Boundary Harnack inequality for the linearized Monge-Amp\`ere equations and applications

TL;DR

This paper establishes boundary Harnack inequalities and comparison results for solutions to linearized Monge-Ampère equations, extending interior estimates to boundary settings and deriving sharp bounds for the Green's function.

Contribution

It provides boundary versions of the interior Harnack inequality for linearized Monge-Ampère equations, with applications to Green's function estimates.

Findings

Boundary Harnack estimates for solutions

Comparison theorem under natural assumptions

Sharp upper bounds for Green's function

Abstract

In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Amp\`ere equations under natural assumptions on the domain, Monge-Amp\`ere measures and boundary data. Our results are boundary versions of Caffarelli and Guti\'errez's interior Harnack inequality for the linearized Monge-Amp\`ere equations. As an application, we obtain sharp upper bound and global $L^p$-integrability for the Green's function of the linearized Monge-Amp\`ere operator.