Partial regularity for singular solutions to the Monge-Ampere equation

Abstract

We prove that solutions to the Monge-Ampere inequality $$\det D^{2}u\ge 1$$ 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 $\det D^{2}u=1$ 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.

arXiv commentFinal version, to appear in Comm. Pure Appl. Math