Asympotitcs for Some Singular Monge-Amp\`{e}re Equations

TL;DR

This paper investigates the asymptotic behavior of solutions to certain singular Monge-Ampère equations, establishing bounds and comparability results that extend to complex Hessian equations and aid in constructing Green's functions.

Contribution

It provides new bounds and comparability results for solutions to singular Monge-Ampère equations under natural conditions, extending to complex Hessian equations and Green's functions.

Findings

Solutions are comparable to scaled original functions on large parts of the domain.

Solutions are bounded within the interior of the zero set of the weight function.

Results extend to complex Hessian equations and facilitate Green's function construction.

Abstract

Given a psh function $\varphi\in\mathcal{E}(\Omega)$ and a smooth, bounded $\theta\geq 0$, it is known that one can solve the Monge-Amp\`{e}re equation $\mathrm{MA}(\varphi_\theta)=\theta^n\mathrm{MA}(\varphi)$, with some form of Dirichlet boundary values, by work of Ahag--Cegrell--Czy\.{z}--Hiep. Under some natural conditions, we show that $\varphi_\theta$ is comparable to $\theta\varphi$ on much of $\Omega$; especially, it is bounded on the interior of $\{\theta = 0\}$. Our results also apply to complex Hessian equations, and can be used to produce interesting Green's functions.