Continuity and injectivity of optimal maps
- 134 Downloads
Figalli–Kim–McCann proved in (2009) the continuity and injectivity of optimal maps under the assumption (B3) of nonnegative cross-curvature. In the recent Figalli et al. (Arch Ration Mech Anal 209:747–795, 2013), Figalli et al. (Methods Appl Anal 2013), they extend their results to the assumption (A3w) of Trudinger–Wang (Ann Sc Norm Super Pisa Cl Sci 8:143–174, 2009), and they prove, moreover, the Hölder continuity of these maps. We give here an alternative and independent proof of the extension to (A3w) of the continuity and injectivity of optimal maps based on the sole arguments of Figalli et al. (2009) and on new Alexandrov-type estimates for lower bounds.
Mathematics Subject Classification35J96
The author is very grateful to the Australian National University for its generous hospitality during the period when the majority of this work has been carried out. The author wishes to express his gratitude to Philippe Delanoë, Neil Trudinger, and Xu-Jia Wang for supporting his visit to Canberra, and for several stimulating discussions on optimal transportation. The author is indebted to Neil Trudinger for having introduced him to the problem of regularity of optimal maps. The author wishes also to express his gratitude to Emmanuel Hebey for helpful support and advice during the redaction, and to Jiakun Liu and, again, Neil Trudinger for their valuable comments and suggestions on the manuscript.
- 2.Borwein, J.M., Lewis,.: Convex Analysis and Nonlinear Optimization. Theory and Examples. CMS Books in Mathematics, 3, Springer, New York (2003)Google Scholar
- 9.Caffarelli, L.A.: Allocation Maps with General Cost Functions, Partial Differential Equations and Applications, Lecture Notes in Pure and Applied Mathematics, 177, pp. 29–35. Dekker, New York (1996)Google Scholar
- 14.Figalli, A., Kim, Y.-H., McCann, R.J.: Continuity and injectivity of optimal maps for non-negatively cross-curved costs. (2009) (Preprint at arXiv:0911:3952)Google Scholar
- 16.Figalli, A., Kim, Y.-H., McCann, R.J.: On supporting hyperplanes to convex bodies. Methods Appl. Anal. 20(3), 261–272 (2013)Google Scholar
- 23.John, F.: Extremum Problems with Inequalities as Subsidiary Conditions, Studies and Essays Presented to R. Courant on his 60th Birthday, January 8, pp. 187–204. Interscience Publishers Inc, New York (1948)Google Scholar
- 33.Trudinger, N.S., Wang, X.-J.: On convexity notions in optimal transportation, 2008 (Preprint)Google Scholar
- 37.Urbas, J.: Mass Transfer Problems, Lecture Notes. Univ, Bonn (1998)Google Scholar
- 38.Villani, C.: Optimal Transport. Old and new, Grundlehren der Mathematischen Wissenschaften, 338. Springer, Berlin (2009)Google Scholar