Calculus of Variations and Geometric Measure Theory
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F. Cavalletti

Monge problem in metric measure spaces with Riemannian Curvature-Dimension condition

created by cavallett on 22 Oct 2013
modified on 28 Jan 2014


Published Paper

Inserted: 22 oct 2013
Last Updated: 28 jan 2014

Journal: Nonlinear Analysis
Year: 2013
Doi: 10.1016/


We prove the existence of solutions for the Monge minimization problem, addressed in a metric measure space (X,d,m) enjoying the Riemannian curvature-dimension condition RCD∗(K,N), with N < ∞. For the first marginal measure, we assume that μ0 ≪ m. As a corollary, we obtain that the Monge problem and its relaxed version, the Monge-Kantorovich problem, attain the same minimal value.

Moreover we prove a structure theorem for d-cyclically monotone sets: neglecting a set of zero m- measure they do not contain any branching structures, that is, they can be written as the disjoint union of the image of a disjoint family of geodesics.


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