Regularity of Optimal Transport Maps and Applications [electronic resource] /
Philippis, Guido.author.
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.
Description based upon print version of record.
Cover; Title Page; Copyright Page; Table of Contents; Introduction; 1. Regularity of optimal transport maps and applications; 2. Other papers; 1. Г-convergence of non-local perimeter; 2. Sobolev regularity of optimal transport map and differential inclusions; 3. A non-autonomous chain rule in W1,p and BV; 4. Aleksandrov-Bakelman-Pucci estimate for the infinity Laplacian; 5. Stability for the Plateau problem; 6. Stability for the second eigenvalue of the Stekloff-Laplacian; 7. Regularity of the convex envelope; Chapter 1 An overview on optimal transportation
1.1. The case of the quadratic cost and Brenier Polar Factorization Theorem1.2. Brenier vs Aleksandrov solutions to the Monge-Ampère equation; 1.2.1. Brenier solutions; 1.2.2. Aleksandrov solutions; 1.3. The case of a general cost c(x, y); 1.3.1. Existence of optimal maps; 1.3.2. Regularity of optimal maps and the MTW condition; Chapter 2 The Monge-Ampère equation; 2.1. Aleksandrov maximum principle; 2.2. Sections of solutions of the Monge-Ampère equation and Caffarelli regularity theorems; 2.3. Existence of smooth solutions to the Monge-Amp`ereequation
Chapter 3 Sobolev regularity of solutions to the Monge Ampère equation3.1. Proof of Theorem 3.1; 3.2. Proof of Theorem 3.2; 3.2.1. A direct proof of Theorem 3.8; 3.2.2. A proof by iteration of the L log L estimate; 3.3. A simple proof of Caffarelli W2,p estimates; Chapter 4 Second order stability for the Monge-Ampère equation and applications; 4.1. Proof of Theorem 4.1; 4.2. Proof of Theorem 4.2; Chapter 5 The semigeostrophic equations; 5.1. The semigeostrophic equations in physical and dual variables; 5.2. The 2-dimensional periodic case; 5.2.1. The regularity of the velocity field
5.2.2. Existence of an Eulerian solution5.2.3. Existence of a Regular Lagrangian Flow for the semigeostrophic velocity field; 5.3. The 3-dimensional case; Chapter 6 Partial regularity of optimal transport maps; 6.1. The localization argument and proof of the results; 6.2. C1,β regularity and strict c - convexity; 6.3. Comparison principle and C2,α regularity; Appendix A Properties of convex functions; Appendix B A proof of John lemma; References; THESES; Published volumes; Volumes published earlier
Includes bibliographical references.
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.
Calculus of variations.
Calculus of Variations and Optimal Control; Optimization.