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arXiv · 2410.15202

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

Nicholas McCleerey
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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.

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Nicholas McCleerey. 2024-10-19. Asympotitcs for Some Singular Monge-Amp\`{e}re Equations. https://arxiv.org/abs/2410.15202

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