2015/07/21 by Abbas Moameni, Moameni, Abbas, Brendan Pass +1
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Nonlinear Partial Differential Equations #Optimization and Control (math.OC)
paper · pdf · doi:10.48550/arxiv.1507.05923
openalex publication_date 2015/07/21 · openalex created_date 2022/10/02 · openalex updated_date 2026/07/28
We study solutions to the multi-marginal Monge-Kantorovich problem which are\nconcentrated on several graphs over the first marginal. We first present two\ngeneral conditions on the cost function which ensure, respectively, that any\nsolution must concentrate on either finitely many or countably many graphs. We\nshow that local differential conditions on the cost, known to imply local\nd-rectifiability of the solution, are sufficient to imply a local version of\nthe first of our conditions. We exhibit two examples of cost functions\nsatisfying our conditions, including the Coulomb cost from density functional\ntheory in one dimension. We also prove a number of results relating to the\nuniqueness and extremality of optimal measures. These include a sufficient\ncondition on a collection of graphs for any competitor in the Monge-Kantorovich\nproblem concentrated on them to be extremal, and a general negative result,\nwhich shows that when the problem is symmetric with respect to permutations of\nthe variables, uniqueness cannot occur except under very special circumstances.\n