Archive for Rational Mechanics and Analysis

, Volume 199, Issue 2, pp 563–596 | Cite as

Convex Sobolev Inequalities Derived from Entropy Dissipation

  • Daniel MatthesEmail author
  • Ansgar Jüngel
  • Giuseppe Toscani


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.


Sobolev Inequality Planck Equation Logarithmic Sobolev Inequality Functional Inequality Optimal Constant 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 2010

Authors and Affiliations

  • Daniel Matthes
    • 1
    Email author
  • Ansgar Jüngel
    • 1
  • Giuseppe Toscani
    • 2
  1. 1.Institute for Analysis and Scientific ComputingVienna University of TechnologyViennaAustria
  2. 2.Department of MathematicsUniversity of PaviaPaviaItaly

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