Continuity and injectivity of optimal maps for non-negatively cross-curved costs

Mathematics – Analysis of PDEs

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

37 pages, 7 figures

Scientific paper

Consider transportation of one distribution of mass onto another, chosen to optimize the total expected cost, where cost per unit mass transported from x to y is given by a smooth function c(x,y). If the source density f^+(x) is bounded away from zero and infinity in an open region U' \subset R^n, and the target density f^-(y) is bounded away from zero and infinity on its support clV \subset R^n, which is strongly c-convex with respect to U', and the transportation cost c is non-negatively cross-curved, we deduce continuity and injectivity of the optimal map inside U' (so that the associated potential u belongs to C^1(U')). This result provides a crucial step in the low/interior regularity setting: in a subsequent paper [15], we use it to establish regularity of optimal maps with respect to the Riemannian distance squared on arbitrary products of spheres. The present paper also provides an argument required by Figalli and Loeper to conclude in two dimensions continuity of optimal maps under the weaker (in fact, necessary) hypothesis A3w [17]. In higher dimensions, if the densities f^\pm are H\"older continuous, our result permits continuous differentiability of the map inside U' (in fact, C^{2,\alpha}_{loc} regularity of the associated potential) to be deduced from the work of Liu, Trudinger and Wang [33].

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Continuity and injectivity of optimal maps for non-negatively cross-curved costs does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Continuity and injectivity of optimal maps for non-negatively cross-curved costs, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Continuity and injectivity of optimal maps for non-negatively cross-curved costs will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-431817

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.