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Singular Solutions of the Loewner-Nirenberg Problem in Conic Domains with Prescribed Singularity at Vertices

Stephen Zhou

TL;DR

This work extends the analysis of singular solutions to the Loewner–Nirenberg problem on conical domains by prescribing higher-order vertex asymptotics and constructing actual solutions realizing those prescribed data. The authors first recast the problem in cylindrical coordinates, identify the radial cone solution $u_V(x)=|x|^{-\frac{n-2}{2}}\xi(\theta)$, and perform a linearization around it, yielding the linear operator $\mathcal{L}$ with a singular angular term. They develop a robust weighted Hölder framework and perform a spectral decomposition on the angular domain, enabling a contraction mapping to upgrade approximate order-$\mu$ solutions to genuine solutions with controlled decay near the vertex. They also provide a general scheme to generate approximate solutions by assembling linearized modes with nonlinear corrections, parameterized by a finite set of free data from the index set $\mathcal{I}$, thereby achieving a canonical higher-order asymptotic expansion for solutions in conic domains. The results unify and extend prior asymptotic analyses (e.g., for cones) and offer a concrete method to realize prescribed vertex singularities in conformally invariant Yamabe-type problems.

Abstract

We study positive singular solutions of the Loewner-Nirenberg problem on conical domains and establish the existence of solutions that admit prescribed asymptotic expansions near vertices, valid to arbitrarily high order of approximation.

Singular Solutions of the Loewner-Nirenberg Problem in Conic Domains with Prescribed Singularity at Vertices

TL;DR

This work extends the analysis of singular solutions to the Loewner–Nirenberg problem on conical domains by prescribing higher-order vertex asymptotics and constructing actual solutions realizing those prescribed data. The authors first recast the problem in cylindrical coordinates, identify the radial cone solution , and perform a linearization around it, yielding the linear operator with a singular angular term. They develop a robust weighted Hölder framework and perform a spectral decomposition on the angular domain, enabling a contraction mapping to upgrade approximate order- solutions to genuine solutions with controlled decay near the vertex. They also provide a general scheme to generate approximate solutions by assembling linearized modes with nonlinear corrections, parameterized by a finite set of free data from the index set , thereby achieving a canonical higher-order asymptotic expansion for solutions in conic domains. The results unify and extend prior asymptotic analyses (e.g., for cones) and offer a concrete method to realize prescribed vertex singularities in conformally invariant Yamabe-type problems.

Abstract

We study positive singular solutions of the Loewner-Nirenberg problem on conical domains and establish the existence of solutions that admit prescribed asymptotic expansions near vertices, valid to arbitrarily high order of approximation.

Paper Structure

This paper contains 4 sections, 18 theorems, 265 equations.

Table of Contents

  1. Introduction
  2. Linearized Yamabe Equations
  3. Convergence of Approximate Solutions via Contraction
  4. Approximate Solutions

Key Result

Theorem 1.1

For $n \geq 3$, let $V$ be an infinite Euclidean cone over some Lipschitz domain $\Sigma \subsetneq S^{n-1}$ and $u_V \in C^\infty(V)$ the unique positive solution of eq:LNINFCONE-eq:LNINFCONEB. Then there exist a constant $\tau$ and an increasing sequence of positive constants $\{\gamma_i\}_{i=1}^\ where $C>0$ is a constant, $d_\Sigma$ denotes the distance function on $\Sigma$ to $\partial\Sigma$

Theorems & Definitions (27)

  • Theorem 1.1: HJS2024
  • Theorem 1.2
  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Theorem 2.7
  • Definition 2.8
  • ...and 17 more