The Loewner-Nirenberg problem in singular domains

Han, Qing‐Long; Shen, Weiming

doi:10.1016/j.jfa.2020.108604

Cited by 14 publications

(12 citation statements)

References 30 publications

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“…All these results require ∂Ω to have some degree of regularity. If Ω is a Lipschitz domain, Han and Shen [9] studied asymptotic behaviors of solutions near singular points on ∂Ω, and proved an estimate similar as (1.3), under appropriate conditions of the domain near singular points.…”

Section: Introductionmentioning

confidence: 94%

The Loewner-Nirenberg Problem in Cones

Han¹,

Jiang²,

Shen³

2020

Preprint

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We study asymptotic behaviors of solutions to the Loewner-Nirenberg problem in finite cones and establish optimal asymptotic expansions in terms of the corresponding solutions in infinite cones. The spherical domains over which cones are formed are allowed to have singularities. An elliptic operator on such spherical domains with coefficients singular on boundary play an important role. Due to the singularity of the spherical domains, extra cares are needed for the study of the global regularity of the eigenfunctions and solutions of the associated singular Dirichlet problem.

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Section: Introductionmentioning

confidence: 94%

The Loewner-Nirenberg Problem in Cones

Han¹,

Jiang²,

Shen³

2020

Preprint

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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%

“…When k = 1, Theorem 1.3 was proved by Andersson, Chruściel and Friedrich [3] and Mazzeo [30]. See Han, Jiang and Shen [19] and Han and Shen [20] for results when Ω is not a smooth domain. Also in the case k = 1, it was shown by Gover and Waldron [13] and by Graham [14] via a volume renormalization procedure that the obstruction for the smoothness of du 2 n−2 (i.e.…”

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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“…For regularity of the Loewner-Nirenberg metric in the conformal class of a smooth compact manifold (M n , g) with boundary, an nice expansion of the solution near the boundary is given in [1,38]. Recently, Xumin Jiang and Qing Han developed a type of weighted Schauder estimates near the boundary, see [25] (see also [30]), which fits in this expansion well, and for expansion of the solution near the boundary for manifolds with corners on the boundary see [27] and for more references on this topic one is referred to [26].…”

Section: Introductionmentioning

confidence: 99%

Two Flow Approaches to the Loewner–Nirenberg Problem on Manifolds

2021

J Geom Anal

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We introduce two flow approaches to the Loewner-Nirenberg problem on compact Riemannian manifolds (M n , g) with boundary and establish the convergence of the corresponding Cauchy-Dirichlet problems to the solution of the Loewner-Nirenberg problem. In particular, when the initial data u 0 is a subsolution to (1.1), the convergence holds for both the direct flow (1.3)-(1.5) and the Yamabe flow (1.6). Moreover, when the background metric satisfies R g ≥ 0, the convergence holds for any positive initial data u 0 ∈ C 2,α (M) for the direct flow; while for the case the first eigenvalue λ 1 < 0 for the Dirichlet problem of the conformal Laplacian L g , the convergence holds for u 0 > v 0 where v 0 is the largest solution to the homogeneous Dirichlet boundary value problem of (1.1) and v 0 > 0 in M • . We also give an equivalent description between the existence of a metric of positive scalar curvature in the conformal class of (M, g) and inf u∈C 1 (M)−{0} Q(u) > −∞ when (M, g) is smooth, provided that the positive mass theorem holds, where Q is the energy functional (see (3.2)) of the second type Escobar-Yamabe problem.

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The Loewner-Nirenberg problem in singular domains

Cited by 14 publications

References 30 publications

The Loewner-Nirenberg Problem in Cones

The Loewner-Nirenberg Problem in Cones

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

Two Flow Approaches to the Loewner–Nirenberg Problem on Manifolds

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