Calculus of Variations and Geometric Measure Theory

L. Nenna - B. Pass

Variational problems involving unequal dimensional optimal transport

created by nenna on 01 Apr 2019
modified on 19 Feb 2020


Accepted Paper

Inserted: 1 apr 2019
Last Updated: 19 feb 2020

Journal: Journal de Mathématiques Pures et Appliquées
Year: 2020


This paper is devoted to variational problems on the set of probability measures which involve optimal transport between unequal dimensional spaces. In particular, we study the minimization of a functional consisting of the sum of a term reflecting the cost of (unequal dimensional) optimal transport between one fixed and one free marginal, and another functional of the free marginal (of various forms). Motivating applications include Cournot-Nash equilibria where the strategy space is lower dimensional than the space of agent types. For a variety of different forms of the term described above, we show that a nestedness condition, which is known to yield much improved tractability of the optimal transport problem, holds for any minimizer. Depending on the exact form of the functional, we exploit this to find local differential equations characterizing solutions, prove convergence of an iterative scheme to compute the solution, and prove regularity results.