Regularity of Optimal Transport Maps and Applications

  • Guido De Philippis

Part of the Publications of the Scuola Normale Superiore book series (PSNS, volume 17)

Table of contents

  1. Front Matter
    Pages i-xix
  2. Guido De Philippis
    Pages 1-27
  3. Guido De Philippis
    Pages 29-54
  4. Guido De Philippis
    Pages 81-118
  5. Guido De Philippis
    Pages 119-146
  6. Back Matter
    Pages 147-169

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.


Monge-Ampère equation Sobolev regularity, Sobolev stability for optimal maps general cost function optimal transportation semi-geostrophic system

Authors and affiliations

  • Guido De Philippis
    • 1
  1. 1.Hausdorff Center for MathematicsBonnGermany

Bibliographic information