D. Matthes, A. Jüngel, G. Toscani:

"Convex Sobolev inequalities derived from entropy dissipation";

in: "ASC Report 19/2009", issued by: Institute for Analysis and Scientific Computing; Vienna University of Technology, Wien, 2009, ISBN: 978-3-902627-02-5.

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 rst two authors, a re nement of the Bakry- Emery method is established, which allows us to

prove non-trivial inequalities even in situations where the classical Bakry- Emery criterion fails.

The main application of our theory concerns the linearized fast di usion equation in dimensions

d 1, which admits a Poincar e, 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 speci ed 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 rst obtained by Denzler and McCann. In further applications of our method,

we prove convex Sobolev inequalities for a mean eld model for the redistribution of wealth in

a simple market economy, and the Lasota model for blood cell production.

http://www.asc.tuwien.ac.at/preprint/2009/asc19x2009.pdf

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