W^{2,1} estimate for singular solutions to the Monge-Ampere equation

TL;DR

This paper establishes an optimal interior W^{2,1} regularity estimate for singular solutions to the Monge-Ampère equation, advancing understanding of their regularity properties.

Contribution

It provides the first interior W^{2,1} estimate for singular solutions and demonstrates its optimality through explicit examples.

Findings

Proved interior W^{2,1} regularity for singular solutions

Constructed examples showing the estimate's optimality

Enhanced understanding of solution regularity in Monge-Ampère equations

Abstract

We prove an interior $W^{2,1}$ estimate for singular solutions to the Monge-Ampere equation, and construct an example to show our results are optimal.