Regularity of Optimal Transport Maps and the Monge-Ampére Equation

When considering the Monge Problem, it is natural to ask about the regularity of optimal transport maps (when they exist). In particular, we can consider the Monge Problem variant

$\inf _{T}\left\{F(T):=\int _{X}|x-T(x)|^{2}d\mu \right\}$

where we are taking the infimum over all transport plans for our associated measure. This allows us to reformulate the problem into a boundary value problem for a specific partial differential equation. From there, one can ask about the regularity of solutions to the PDE, which are associated with optimal transport plans.