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

Abstract

We study the boundary behavior of solutions to the Loewner-Nirenberg problem in domains with conic singularities. To analyze the boundary behavior of solutions with respect to multiple normal directions, we first derive certain eigenvalue growth estimates for singular elliptic operators.

The Loewner-Nirenberg problem in singular domains

“…Then, for any m ≥ n + 1, α ∈ (0, 1), and any θ ∈ Σ near ∂Σ, Similar expansions hold for the coefficients c ij in (7.23) near ∂Σ. As a consequence, we can expand v as a series in terms of t j e −µ i t ρ k+s (log ρ) l with coefficients defined on ∂Σ, for positive integer i and nonnegative integers j, k and l. Refer to [12] for details.…”

“…Here, particular solutions were constructed as a simple application of the Fredholm alternative. In [12], similar particular solutions were constructed as infinite series and the convergence of such series is based on a growth estimate of eigenvalues.…”

“…Jiang [12] proved Theorem 1.1 with τ = n for the case that Σ is a star-shaped smooth spherical domain, and also discussed asymptotic expansions near ∂Σ. If Σ is at least C 2 , the distance function d Σ is at least C 2 near ∂Σ and hence can be used to construct various barrier functions.…”

“…If Σ is at least C 2 , the distance function d Σ is at least C 2 near ∂Σ and hence can be used to construct various barrier functions. This is the strategy adopted in [12].…”