D. Matthes, A. Jüngel, G. Toscani:
"Convex Sobolev Inequalities Derived from Entropy Dissipation";
Archive for Rational Mechanics and Analysis, 199 (2011), S. 563 - 596.

Kurzfassung deutsch:
Siehe englisches Abstract.

Kurzfassung englisch:
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-Emery
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.

Entropy method; Fokker-Planck equations; logarithmic Sobolev inequality

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