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Inequalities for generalized entropy and optimal transportation. (English) Zbl 1135.49026
Carvalho, M. C. (ed.) et al., Recent advances in the theory and applications of mass transport. Papers from the summer school on mass transportation methods in kinetic theory and hydrodynamics, Ponta Delgada, Azores, Portugal, September 4–9, 2000. Providence, RI: American Mathematical Society (AMS) (ISBN 0-8218-3278-6/pbk). Contemporary Mathematics 353, 73-94 (2004).
Summary: A new concept of Fisher-information is introduction through a cost function. That concept is used to obtain extensions and variants of transport and logarithmic Sobolev inequalities for general entropy functionals and transport costs. Our proofs rely on optimal mass transport from the Monge-Kantorovich theory. They express the convexity of entropy functionals with respect to suitably chosen paths on the set of probability measures.
For the entire collection see [Zbl 1052.35003].

MSC:
49Q20 Variational problems in a geometric measure-theoretic setting
35K50 Systems of parabolic equations, boundary value problems (MSC2000)
35K55 Nonlinear parabolic equations
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
90B06 Transportation, logistics and supply chain management
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