Regularity of Optimal Transport Maps and Applications, 1st Edition

  • Published By:
  • ISBN-10: 887642458X
  • ISBN-13: 9788876424588
  • DDC: 515.353
  • Grade Level Range: College Freshman - College Senior
  • 190 Pages | eBook
  • Original Copyright 2013 | Published/Released May 2014
  • This publication's content originally published in print form: 2013

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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.

Table of Contents

Front Cover.
Other Frontmatter.
Other Frontmatter.
Title Page.
Copyright Page.
1: An Overview on Optimal Transportation.
2: The Monge-Ampère Equation.
3: Sobolev Regularity of Solutions to the Monge Ampère Equation.
4: Second Order Stability for the Monge-Ampère 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.