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 L ϕ u = 0 in any domain ⊂ R n , where L ϕ 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.

Harnack inequality for the fractional nonlocal linearized Monge–Ampère equation

“…The function Φ and the (DC) doubling property. By defining (6.1) for Φ as in (7.1), for X = (x, z), X 0 = (x 0 , z 0 ) ∈ R n+1 , we have for every z, z , z ∈ R. Now, since µ ϕ ∈ (DC) ϕ and h s ∈ (DC) hs , from [11,Lemma 6] it follows that Φ, being the tensor sum of ϕ and h s , satisfies Φ ∈ (DC) Φ with constants depending only on the (DC) constants for µ ϕ , dimension n, and s. In addition, the condition µ Φ ∈ (DC) Φ is quantitatively equivalent to the existence of K ≥ 1 such that…”

“…for every X, Y, Z ∈ R n+1 . By [11,Lemma 6] the sections of Φ are related to the ones of ϕ and h s by…”

“…uniformly in ε > 0. By [11,Lemma 6], it follows that the (DC) constants for Φ ε are controlled by the one for Φ uniformly in ε > 0. Next, let Ψ ε denote the convex conjugate of Φ ε .…”

Poincaré and Sobolev inequalities in the Monge-Ampère quasi-metric structure

“…This establishes the inclusion A ∞ (Ω, δ ϕ ) ⊆ DC(Ω, δ ϕ ). In fact, the inclusion is strict and we refer the readers to see Section 3 in [27] in order mark this gap. Now if the assumption det D 2 ϕ ∈ DC(Ω, δ ϕ ) used in Chapter 3 is replaced with the strictly stronger condition det D 2 ϕ ∈ A ∞ (Ω, δ ϕ ), then the exponent on the right-hand sides of the Poincaré inequalities (3.4) and (3.5) can be improved from 2 to 2 − for some structural 0 < < 1.…”

Analysis of nonlocal equations: One-sided weighted fractional Sobolev spaces and Harnack inequality for fractional nondivergence form elliptic equations

“…Forzani-Maldonado in [29] further remark that µ ψ ∈ (DC) ψ is quantitatively equivalent to δ ψ satisfying the quasi-triangle inequality.…”