Titlebook: Regularity of Optimal Transport Maps and Applications; Guido Philippis Book 2013 The Editor(s) (if applicable) and The Author(s), under ex

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Regularity of Optimal Transport Maps and Applications978-88-7642-458-8Series ISSN 2239-1460 Series E-ISSN 2532-1668

生氣地

An overview on optimal transportation, ? (Y) and a .: . × . → ? we look for a map . such that .. = .. and that minimize . In general, there could be no solution to the above problem both because the class of admissible maps is empty (for instance in the case in which μ is a Dirac mass and . is not) or because the infimum is not attained (see [95, Example 4.9]).

Callus

,The Monge-Ampère equation,a proof of Caffarelli .. regularity theorem [18, 20]. Many of the tools developed in this Chapter will play a crucial role in the proof of the Sobolev regularity in Chapter 3. In the last Section we show, without proofs, how to build smooth solutions to the Monge-Ampère equation throughout the method of continuity.

LEER

,Sobolev regularity of solutions to the Monge Ampère equation,In this Chapter we prove the .. regularity of solutions of (2.1). This has been first shown in [40] in collaboration with Alessio Figalli, where actually the following higher integrability result was proved

Nefarious

Certainty

Partial regularity of optimal transport maps,The goal of this chapter (based on a joint work with Alessio Figalli [43]) is to prove partial regularity of optimal transport maps under mild assumptions on the cost function c and on the densities f and g, Theorems 6.1 and 6.2 below.

Gobble

Guido PhilippisEssentially 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