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Interpolation Inequalities, Nonlinear Flows, Boundary Terms, Optimality and Linearization

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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 carré du champ / Bakry-Emery / Rényi 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–en- tropy production inequality, the optimal constant in the information–information production inequality, the asymptotic growth rate of generalized Rényi 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 carré du champ and Rényi 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.

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Authors and Affiliations

  1. CEREMADE (CNRS UMR no 7534), PSL research university, Universitée Paris-Dauphine, Place de Lattre de Tassigny, 75775, Paris 16, France

    Jean Dolbeault & Maria J. Esteban

  2. School of Mathematics, Georgia Tech, 686 Cherry St., Atlanta, GA, 30332-0160, USA

    Michael Loss

Authors
  1. Jean Dolbeault
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  2. Maria J. Esteban
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  3. Michael Loss
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Correspondence to Jean Dolbeault.

Additional information

Research partially supported by the Projects STAB and Kibord (J.D.) of the French National Research Agency (ANR), and by NSF Grant DMS- 1600560 and the Humboldt Foundation (M.L.). Part of this work was done at the Institute Mittag-Leffler during the fall program Interactions between Partial Differential Equations & Functional Inequalities.

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Dolbeault, J., Esteban, M.J. & Loss, M. Interpolation Inequalities, Nonlinear Flows, Boundary Terms, Optimality and Linearization. J Elliptic Parabol Equ 2, 267–295 (2016). https://doi.org/10.1007/BF03377405

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