Optimal Transport and the Monge Ampère equation

The Monge Ampère equation is an nonlinear second-order elliptic partial differential equation. Given sufficient conditions on an optimal transport problem, the Monge Ampère equation defines a change of variables formula for the pushforward of a transport map. In particular, the Monge Ampère equation applies to a variant of the Monge Problem where the cost function is quadratic.

Deriving the Monge Ampère equation from the Monge Problem

The appropriate variant of the Monge Problem for this situation is

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

Consider the case when $\mu ,\nu$ are absolutely continuous, where $\mu$ is the starting measure, and $\nu =T_{\#}\mu$. Say $f$ and $g$ are the densities of $\mu$ and $\nu >$ respectively. Moreover, let $T:X\to X$ be $C^{1}$ and injective. The change of variables formula tells us that

$g(y)={\frac {f(T^{-1}(y))}{\det(DT(T^{-1}(y)))}}$

For this particular variant of the Monge problem, it can be shown that $T=\nabla u$ where $u$ is convex. If we require $u$ to be strictly convex, $T$ is guaranteed to be injective. Once $\nabla u$ is substituted for $T$ in the change of variables formula, one obtains

$\det(D^{2}u(x))={\frac {f(x)}{g(\nabla u(x))}}$

The above equation is the Monge Ampère equation. Note that since $u$ from above is assumed to be convex, the Jacobian term is positive. Moreover, $\det D^{2}(u(x))$ has a monotonicity property when $D^{2}(u(x))\leq D^{2}(u(y))$.