While I don't work on the regularity theory for the optimal transport map, I was curious about the open problem 1.28 listed in Ambrosio and Gigli's *User's guide*: the problem to determine whether we have $T\in W^{1,1}$ for the transport map $T$ is still open?


More precisely, assume $\Omega\_1,\Omega\_2\subset \mathbb{R}^d$ are two bounded and connected open sets, $\mu=\rho \mathcal{L}^d |\_{\Omega\_1}, \nu=\eta \mathcal{L}^d |\_{\Omega\_2}$ with $0<c\leqslant \rho,\eta \leqslant C$ for some $c,C\in\mathbb{R}$. Assume also that $\Omega\_2$ is convex, then the optimal transport map $T\in W^{1,1}(\Omega\_1)$?



Open problem 1.28: $W^{1,1}$ regularity for optimal transport map
The problem has indeed been solved by De Philippis-Figalli. The reference is:


<https://link.springer.com/article/10.1007/s00222-012-0405-4>


the paper linked above is a subsequent, related, development of the theory and concerns stability of optimal maps.


