Niveau: Supérieur, Doctorat, Bac+8
OPTIMAL TRANSPORTATION, DISSIPATIVE PDE'S AND FUNCTIONAL INEQUALITIES C. VILLANI Recent research has shown the emergence of an intricate pattern of tight links between certain classes of optimal transportation problems, certain classes of evolution PDE's and certain classes of functional inequalities. It is my purpose in these notes to convey an idea of these links through (hopefully) pedagogical examples taken from recent works by various authors. During this process, we shall encounter such diverse areas as fluid me- chanics, granular material physics, mean-field limits in statistical mechanics, and optimal Sobolev inequalities. I have written two other texts dealing with mass transportation techniques, which may complement the present set of notes. One [41] is a set of lectures notes for a graduate course taught in Georgia Tech, Atlanta; the other one [40] is a short contribution to the proceedings of a summer school in the Azores, organized by Maria Carvalho; I have tried to avoid repetition. With respect to both abovementioned sources, the present notes aim at giving a more impressionist picture, with priority given to the diversity of applications rather than to the systematic nature of the exposition. The plan here is the opposite of the one that you would expect in a course: it starts with applications and ends up with theoretical background. There is a lot of overlapping with the proceedings of the Azores summer school, however the latter was mainly focusing on the problem of trend to equilibrium for dissipative equations.
- fast trend
- optimal transportation
- sobolev inequality
- partial differential
- recover explicit estimates
- dx only
- inequality holds
- ref- erence measure
- equation