Boundary expansion for the Loewner-Nirenberg problem in domains with conic singularities

Jiang, Xumin

doi:10.1016/j.jfa.2021.109122

Cited by 6 publications

(3 citation statements)

References 18 publications

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“…Important progress has been made by Andersson, Chruściel and Friedrich [1], Gover and Waldron [24], Graham [22] and Loewner and Nirenberg [34]. For further literature, see e.g., [3][4][5]8,[16][17][18]25,[27][28][29]33,34,39,40,42,44,46] and references therein.…”

Section: Introductionmentioning

confidence: 99%

Existence and Uniqueness for the Non-Compact Yamabe Problem of Negative Curvature Type

Joseph Hogg,

Luc Nguyen

2024

ATA

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We study existence and uniqueness results for the Yamabe problem on noncompact manifolds of negative curvature type. Our first existence and uniqueness result concerns those such manifolds which are asymptotically locally hyperbolic. In this context, our result requires only a partial C 2 decay of the metric, namely the full decay of the metric in C 1 and the decay of the scalar curvature. In particular, no decay of the Ricci curvature is assumed. In our second result we establish that a local volume ratio condition, when combined with negativity of the scalar curvature at infinity, is sufficient for existence of a solution. Our volume ratio condition appears tight. This paper is based on the DPhil thesis of the first author.

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Section: Introductionmentioning

confidence: 99%

Existence and Uniqueness for the Non-Compact Yamabe Problem of Negative Curvature Type

Joseph Hogg,

Luc Nguyen

2024

ATA

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“…For 1 ⩽ 𝑘 ⩽ 𝑛, let 𝜎 𝑘 ∶ ℝ 𝑛 → ℝ be the 𝑘th elementary symmetric function 𝜎 𝑘 (𝜆) = ∑ This problem was first studied in the classical paper of Loewner and Nirenberg [28] where, among other results, the existence of a unique smooth positive solution was proved when the boundary of Ω is smooth and compact. Since then, further studies of problem (1.3) and its generalization in manifold settings have been done by many authors; see, for example, Allen, Isenberg, Lee, and Allen [1], Andersson, Chruściel, and Friedrich [3], Aviles [4], Aviles and McOwen [5], Finn [9], Gover and Waldron [13], Graham [14], Han, Jiang, and Shen [19], Han and Shen [20], Jiang [21], Mazzeo [30], Véron [35], and the references therein. When 2 ⩽ 𝑘 ⩽ 𝑛, the 𝜎 𝑘 -Loewner-Nirenberg problem (1.1)-(1.2) is a fully nonlinear (nonuniformly) elliptic problem of Hessian type.…”

Section: Introductionmentioning

confidence: 99%

Regularity of viscosity solutions of the σk$\sigma _k$‐Loewner–Nirenberg problem

Nguyen

Xiong

2023

Proceedings of London Math Soc

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We study the regularity of the viscosity solution u$u$ of the σk$\sigma _k$‐Loewner–Nirenberg problem on a bounded smooth domain normalΩ⊂Rn$\Omega \subset \mathbb {R}^n$ for k⩾2$k \geqslant 2$. It was known that u$u$ is locally Lipschitz in Ω$\Omega$. We prove that, with d$d$ being the distance function to ∂normalΩ$\partial \Omega$ and δ>0$\delta > 0$ sufficiently small, u$u$ is smooth in false{0

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“…Since then, further studies of problem (1.3) and its generalization in manifold settings have been done by many authors; see e.g. Allen, Isenberg, Lee and Allen [1], Andersson, Chruściel and Friedrich [3], Aviles [4], Aviles and McOwen [5], Finn [9], Gover and Waldron [13], Graham [14], Han, Jiang and Shen [19], Han and Shen [20], Jiang [21], Mazzeo [30], Véron [35] and the references therein. When 2 ≤ k ≤ n, the σ k -Loewner-Nirenberg problem (1.1)-(1.2) is a fully nonlinear (non-uniformly) elliptic problem of Hessian type.…”

Section: Introductionmentioning

confidence: 99%

Regularity of viscosity solutions of the $σ_k$-Loewner-Nirenberg problem

Li¹,

Nguyen²,

Xiong³

2022

Preprint

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We study the regularity of the viscosity solution u of the σ k -Loewner-Nirenberg problem on a bounded smooth domain Ω ⊂ R n for k ≥ 2. It was known that u is locally Lipschitz in Ω. We prove that, with d being the distance function to ∂Ω and δ > 0 sufficiently small, u is smooth in {0 < d(x) < δ} and the first (n−1) derivatives of d n−2 2 u are Hölder continuous in {0 ≤ d(x) < δ}. Moreover, we identify a boundary invariant which is a polynomial of the principal curvatures of ∂Ω and its covariant derivatives and vanishes if and only ifUsing a relation between the Schouten tensor of the ambient manifold and the mean curvature of a submanifold and related tools from geometric measure theory, we further prove that, when ∂Ω contains more than one connected components, u is not differentiable in Ω.

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Boundary expansion for the Loewner-Nirenberg problem in domains with conic singularities

Cited by 6 publications

References 18 publications

Existence and Uniqueness for the Non-Compact Yamabe Problem of Negative Curvature Type

Existence and Uniqueness for the Non-Compact Yamabe Problem of Negative Curvature Type

Regularity of viscosity solutions of the σk$\sigma _k$‐Loewner–Nirenberg problem

Regularity of viscosity solutions of the $σ_k$-Loewner-Nirenberg problem

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