The optimal evolution of the free energy of interacting gases and its applications to geometric inequalities new and old

November 3- 5- 6, 2003

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

November 3 : 2 p.m. - 3 : 30 p.m

November 5 : 3 p.m. - 4 : 30 p.m.

November 10 : 2 p.m. - 3 : 30 p.m.

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.

September 22, 2003