Springer celebrates 175 years of publishing excellence! Join us >>

Theses (Scuola Normale Superiore)

Regularity of Optimal Transport Maps and Applications

Authors: Philippis, Guido

  • Essentially self-contained account of the known regularity theory of optimal maps in the case of quadratic cost
  • Presents proofs of some recent results like Sobolev regularity and Sobolev stability for optimal maps and their applications too the semi-geostrophic system
  • Proves for the first time a partial regularity theorem for optimal map with respect to a general cost function
see more benefits

Buy this book

eBook $19.99
price for USA (gross)
  • ISBN 978-88-7642-458-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $29.99
price for USA
  • ISBN 978-88-7642-456-4
  • Free shipping for individuals worldwide
  • Online orders shipping within 2-3 days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
About this book

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 (6 chapters)

  • An overview on optimal transportation

    Philippis, Guido

    Pages 1-27

  • The Monge-Ampère equation

    Philippis, Guido

    Pages 29-54

  • Sobolev regularity of solutions to the Monge Ampère equation

    Philippis, Guido

    Pages 55-72

  • Second order stability for the Monge-Ampère equation and applications

    Philippis, Guido

    Pages 73-80

  • The semigeostrophic equations

    Philippis, Guido

    Pages 81-118

Buy this book

eBook $19.99
price for USA (gross)
  • ISBN 978-88-7642-458-8
  • Digitally watermarked, DRM-free
  • Included format: PDF
  • ebooks can be used on all reading devices
  • Immediate eBook download after purchase
Softcover $29.99
price for USA
  • ISBN 978-88-7642-456-4
  • Free shipping for individuals worldwide
  • Online orders shipping within 2-3 days.
Rent the ebook  
  • Rental duration: 1 or 6 month
  • low-cost access
  • online reader with highlighting and note-making option
  • can be used across all devices
Loading...

Recommended for you

Loading...

Bibliographic Information

Bibliographic Information
Book Title
Regularity of Optimal Transport Maps and Applications
Authors
Series Title
Theses (Scuola Normale Superiore)
Series Volume
17
Copyright
2013
Publisher
Edizioni della Normale
Copyright Holder
Scuola Normale Superiore
eBook ISBN
978-88-7642-458-8
DOI
10.1007/978-88-7642-458-8
Softcover ISBN
978-88-7642-456-4
Edition Number
1
Topics