Asympotitcs for Some Singular Monge-Amp\`ere Equations
Abstract
Given a psh function ∈E() and a smooth, bounded θ≥ 0, it is known that one can solve the Monge-Amp\`ere equation MA(θ)=θnMA(), with some form of Dirichlet boundary values, by work of Ahag--Cegrell--Czy\.z--Hiep. Under some natural conditions, we show that θ is comparable to θ on much of ; especially, it is bounded on the interior of \θ = 0\. Our results also apply to complex Hessian equations, and can be used to produce interesting Green's functions.
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