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Regularity of Optimal Transport Maps and Applications

  • Book
  • © 2013

Overview

Authors:
  1. Guido Philippis

    1. Hausdorff Center for Mathematics, Bonn, Germany

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  • 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

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

Part of the book sub series: Theses (Scuola Normale Superiore) (TSNS)

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Table of contents (6 chapters)

  1. Front Matter

    Pages i-xix
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  2. An overview on optimal transportation

    • Guido De Philippis
    Pages 1-27

  3. The Monge-Ampère equation

    • Guido De Philippis
    Pages 29-54

  4. Sobolev regularity of solutions to the Monge Ampère equation

    • Guido De Philippis
    Pages 55-72

  5. Second order stability for the Monge-Ampère equation and applications

    • Guido De Philippis
    Pages 73-80

  6. The semigeostrophic equations

    • Guido De Philippis
    Pages 81-118

  7. Partial regularity of optimal transport maps

    • Guido De Philippis
    Pages 119-146

  8. Back Matter

    Pages 147-169
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Keywords

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.

Authors and Affiliations

  • Hausdorff Center for Mathematics, Bonn, Germany

    Guido Philippis

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