Free boundaries in optimal transport and Monge-Ampère obstacle problems

Abstract

Given compactly supported $0\le f,g \in L^1(\mathbb{R}^n)$, the problem of transporting a fraction $m \le \min\{\|f\|_{L^1},\|g\|_{L^1}\}$ of the mass of $f$ onto $g$ as cheaply as possible is considered, where cost per unit mass transported is given by a cost function $c$, typically quadratic $c(\mathbf{x},\mathbf{y}) = |\mathbf{x}-\mathbf{y}|^2/2$. This question is shown to be equivalent to a double obstacle problem for the Monge-Ampère equation, for which sufficient conditions are given to guarantee uniqueness of the solution, such as $f$ vanishing on $\operatorname{spt}\, g$ in the quadratic case. The part of $f$ to be transported increases monotonically with $m$, and if $\operatorname{spt}\, f$ and $\operatorname{spt}\, g$ are separated by a hyperplane $H$, then this part will be separated from the balance of $f$ by a semiconcave Lipschitz graph over the hyperplane. If $f= f\chi_{\Omega}$ and $g=g\chi_{\Lambda}$ are bounded away from zero and infinity on separated strictly convex domains $\Omega,\Lambda \subset \mathbf{R}^n$, for the quadratic cost this graph is shown to be a $C^{1,\alpha}_{\rm loc}$ hypersurface in $\Omega$ whose normal coincides with the direction transported; the optimal map between $f$ and $g$ is shown to be Hölder continuous up to this free boundary, and to those parts of the fixed boundary $\partial\Omega$ which map to locally convex parts of the path-connected target region.