Inégalités de Sobolev logarithmiques et hypercontractivité en mécanique statistique et en E.D.P.

Abstract : In this thesis we are interested in functional inequalities as inequalities of Poincaré, logarithmic Sobolev, Sobolev, and the so called inequalities of transport. At first, we study the inequalities of Poincaré and logarithmic Sobo\-lev for models of statistical mechanics. We then propose a new class of phases such that the associated Gibbs measure satisfy these two inequalities. In a 2nd part, the inequalities of logarithmic Sobolev and Sobolev are explored via Hamilton-Jacobi's equations. We show, in a similar way as Gross did in 1975 for diffusions semigroups, the equivalence between the inequality of logarithmic Sobolev and the hypercontractivity of Hamilton-Jacobi's equations. This equivalence allows to show, using a new method proposed by Otto and Villani, that the inequality of logarithmic Sobolev implies a quadratic inequality of transport. In the same way as Varopoulos in 1985 for the diffusions semigroups, we establish the link between the Sobolev's inequality and ultracontractivity of the Hamilton-Jacobi's equations. Finally, we study the inequalities of transport in a general frame. This study allows on one hand to give the link between inequalities of modified logarithmic Sobolev and some particular inequalities of transport, and on the other hand to give an example of quadratic inequality of transport for a measure in infinite dimension, namely the Wiener's measure.
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https://tel.archives-ouvertes.fr/tel-00001145
Contributor : Ivan Gentil <>
Submitted on : Tuesday, February 26, 2002 - 1:40:41 PM
Last modification on : Monday, April 29, 2019 - 3:22:51 PM
Long-term archiving on : Friday, April 2, 2010 - 7:52:18 PM

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Ivan Gentil. Inégalités de Sobolev logarithmiques et hypercontractivité en mécanique statistique et en E.D.P.. Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2001. Français. ⟨tel-00001145⟩

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