Symmetric Monge-Kantorovich problems and polar decompositions of vector\n fields

2013/02/12 by Nassif Ghoussoub, Ghoussoub, Nassif, Abbas Moameni +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Mathematical Dynamics and Fractals

paper · pdf · doi:10.48550/arxiv.1302.2886

openalex publication_date 2013/02/12 · openalex created_date 2025/10/24 · openalex updated_date 2026/07/28

Abstract

For any given integer N\≥ 2, we show that every bounded measurable vector\nfield from a bounded domain \Ω into R^d is N-cyclically monotone up\nto a measure preserving N-involution. The proof involves the solution of a\nmultidimensional symmetric Monge-Kantorovich problem, which we first study in\nthe case of a general cost function on a product domain \Ω^N. The polar\ndecomposition described above corresponds to a special cost function derived\nfrom the vector field in question (actually N-1 of them). In this case, we\nshow that the supremum over all probability measures on \Ω^N which are\ninvariant under cyclic permutations and with a given first marginal \μ, is\nattained on a probability measure that is supported on the graph of a function\nof the form x\→ (x, Sx, S²x,..., S^N-1x), where S is a \μ-measure\npreserving transformation on \Ω such that S^N=I a.e. The proof exploits\na remarkable duality between such involutions and those Hamiltonians that are\nN-cyclically antisymmetric.\n

Citations

• Polar factorization and monotone rearrangement of vector‐valued functions

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• Pair densities in density functional theory

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