Title: Regularity of Optimal Transport Maps and Applications: 17 (Publications of the Scuola Normale Superiore, 17)
Publisher: Edizioni della Normale
Publication Date: 2013
Language: English
ISBN 10: 8876424563
ISBN 13: 9788876424564
Binding: Soft cover
Condition: New

About this title

Synopsis

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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