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 $D^{2}(u(x))\leq D^{2}(u(y))$.

Notable Properties of the Monge Ampère equation

The specific boundary problem that is relevant to our Monge Problem is $\nabla u(\Omega )=\Omega '$. In this case, $\Omega$ is the domain for the measure $\mu$, and $\Omega '$ is the domain for the measure $\nu$. When $u$ is a homeomorphism, this implies that $u(\partial \Omega )=\partial \Omega '$. Solutions to this boundary value problem and the Monge Ampère equation gives us information about the optimal transport map. For example, regularity of the solutions to our boundary value problem implies regularity for the optimal transport map. This is given by the following theorem

Theorem. If $f$ and $g$ are $C^{0,\alpha }(\Omega )$ and are both bounded from above and from below on the whole $\Omega$ by positive constants and $\Omega$ is a convex open set, then the unique Brenier solution of $u$ of the Monge Ampère equation belongs to $C^{2,\alpha }(\Omega )\cap C^{1,\alpha }{\overline {\Omega }}$, and $u$ satisfies the equation in the classical sense.