Boundary behavior in the Loewner-Nirenberg problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866915370375315456 |
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| author | Kichenassamy, Satyanad |
| author_facet | Kichenassamy, Satyanad |
| contents | Let $Ω\subset\mathbb R^n$ be a bounded domain of class $C^{2+α}$, $0<α<1$. We show that if $n\geq 3$ and $u_Ω$ is the maximal solution of equation $Δu = n(n-2)u^{(n+2)/(n-2)}$ in $Ω$, then the hyperbolic radius $v_Ω=u_Ω^{-2/(n-2)}$ is of class $C^{2+α}$ up to the boundary. The argument rests on a reduction to a nonlinear Fuchsian elliptic PDE. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_02484 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Boundary behavior in the Loewner-Nirenberg problem Kichenassamy, Satyanad Complex Variables Differential Geometry Functional Analysis Let $Ω\subset\mathbb R^n$ be a bounded domain of class $C^{2+α}$, $0<α<1$. We show that if $n\geq 3$ and $u_Ω$ is the maximal solution of equation $Δu = n(n-2)u^{(n+2)/(n-2)}$ in $Ω$, then the hyperbolic radius $v_Ω=u_Ω^{-2/(n-2)}$ is of class $C^{2+α}$ up to the boundary. The argument rests on a reduction to a nonlinear Fuchsian elliptic PDE. |
| title | Boundary behavior in the Loewner-Nirenberg problem |
| topic | Complex Variables Differential Geometry Functional Analysis |
| url | https://arxiv.org/abs/2507.02484 |