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)}.
Singularities of the solution to a Monge--Amp\`ere equation on the boundary of the 3-simplex
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We show that the metric defined by the solution to the tropical Monge-Amp\`ere equation, as defined by Hultgren, Mazzon, and the first two authors, on the boundary of the 3-simplex is asymptotic to the Gross-Wilson metric on $S^2$ near each of the 6 singular points. We deduce in addition that the solution is not $C^{1,1}$ across the singular points. Compared to previous works, our starting point is the real Monge-Amp\`ere equation, as opposed to the complex structure.
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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)}.