ENGLISH

Regularity of Optimal Transport Maps and Applications

By Guido De Philippis (auth.)

Book information

Publisher
Edizioni della Normale
Year
2013
ISBN
978-88-7642-456-4, 978-88-7642-458-8
DOI
10.1007/978-88-7642-458-8
Language
english
Format
PDF
Filesize
1 MB (1125329 bytes)
Series
Publications of the Scuola Normale Superiore 17
Edition
1
Pages
0\186
Orientation
yes
Scanned
yes
Time added
2013-08-01 04:00:00
Calculus of Variations and Optimal Control; Optimi

Description

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 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.

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