Asympotitcs for Some Singular Monge-Ampère Equations

Abstract

Given a psh function $\phi \in \mathcal{E}(\Omega )$ and a smooth, bounded $\theta \ge 0$, it is known that one can solve the Monge-Ampère equation $\mathrm{MA}({\phi }_{\theta })={\theta }^n\mathrm{MA}(\phi )$, with some form of Dirichlet boundary values, by work of Ahag--Cegrell--Czyż--Hiep. Under some natural conditions, we show that ${\phi }_{\theta }$ is comparable to $\theta \phi$ 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.

arXiv comment13 pages