Partial regularity for singular solutions to the Monge-Ampere equation

TL;DR

This paper establishes partial regularity results for solutions to the Monge-Ampere inequality, showing they are strictly convex outside a small singular set, and constructs solutions with nearly maximal singular sets, advancing understanding of solution regularity.

Contribution

It proves strict convexity away from a small singular set for solutions to the Monge-Ampere inequality and constructs solutions with near-maximal singular sets, demonstrating optimality.

Findings

Solutions are strictly convex outside a Hausdorff measure zero set.

Constructed solutions with singular sets of dimension arbitrarily close to n-1.

Established $W^{2,1}$ regularity and unique continuation properties.

Abstract

We prove that solutions to the Monge-Ampere inequality $$\det D^2u \geq 1$$ in $\mathbb{R}^n$ are strictly convex away from a singular set of Hausdorff $n-1$ dimensional measure zero. Furthermore, we show this is optimal by constructing solutions to $\det D^2u = 1$ with singular set of Hausdorff dimension as close as we like to $n-1$. As a consequence we obtain $W^{2,1}$ regularity for the Monge-Ampere equation with bounded right hand side and unique continuation for the Monge-Ampere equation with sufficiently regular right hand side.