Calculus of Variations and Geometric Measure Theory

H. Lavenant

Lifting functionals defined on maps to measure-valued maps via optimal transport

created by lavenant on 17 Jul 2023
modified on 06 Sep 2023



Inserted: 17 jul 2023
Last Updated: 6 sep 2023

Year: 2023

ArXiv: 2309.02260 PDF


How can one lift a functional defined on maps from a space X to a space Y into a functional defined on maps from X into P(Y) the space of probability distributions over Y? Looking at measure-valued maps can be interpreted as knowing a classical map with uncertainty, and from an optimization point of view the main gain is the convexification of Y into P(Y). We will explain why trying to single out the largest convex lifting amounts to solve an optimal transport problem with an infinity of marginals which can be interesting by itself. Moreover we will show that, to recover previously proposed liftings for functionals depending on the Jacobian of the map, one needs to add a restriction of additivity to the lifted functional.