Alexandrov's estimate revisited

Abstract

Alexandrov's estimate states that if $\Omega$ is a bounded open convex domain in ${\mathbb{R}}^n$ and $u:\bar{\Omega }\to \mathbb{R}$ is a convex solution of the Monge-Ampere equation $\det D^{2}u=f$ that vanishes on $\partial \Omega$, 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 $\partial \Omega$. For example, we show that if the curvature is bounded away from $0$, then the estimate remains valid if $\omega (\delta )$ is replaced by $C_{\Omega }{\delta }^{\frac{1}{2}+\frac{1}{2n}}$. We determine the sharp constant $C_{\Omega }$ when $n=2$, and when $n\ge 3$ and $\partial \Omega$ is $C^2$, we determine the sharp asymptotics of the optimal modulus of continuity ${\omega }_{\Omega }(\delta )$ as $\delta \to 0$. For arbitrary convex domains, we characterize the scaling of the optimal modulus ${\omega }_{\Omega }$. Under very mild nondegeneracy conditions, our results yield the improved Holder estimate, ${\omega }_{\Omega }(\delta )\le C{\delta }^{\alpha }$ for some $\alpha >1/n$.