Optimal Transportation

Regularity of Optimal Transport Maps

A classical variational problem at the crossroads between many areas of mathematics and applications is the optimal transportation problem. Given two measures \(\mu, \lambda\) on \(\mathbb{R}^d\) and a cost function \(c: \mathbb{R}^d \times \mathbb{R}^d \to \mathbb{R}\), we look for a transfer plan (coupling) \(\pi\), i.e., a measure on the product space with marginals \(\mu\) and \(\lambda\), that minimises the transport cost $$ \int_{\mathbb{R}^d \times \mathbb{R}^d} c(x,y) \,\pi(\mathrm{d} x, \mathrm{d} y) $$ among all admissible transport plans. In the case of quadratic cost \(c(x,y) = |x-y|^2\) the minimum is the squared Wasserstein distance between the measures \(\mu\) and \(\lambda\). Let us briefly focus on this case with the target measure \(\lambda\) being Lebesgue measure on \(\mathbb{R}^d\). One can show that minimality in that setting implies that the support \(\mathrm{Spt}\pi\) of the optimal coupling is a cyclically monotone set, hence there exists a convex function \(\psi\) (the so-called Kantorovich potential) such that \(\mathrm{Spt} \pi \subseteq \left\{ (x,y): y \in \text{subgradient } \partial \psi(x) \right\}\). Together with the transport condition $$\int \zeta(\nabla \psi)\,\mathrm{d} \mu = \int \zeta \,\mathrm{d}\lambda \quad \text{for all test functions } \zeta,$$ this shows that - in an appropriate weak sense - \(\psi\) satisfies the Monge-Ampère equation $$\det D^2 \psi = \mu.$$

Singularities appear quite generically in the optimal transportation problem: even if the data \(\mu\) and \(\lambda\) are nice (e.g. with smooth densities with respect to Lebesgue measure), the optimal transport map does not need to be smooth, as a classical example by Caffarelli shows. Only under convexity assumptions on the supports of the measures \(\mu\) and \(\lambda\) can such singularities be avoided. In the case of optimal transport on manifolds, there is yet another, geometric mechanism due to curvature that can create singularities, as highlighted by Loeper .

Recently, Goldman-Otto showed that one can avoid the use of maximum principles for the Monge-Ampère equation, and instead work within a completely variational framework to obtain an \(\epsilon\)-regularity result for the Kantorovich potential \(\psi\): if \(\int |x-y|^2\,\mathrm{d} \pi \leq \epsilon\) locally and the data \(\mu, \lambda\) are smooth locally, then also \(\psi\) is locally smooth. Combined with deep results in measure theory (Alexandrov's theorem about a.e. twice differentiability of convex functions), this gives partial regularity for optimal transport maps, without any restrictions on the gemetry of the supports of \(\mu\), \(\lambda\). The approach is in spirit very similar to De Giorgi's approach to the regularity of minimal surfaces: it draws from a harmonic approximation result, which through elliptic regularity and a suitable change of coordinates (making use of the affine invariance of the minimality property) feeds into a one-step improvement result. This can be iterated to obtain, via Campanato's characterisation of Hölder regularity, \(\epsilon\)-regularity for \(\psi\).

We could show that, making use of the concept of almost-minimisers, one can also develop an \(\epsilon\)-regularity for optimal transport maps in the case of general cost functions \(c\) with Hölder continuous second derivatives, which also applies to optimal transport between measures on Riemannian manifolds and cost functions given by the squared geodesic distance.

Related publications

  1. Variational approach to partial regularity of optimal transport maps: general cost functions. Annals of PDE 7 (2021), Article 17.
    (with F. Otto, M. Prod'homme)