LEADER 04753nam 2200541 a 450 001 9910438034503321 005 20200520144314.0 010 $a88-7642-458-X 024 7 $a10.1007/978-88-7642-458-8 035 $a(OCoLC)852840033 035 $a(MiFhGG)GVRL6XMQ 035 $a(CKB)3710000000002682 035 $a(MiAaPQ)EBC1474384 035 $a(EXLCZ)993710000000002682 100 $a20130718d2013 uy 0 101 0 $aeng 135 $aurun|---uuuua 181 $ctxt 182 $cc 183 $acr 200 10$aRegularity of optimal transport maps and applications /$fGuido de Philippis 205 $a1st ed. 2013. 210 $aPisa [Italy] $cEdizioni della Normale$d2013 215 $a1 online resource (xix, 169 pages) $cillustrations 225 1 $aTheses (Scuola Normale Superiore),$x2239-1460 ;$v17 300 $aDescription based upon print version of record. 311 $a88-7642-456-3 320 $aIncludes bibliographical references. 327 $aCover; Title Page; Copyright Page; Table of Contents; Introduction; 1. Regularity of optimal transport maps and applications; 2. Other papers; 1. ?-convergence of non-local perimeter; 2. Sobolev regularity of optimal transport map and differential inclusions; 3. A non-autonomous chain rule in W1,p and BV; 4. Aleksandrov-Bakelman-Pucci estimate for the infinity Laplacian; 5. Stability for the Plateau problem; 6. Stability for the second eigenvalue of the Stekloff-Laplacian; 7. Regularity of the convex envelope; Chapter 1 An overview on optimal transportation 327 $a1.1. The case of the quadratic cost and Brenier Polar Factorization Theorem1.2. Brenier vs Aleksandrov solutions to the Monge-Ampe?re equation; 1.2.1. Brenier solutions; 1.2.2. Aleksandrov solutions; 1.3. The case of a general cost c(x, y); 1.3.1. Existence of optimal maps; 1.3.2. Regularity of optimal maps and the MTW condition; Chapter 2 The Monge-Ampe?re equation; 2.1. Aleksandrov maximum principle; 2.2. Sections of solutions of the Monge-Ampe?re equation and Caffarelli regularity theorems; 2.3. Existence of smooth solutions to the Monge-Amp`ereequation 327 $aChapter 3 Sobolev regularity of solutions to the Monge Ampe?re equation3.1. Proof of Theorem 3.1; 3.2. Proof of Theorem 3.2; 3.2.1. A direct proof of Theorem 3.8; 3.2.2. A proof by iteration of the L log L estimate; 3.3. A simple proof of Caffarelli W2,p estimates; Chapter 4 Second order stability for the Monge-Ampe?re equation and applications; 4.1. Proof of Theorem 4.1; 4.2. Proof of Theorem 4.2; Chapter 5 The semigeostrophic equations; 5.1. The semigeostrophic equations in physical and dual variables; 5.2. The 2-dimensional periodic case; 5.2.1. The regularity of the velocity field 327 $a5.2.2. Existence of an Eulerian solution5.2.3. Existence of a Regular Lagrangian Flow for the semigeostrophic velocity field; 5.3. The 3-dimensional case; Chapter 6 Partial regularity of optimal transport maps; 6.1. The localization argument and proof of the results; 6.2. C1,? regularity and strict c - convexity; 6.3. Comparison principle and C2,? regularity; Appendix A Properties of convex functions; Appendix B A proof of John lemma; References; THESES; Published volumes; Volumes published earlier 330 $aIn this thesis, we study the regularity of optimal transport maps and its applications to the semi-geostrophic system. The first two chapters survey the known theory, in particular there is a self-contained proof of Brenier? theorem on existence of optimal transport maps and of Caffarelli?s Theorem on Holder continuity of optimal maps. In the third and fourth chapter we start investigating Sobolev regularity of optimal transport maps, while in Chapter 5 we show how the above mentioned results allows to prove the existence of Eulerian solution to the semi-geostrophic equation. In Chapter 6 we prove partial regularity of optimal maps with respect to a generic cost functions (it is well known that in this case global regularity can not be expected). More precisely we show that if the target and source measure have smooth densities the optimal map is always smooth outside a closed set of measure zero. 410 0$aTheses (Scuola Normale Superiore),$x2239-1460 ;$v17 606 $aTransportation problems (Programming) 606 $aMathematical optimization 615 0$aTransportation problems (Programming) 615 0$aMathematical optimization. 676 $a510 676 $a514 700 $aDe Philippis$b Guido$0854265 801 0$bMiAaPQ 801 1$bMiAaPQ 801 2$bMiAaPQ 906 $aBOOK 912 $a9910438034503321 996 $aRegularity of optimal transport maps and applications$94203731 997 $aUNINA