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

摘要

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