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.

The Monge Ampère Equation

The Monge Ampère Equation is a nonlinear second-order elliptic partial differential equation. When we consider the kind of Monge Problem discussed earlier, we can show that the optimal transport plan, $T$ must satisfy the equation

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

where $g(y)$ and $f(y)$ are the respective target and starting measures for the transport problem. Note that these measures are assumed to be absolutely continuous to the Lebesgue measure. The relevant Monge Problem has a quadratic cost problem, which can be shown to imply that $T=\nabla u$, where $u$ is a convex function. If we require $u$ to be strictly convex, a change of variables gives us the Monge Ampère equation

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

From here, we can ask about regularity of solutions of to the PDE, which in turns gives us regularity on $T=\nabla u$. For example, we have 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.

Here, a Brenier solution simply implies that $T$ is a transport plan from $f$ to $g$. Note that the support of $g$ is very relevant when trying to show regularity of our transport plan. Without any conditions on the support of $g$, $T$ may have singularities. On the other hand, conditions on the support of $g$ can eliminate such singularities. For example, Caffarelli has shown that if $f,g$ are smooth and strictly positive on their support, and the support of $g$ is convex, the optimal transport plan will be smooth in the support of $f$. Moreover, if both supports are smooth and uniformly convex, one can show that the optimal transport plan is a smooth diffeomorphism between the support of $f$ and the support of $g$.

Existence and Uniqueness on Riemannian Manifolds

Parts of the previous section can be summarized into the following theorem.

Theorem. Let $\mu$ and $\nu$ be two compactly supported probability measures on $R^{n}$. If $\mu$ is absolutely continuous with respect to the Lebesgue measure, then:

• There exists a unique solution $T$ to the Monge problem.
• The optimal map $T$ is characterized by the structure $T(x)=\nabla u(x)$, for some convex function $u:R^{n}\to R$.

Furthermore, if $\mu (dx)=f(x)dx$, and $\nu (dy)=g(y)dy$,

$\|\mathrm {det} (\nabla T(x))\|={\frac {f(x)}{g(T(x))}}$ for $\mu$-a.e. $x\in R^{n}$

Note that many of these notions, such as Lebesgue measure and gradients, all have well-defined generalizations on all Riemannian manifolds. In particular, the above theorem was able to be extended to compact Riemannian manifolds. To state the theorem for Riemannian manifolds, we need the following definitions.

A function is $\psi :X\to R\cup \{+\infty \}$ is c-convex if