OPTIMAL TRANSPORTATION DISSIPATIVE PDE S AND FUNCTIONAL INEQUALITIES
34 pages
English

OPTIMAL TRANSPORTATION DISSIPATIVE PDE'S AND FUNCTIONAL INEQUALITIES

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34 pages
English
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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


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Nombre de lectures 17
Langue English

Exrait

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 variousauthors.Duringthisprocess,weshallencountersuchdiverseareasas uidme-chanics,granularmaterialphysics,mean- eldlimitsinstatisticalmechanics,andoptimal 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. Most of the material in sections III and IV is absent from the Atlanta lecture notes. Ichosenottostartbygivingprecisede nitionsofmasstransportation;infact,each lecturewillpresentaslightlydi erentviewonmasstransportation. It is a pleasure to thank the CIME organization for their beautiful work on the occasion of the summer school in Martina Franca, in which these lectures have been given. I also thank Yann Brenier for his suggestions during the preparation of these lectures. Contents I. Some motivations 2 II. A study of fast trend to equilibrium 4 III. A study of slow trend to equilibrium 12 IV. Estimates in a mean- eld limit problem 19 V. Otto’s di eren tial point of view 27 References 33
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All three problems have been studied by various authors and solved in certain cases. Some of them are quite old, like the study of optimal Young inequalities, which goes back to the seventies. Among recent works, let me mention Carrillo, McCann and Villani [12] for problem 1; Malrieu [24] for problem 2; Dolbeault and del Pino [19] for the Gagliardo-Nirenberg inequality in problem 3; Barthe [4] for the Young inequality in the same
problem. It turns out that in all these cases, either optimal mass transportation was explicit in the solution, or it has been found to provide much more transparent proofs thanthosewhichhavebeen rstsuggested.Mygoalhereisnottoexplain why mass transportation is ecien t in such problems, but to give an idea of how it can be used in these various contexts.
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