Some counterexamples to Sobolev regularity for degenerate Monge–Ampère equations

Abstract

We construct a counterexample to $W^{2,1}$ regularity for convex solutions to

$$det{D}^{2}u\le 1,u{|}_{\partial \Omega }=\text{ const.}$$

in two dimensions. We also prove a result on the propagation of singularities of the form $\left|x_2\right|∕|log{x}_2|$ in two dimensions. This generalizes a classical result of Alexandrov and is optimal by example.