Aleksandrov-type estimates for a parabolic Monge-Ampere equation

Abstract

A classical result of Aleksandrov allows us to estimate the size of a convex function u at a point x in a bounded domain Ω in terms of the distance from x to the boundary of Ω if ∫Ω det D2u dx < ∞. This estimate plays a prominent role in the existence and regularity theory of the Monge-Ampère equation. Jerison proved an extension of Aleksandrov's result that provides a similar estimate, in some cases for which this integral is infinite. Gutiérrez and Huang proved a variant of the Aleksandrov estimate, relevant to solutions of a parabolic Monge-Ampère equation. In this paper, we prove Jerison-like extensions to this parabolic estimate.