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

TL;DR

This paper constructs a counterexample demonstrating the failure of $W^{2,1}$ regularity for convex solutions to a degenerate Monge-Ampère equation in two dimensions and explores the propagation of singularities.

Contribution

It provides the first explicit counterexample to $W^{2,1}$ regularity in this context and generalizes classical results on singularity propagation with optimality.

Findings

Counterexample to $W^{2,1}$ regularity in 2D

Propagation of singularities slower than Lipschitz

Generalization of Alexandrov's classical result

Abstract

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