Blick ins Buch

Regularity of Optimal Transport Maps and Applications

Guido Philippis (Autor)

Buch | Softcover

2013 | 2013 ed.
Scuola Normale Superiore (Verlag)
978-88-7642-456-4 (ISBN)

Lese- und Medienproben

Blick ins Buch (midvox)

Artikel merken

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.

Introduction.- 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.- A. Properties of convex functions.- B. A proof of John Lemma.- Bibliography.

Erscheint lt. Verlag	5.9.2013
Reihe/Serie	Publications of the Scuola Normale Superiore ; 17
Reihe/Serie	Theses (Scuola Normale Superiore)
Zusatzinfo	Approx. 190 p.
Verlagsort	Pisa
Sprache	englisch
Maße	150 x 240 mm
Themenwelt	Mathematik / Informatik ► Mathematik ► Analysis
	Mathematik / Informatik ► Mathematik ► Angewandte Mathematik
	Mathematik / Informatik ► Mathematik ► Finanz- / Wirtschaftsmathematik
Schlagworte	general cost function • Monge-Ampère equation • Optimal Transportation • semi-geostrophic system • Sobolev regularity, Sobolev stability for optimal maps
ISBN-10	88-7642-456-3 / 8876424563
ISBN-13	978-88-7642-456-4 / 9788876424564
Zustand	Neuware