On global $W^{2,\delta }$ estimates for the Monge-Ampère equation on general bounded convex domains

Abstract

We establish global $W^{2,\delta }$ estimates, for all $\delta <\frac{1}{n-1}$, for convex solutions to the Monge-Ampère equation with positive $C^{2,\beta }$ right-hand side and zero boundary values on general bounded convex domains in ${\mathbb{R}}^n$ ($n\ge 2$). We exhibit examples showing that global $W^{2,\frac{n}{2(n-1)}}$ estimates fail in all dimensions, so the range of $\delta$ is sharp in two dimensions.