Harnack's inequality for solutions to the linearized Monge-Ampere equation under minimal geometric assumptions

TL;DR

This paper establishes a Harnack inequality for solutions to a linearized Monge-Ampère equation under minimal geometric assumptions on the associated convex function, with implications for Sobolev inequalities.

Contribution

It introduces a Harnack inequality for linearized Monge-Ampère equations with minimal geometric conditions on the convex function.

Findings

Harnack inequality proven for solutions to the linearized Monge-Ampère equation

Application to Sobolev inequalities included

Results hold under minimal geometric assumptions

Abstract

We prove a Harnack inequality for solutions to $L_A u = 0$ where the elliptic matrix $A$ is adapted to a convex function satisfying minimal geometric conditions. An application to Sobolev inequalities is included.