A Mean-Value Inequality for Non-negative Solutions to the Linearized Monge–Ampère Equation

Abstract

We prove a mean value inequality for non-negative solutions to $\mathcal{L}_\varphi u = 0$ in any domain Ω ⊂ ℝ n , where $\mathcal{L}_\varphi$ is the Monge–Ampere operator linearized at a convex function ϕ, under minimal assumptions on the Monge–Ampere measure of ϕ. An application to the Harnack inequality for affine maximal hypersurfaces is included.