LEADER 04476nam 22006855 450 001 9910919815603321 005 20251113180006.0 010 $a9783031768347 010 $a3031768345 024 7 $a10.1007/978-3-031-76834-7 035 $a(CKB)37115973200041 035 $a(DE-He213)978-3-031-76834-7 035 $a(MiAaPQ)EBC31870179 035 $a(Au-PeEL)EBL31870179 035 $a(EXLCZ)9937115973200041 100 $a20241229d2024 u| 0 101 0 $aeng 135 $aur||||||||||| 181 $ctxt$2rdacontent 182 $cc$2rdamedia 183 $acr$2rdacarrier 200 10$aLectures on Optimal Transport /$fby Luigi Ambrosio, Elia Brué, Daniele Semola 205 $a2nd ed. 2024. 210 1$aCham :$cSpringer Nature Switzerland :$cImprint: Springer,$d2024. 215 $a1 online resource (XI, 260 p. 2 illus., 1 illus. in color.) 225 1 $aLa Matematica per il 3+2,$x2038-5757 ;$v169 311 08$a9783031768330 311 08$a3031768337 327 $a- 1. Lecture I. Preliminary notions and the Monge problem -- 2. Lecture II. The Kantorovich problem -- 3. Lecture III. The Kantorovich - Rubinstein duality -- 4. Lecture IV. Necessary and sufficient optimality conditions -- 5. Lecture V. Existence of optimal maps and applications -- 6. Lecture VI. A proof of the isoperimetric inequality and stability in Optimal Transport -- 7. Lecture VII. The Monge-Ampére equation and Optimal Transport on Riemannian manifolds -- 8. Lecture VIII. The metric side of Optimal Transport -- 9. Lecture IX. Analysis on metric spaces and the dynamic formulation of Optimal Transport -- 10. Lecture X.Wasserstein geodesics, nonbranching and curvature -- 11. Lecture XI. Gradient flows: an introduction -- 12. Lecture XII. Gradient flows: the Brézis-Komura theorem -- 13. Lecture XIII. Examples of gradient flows in PDEs -- 14. Lecture XIV. Gradient flows: the EDE and EDI formulations -- 15. Lecture XV. Semicontinuity and convexity of energies in the Wasserstein space -- 16. Lecture XVI. The Continuity Equation and the Hopf-Lax semigroup -- 17. Lecture XVII. The Benamou-Brenier formula -- 18. Lecture XVIII. An introduction to Otto?s calculus -- 19. Lecture XIX. Heat flow, Optimal Transport and Ricci curvature. 330 $aThis textbook is addressed to PhD or senior undergraduate students in mathematics, with interests in analysis, calculus of variations, probability and optimal transport. It originated from the teaching experience of the first author in the Scuola Normale Superiore, where a course on optimal transport and its applications has been given many times during the last 20 years. The topics and the tools were chosen at a sufficiently general and advanced level so that the student or scholar interested in a more specific theme would gain from the book the necessary background to explore it. After a large and detailed introduction to classical theory, more specific attention is devoted to applications to geometric and functional inequalities and to partial differential equations. This is the second edition of the book, first published in 2018. It includes refinement of proofs, an updated bibliography and a more detailed discussion of minmax principles, with the aim of giving two fully self-contained proofs of Kantorovich duality. 410 0$aLa Matematica per il 3+2,$x2038-5757 ;$v169 606 $aMathematical analysis 606 $aMathematical optimization 606 $aCalculus of variations 606 $aMeasure theory 606 $aMathematics 606 $aAnalysis 606 $aCalculus of Variations and Optimization 606 $aMeasure and Integration 606 $aMathematics 615 0$aMathematical analysis. 615 0$aMathematical optimization. 615 0$aCalculus of variations. 615 0$aMeasure theory. 615 0$aMathematics. 615 14$aAnalysis. 615 24$aCalculus of Variations and Optimization. 615 24$aMeasure and Integration. 615 24$aMathematics. 676 $a515 700 $aAmbrosio$b Luigi$4aut$4http://id.loc.gov/vocabulary/relators/aut$044009 702 $aBrue?$b Elia$4aut$4http://id.loc.gov/vocabulary/relators/aut 702 $aSemola$b Daniele$4aut$4http://id.loc.gov/vocabulary/relators/aut 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910919815603321 996 $aLectures on Optimal Transport$92175022 997 $aUNINA