Some counterexamples to Sobolev regularity for degenerate Monge-Ampère equations

We construct a counterexample to $W^{2,1}$ regularity for convex solutions to $\det D^2u \leq 1, \quad u|_{\partial Ω} = 0$ in two dimensions. We also prove a result on the propagation of singularities in two dimensions that are logarithmically slower than Lipschitz. This generalizes a classical result of Alexandrov and is optimal by example.

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