$W^{2,1}$ estimates for the Monge-Ampere equation
$\det D^2u = f$ in $R^n$ were first obtained by De Philippis and Figalli in the case
that $f$ is bounded between positive constants. Motivated by
applications to the semigeostrophic equation, we consider the case
that $f$ is bounded but allowed to be zero on some set. In this case
there are simple counterexamples to $W^{2,1}$ regularity in dimension $n\geq 3$ that have a Lipschitz singularity. In contrast, if $n = 2$ a
classical theorem of Alexandrov on the propagation of Lipschitz
singularities shows that solutions are $C^1$. We will discuss a
counterexample to $W^{2,1}$ regularity in two dimensions whose second
derivatives have nontrivial Cantor part, and also a related result on
the propagation of Lipschitz / log(Lipschitz) singularities that is
optimal by example.