Calculus of Variations and Geometric Measure Theory
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F. Cavalletti

Optimal transport with branching distance cost and the obstacle problem

created by cavallett on 23 Jul 2010
modified on 23 Feb 2012

[BibTeX]

Accepted Paper

Inserted: 23 jul 2010
Last Updated: 23 feb 2012

Journal: SIAM J. Math. Anal.
Year: 2011

Abstract:

We address the Monge problem in metric spaces with a geodesic distance: $(X,d)$ is a Polish space and $d_N$ is a geodesic Borel distance which makes $(X,d_N)$ a possibly branching geodesic space. We show that under some assumptions on the transference plan we can reduce the transport problem to transport problems along family of geodesics.

We introduce two assumptions on the transference plan $\pi$ which imply that the conditional probabilities of the first marginal on each family of geodesics are continuous and that each family of geodesics is a hourglass-like set. We show that this regularity is sufficient for the construction of a transport map.

We apply these results to the Monge problem in $\erre^{d}$ with smooth, convex and compact obstacle obtaining the existence of an optimal map provided the first marginal is absolutely continuous w.r.t. the $d$-dimensional Lebesgue measure.


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