Asympotitcs for Some Singular Monge-Ampère Equations
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866910657893367808 |
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| author | McCleerey, Nicholas |
| author_facet | McCleerey, Nicholas |
| contents | Given a psh function $φ\in\mathcal{E}(Ω)$ and a smooth, bounded $θ\geq 0$, it is known that one can solve the Monge-Ampère equation $\mathrm{MA}(φ_θ)=θ^n\mathrm{MA}(φ)$, with some form of Dirichlet boundary values, by work of Ahag--Cegrell--Czyż--Hiep. Under some natural conditions, we show that $φ_θ$ is comparable to $θφ$ on much of $Ω$; 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_15202 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Asympotitcs for Some Singular Monge-Ampère Equations McCleerey, Nicholas Complex Variables Analysis of PDEs Primary: 32U05, Secondary: 35J60 Given a psh function $φ\in\mathcal{E}(Ω)$ and a smooth, bounded $θ\geq 0$, it is known that one can solve the Monge-Ampère equation $\mathrm{MA}(φ_θ)=θ^n\mathrm{MA}(φ)$, with some form of Dirichlet boundary values, by work of Ahag--Cegrell--Czyż--Hiep. Under some natural conditions, we show that $φ_θ$ is comparable to $θφ$ on much of $Ω$; 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. |
| title | Asympotitcs for Some Singular Monge-Ampère Equations |
| topic | Complex Variables Analysis of PDEs Primary: 32U05, Secondary: 35J60 |
| url | https://arxiv.org/abs/2410.15202 |