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Regularity of Optimal Transport Maps and Applications

PSNS: REGULARITY OF OPTIMAL TRANSPORT MAPS & APPLICATNS

Earth Science

Mathematics & Science

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B2B GT

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B2B & B2C

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.

POST SECONDARY FOUR YEAR

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Front Cover.
Other Frontmatter.
Other Frontmatter.
Title Page.
Copyright Page.
Contents.
Introduction.
1: An Overview on Optimal Transportation.
2: The Monge-Amp&#x00E8;re Equation.
3: Sobolev Regularity of Solutions to the Monge Amp&#x00E8;re Equation.
4: Second Order Stability for the Monge-Amp&#x00E8;re Equation and Applications.
5: The Semigeostrophic Equations.
6: Partial Regularity of Optimal Transport Maps.
Appendix A: Properties of Convex Functions.
Appendix B: A Proof of John Lemma.
References.
Theses.

Regularity of Optimal Transport Maps and Applications.css-1p8rvzy{position:absolute;padding-left:8px;}

Regularity of Optimal Transport Maps and Applications