Alexandrov's estimate revisited

Abstract

Alexandrov's estimate states that if $Ω$ is a bounded open convex domain in ${\mathbb R}^n$ and $u:\bar Ω\to {\mathbb R}$ is a convex solution of the Monge-Ampere equation $\det D^2 u = f$ that vanishes on $\partial Ω$, then \[ |u(x) - u(y)| \le ω(|x-y|)(\int_Ωf)^{1/n} \qquad \mbox{for }ω(δ) = C_n\,\mbox{diam}(Ω)^{\frac{n-1}n} δ^{1/n}. \] We establish a variety of improvements of this, depending on the geometry of $\partial Ω$. For example, we show that if the curvature is bounded away from $0$, then the estimate remains valid if $ω(δ)$ is replaced by $C_Ωδ^{\frac 12 + \frac 1{2n}}$. We determine the sharp constant $C_Ω$ when $n=2$, and when $n\ge 3$ and $\partial Ω$ is $C^2$, we determine the sharp asymptotics of the optimal modulus of continuity $ω_Ω(δ)$ as $δ\to 0$. For arbitrary convex domains, we characterize the scaling of the optimal modulus $ω_Ω$. Under very mild nondegeneracy conditions, our results yield the improved Holder estimate, $ω_Ω(δ) \le C δ^α$ for some $α>1/n$.