For convex domains with Hölder continuous densities, optimal transport potentials are C^{1,1-ε} and W^{2,p}; with C^{1,β} boundaries, they are C^{2,min(α,β)}.
The regularity of mappings with a convex potential
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Boundary regularity of optimal transport maps on convex domains
For convex domains with Hölder continuous densities, optimal transport potentials are C^{1,1-ε} and W^{2,p}; with C^{1,β} boundaries, they are C^{2,min(α,β)}.