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Convex Sobolev Inequalities Derived from Entropy Dissipation

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Abstract

We study families of convex Sobolev inequalities, which arise as entropy–dissipation relations for certain linear Fokker–Planck equations. Extending the ideas recently developed by the first two authors, a refinement of the Bakry–Émery method is established, which allows us to prove non-trivial inequalities even in situations where the classical Bakry–Émery criterion fails. The main application of our theory concerns the linearized fast diffusion equation in dimensions d ≧ 1, which admits a Poincaré, but no logarithmic Sobolev inequality. We calculate bounds on the constants in the interpolating convex Sobolev inequalities, and prove that these bounds are sharp on a specified range. In dimension d = 1, our estimates improve the corresponding results that can be obtained by the measure-theoretic techniques of Barthe and Roberto. As a by-product, we give a short and elementary alternative proof of the sharp spectral gap inequality first obtained by Denzler and McCann. In further applications of our method, we prove convex Sobolev inequalities for a mean field model for the redistribution of wealth in a simple market economy, and the Lasota model for blood cell production.

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Authors and Affiliations

  1. Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8-10, 1040, Vienna, Austria

    Daniel Matthes & Ansgar Jüngel

  2. Department of Mathematics, University of Pavia, Via Ferrata 1, 27100, Pavia, Italy

    Giuseppe Toscani

Authors
  1. Daniel Matthes
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  2. Ansgar Jüngel
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  3. Giuseppe Toscani
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Correspondence to Daniel Matthes.

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Communicated by L. Ambrosio

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Matthes, D., Jüngel, A. & Toscani, G. Convex Sobolev Inequalities Derived from Entropy Dissipation. Arch Rational Mech Anal 199, 563–596 (2011). https://doi.org/10.1007/s00205-010-0331-9

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  • DOI: https://doi.org/10.1007/s00205-010-0331-9

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