Contributions to Books:
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,
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.
Electronic version of the publication:
Created from the Publication Database of the Vienna University of Technology.