Global regularity in the Monge-Amp\`ere obstacle problem

TL;DR

This paper proves the global $W^{2,p}$ regularity of solutions to the Monge-Ampère obstacle problem, showing that the optimal transport map is a $W^{1,p}$ diffeomorphism under certain conditions, advancing understanding in optimal partial transportation.

Contribution

It establishes the sharp regularity result that the optimal map is a $W^{1,p}$ diffeomorphism, improving upon the previous known continuous homeomorphism result.

Findings

The optimal map $Du$ is a $W^{1,p}$ diffeomorphism for any $p extgreater 1$.

The regularity result is sharp; Lipschitz regularity may fail even with smooth densities.

The work connects the obstacle problem to optimal partial transportation theory.

Abstract

In this paper, we establish the global $W^{2,p}$ estimate for the Monge-Amp\`ere obstacle problem: $(Du)_{\sharp}f\chi{_{\{u>\frac{1}{2}|x|^2\}}}=g$, where $f$ and $g$ are positive continuous functions supported in disjoint bounded $C^2$ uniformly convex domains $\overline{\Omega}$ and $\overline{\Omega^*}$, respectively. Furthermore, we assume that $\int_{\Omega}f\geq \int_{\Omega^*}g$. The main result shows that $Du:\overline U\rightarrow\overline{\Omega^*}$, where $ U=\{u>\frac{1}{2}|x|^2\}$, is a $W^{1, p}$ diffeomorphism for any $p\in(1,\infty)$. Previously, it was only known to be a continuous homeomorphism according to Caffarelli and McCann \cite{CM}. It is worth noting that our result is sharp, as we can construct examples showing that even with the additional assumption of smooth densities, the optimal map $Du$ is not Lipschitz. This obstacle problem arises naturally in optimal partial transportation.