Regularity of Optimal Transport Maps and Applications [electronic resource] / by Guido Philippis
- De Philippis, Guido
- Pisa : Scuola Normale Superiore : Imprint: Edizioni della Normale, 2013.
- Physical Description:
- approximately 190 pages : online resource
- Additional Creators:
- SpringerLink (Online service)
- Machine generated contents note: 1.Regularity of optimal transport maps and applications -- 2.Other papers -- 1.An overview on optimal transportation -- 1.1.The case of the quadratic cost and Brenier Polar Factorization Theorem -- 1.2.Brenier vs. Aleksandrov solutions -- 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 -- 2.The Monge-Ampere equation -- 2.1.Aleksandrov maximum principle -- 2.2.Sections of solutions and Caffarelli theorems -- 2.3.Existence of smooth solutions to the Monge-Ampere equation -- 3.Sobolev regularity of solutions to the Monge Ampere equation -- 3.1.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 -- 4.Second order stability for the Monge-Ampere equation and applications -- 4.1.Proof of Theorem 4.1 -- 4.2.Proof of Theorem 4.2 -- 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 -- 5.2.2.Existence of an Eulerian solution -- 5.2.3.Existence of a Regular Lagrangian Flow for the semigeostrophic velocity field -- 5.3.The 3-dimensional case -- 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.
- In 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 Caffarellis 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.
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- text file PDF
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