirenberg-Sobolev inequalities; Improved inequalities; Manifold of optimal functions; Entropy-entropy production method; Fast diffusion equation; Barenblatt solutions; Second moment; Intermediate asymptotics; Sharp rates; Optimal constants
Categories
Funding
- ANR project CBDif-Fr
- ANR project EVOL
- MIUR project Optimal mass transportation, geometrical and functional inequalities with applications
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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. (C) 2013 Elsevier Masson SAS. All rights reserved.
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