L'évolution de l'énergie totale d'un gaz le long d'un transport
optimal et ses applications à des inégalites géometriques

3, 5, 10 novembre 2003

Centre de recherches mathématiques,
Université de Montréal
Montréal, Qc Canada
Pavillon André-Aisenstadt, salle 5340

3 novembre : 14 h- 15 h 30

5 novembre : 15 h -16 h 30

10 novembre : 14 h - 15 h 30

Nassif Ghoussoub
University of British Columbia


Using the Monge-Kantorovich theory of mass transport, we propose an inequality for the relative total energy -- internal, potential and interactive -- of two arbitrary probability densities, their Wasserstein distance, their barycenters and their entropy production functional. This inequality is remarkably encompassing as it implies most known geometrical -- Gaussian and Euclidean -- inequalities as well as new ones, while allowing a direct and unified way for computing best constants and extremals: Sobolev, Gagliardo-Nirenberg, Log Sobolev, HWI, Transport, Concentration, Poincare, etc... As expected, such inequalities also lead to exponential rates of convergence to equilibria for solutions of Fokker-Planck and McKean-Vlasov type equations. The proposed inequality also leads to a remarkable correspondence between ground state solutions of certain quasilinear (or semi-linear) equations and stationary solutions of (non-linear) Fokker-Planck type equations.

22 septembre 2003