04753nam 2200541 a 450 991043803450332120200520144314.088-7642-458-X10.1007/978-88-7642-458-8(OCoLC)852840033(MiFhGG)GVRL6XMQ(CKB)3710000000002682(MiAaPQ)EBC1474384(EXLCZ)99371000000000268220130718d2013 uy 0engurun|---uuuuatxtccrRegularity of optimal transport maps and applications /Guido de Philippis1st ed. 2013.Pisa [Italy] Edizioni della Normale20131 online resource (xix, 169 pages) illustrationsTheses (Scuola Normale Superiore),2239-1460 ;17Description based upon print version of record.88-7642-456-3 Includes bibliographical references.Cover; Title Page; Copyright Page; Table of Contents; Introduction; 1. Regularity of optimal transport maps and applications; 2. Other papers; 1. Г-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 transportation1.1. The case of the quadratic cost and Brenier Polar Factorization Theorem1.2. Brenier vs Aleksandrov solutions to the Monge-Ampère 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-Ampère equation; 2.1. Aleksandrov maximum principle; 2.2. Sections of solutions of the Monge-Ampère equation and Caffarelli regularity theorems; 2.3. Existence of smooth solutions to the Monge-Amp`ereequationChapter 3 Sobolev regularity of solutions to the Monge Ampère 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-Ampère 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 field5.2.2. 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,β regularity and strict c - convexity; 6.3. Comparison principle and C2,α regularity; Appendix A Properties of convex functions; Appendix B A proof of John lemma; References; THESES; Published volumes; Volumes published earlierIn 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.Theses (Scuola Normale Superiore),2239-1460 ;17Transportation problems (Programming)Mathematical optimizationTransportation problems (Programming)Mathematical optimization.510514De Philippis Guido854265MiAaPQMiAaPQMiAaPQBOOK9910438034503321Regularity of optimal transport maps and applications4203731UNINA02085nam 22004093a 450 991076574070332120250203235641.097888551899278855189921https://doi.org/10.26530/OAPEN_356385(CKB)5400000000000270(ScCtBLL)9875b5f2-c9c2-4586-a610-ef412737d606(OCoLC)1163814926(EXLCZ)99540000000000027020250203i20032020 uu itauru||||||||||txtrdacontentcrdamediacrrdacarrierNascita ed evoluzione del distretto orafo di Arezzo, 1947-2001 : Primo studio in una prospettiva ecology based /Luciana Lazzeretti[s.l.] :Firenze University Press,2003.1 online resource (1 p.)Studi e saggiThis book, generated by the creation of a database on the population of enterprises and their characteristics, illustrates the process of development of the Arezzo gold and jewellery system. It is a significant case in view of its international prestige and the use of highly specialised manpower, and at the same time interesting in terms of type, since it represents an emerging district produced by gemmation from a large enterprise. The analysis offers insight into the process of district development and a methodological support for ulterior investigations. At the same time, it also represents a tool for an understanding of the long-term economic dynamics characterising one of the most vital productive areas of Tuscany.Studi e saggiBusiness & Economics / Industries / Fashion & Textile IndustrybisacshEconomicsBusiness & Economics / Industries / Fashion & Textile IndustryEconomics.Lazzeretti Luciana286113ScCtBLLScCtBLLBOOK9910765740703321Nascita ed evoluzione del distretto orafo di Arezzo, 1947-20011802295UNINA