Some counterexamples to Sobolev regularity for degenerate Monge-Amp\`{e}re equations

We construct a counterexample to $W^{2,1}$ regularity for convex solutions to $$\det D^{2}u\le 1,u{\mathrm{\mid }}_{\mathrm{\partial }\mathrm{\Omega }}=0$$ 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.