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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 $T\in W^{1,1}(\Omega_1)$?

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    $\begingroup$ I can't find the version of the User's guide which lists open problem 1.28, so not sure if the following covers your question, but the paper by De Philippis and Figalli treats something along those lines in Theorem 1.2: projecteuclid.org/journals/analysis-and-pde/volume-6/issue-4/… $\endgroup$
    – Steve
    Commented Sep 16, 2021 at 9:26

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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.

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