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Solutions of the singular Yamabe problem near singular boundaries

Weiming Shen, Zhehui Wang, Jiongduo Xie

TL;DR

This work extends the asymptotic analysis of the singular Yamabe problem with negative scalar curvature to domains with singular boundaries and non-conformally flat backgrounds. By introducing a tangent cone framework and a near-identity diffeomorphism $T$, the authors establish a sharp first-order tangential approximation of local positive solutions by the cone model $u_V$, quantified as $|u(x)/u_V(Tx)-1|\le C d_g(x,0)$. They further develop higher-order expansions under enhanced regularity of $T$, reducing the problem to solving angular and outer-inner auxiliary problems and revealing a tripartite behavior dictated by the parameter $\mu_1$, with explicit leading correction terms involving $\xi_1$ and $\phi_1$. The paper also constructs no-tangential-decay examples to show the optimality of the results and delineates the limitations of tangential asymptotics in broader geometric settings. Overall, the results deepen our understanding of how boundary singularities and ambient curvature shape the near-boundary behavior of singular Yamabe solutions through tangent-cone approximations and degenerate elliptic-analytic techniques.

Abstract

In this paper, we investigate the asymptotic behaviors of solutions to the singular Yamabe problem with negative constant scalar curvature near singular boundaries and derive optimal estimates, where the background metrics are not assumed to be conformally flat. Specifically, we demonstrate that for a wide class of Lipschitz domains with asymptotic conical structure, the local positive solutions are well approximated by the positive solutions in the tangent cones at singular boundary points. This extends the results of [10, 12, 26].

Solutions of the singular Yamabe problem near singular boundaries

TL;DR

This work extends the asymptotic analysis of the singular Yamabe problem with negative scalar curvature to domains with singular boundaries and non-conformally flat backgrounds. By introducing a tangent cone framework and a near-identity diffeomorphism , the authors establish a sharp first-order tangential approximation of local positive solutions by the cone model , quantified as . They further develop higher-order expansions under enhanced regularity of , reducing the problem to solving angular and outer-inner auxiliary problems and revealing a tripartite behavior dictated by the parameter , with explicit leading correction terms involving and . The paper also constructs no-tangential-decay examples to show the optimality of the results and delineates the limitations of tangential asymptotics in broader geometric settings. Overall, the results deepen our understanding of how boundary singularities and ambient curvature shape the near-boundary behavior of singular Yamabe solutions through tangent-cone approximations and degenerate elliptic-analytic techniques.

Abstract

In this paper, we investigate the asymptotic behaviors of solutions to the singular Yamabe problem with negative constant scalar curvature near singular boundaries and derive optimal estimates, where the background metrics are not assumed to be conformally flat. Specifically, we demonstrate that for a wide class of Lipschitz domains with asymptotic conical structure, the local positive solutions are well approximated by the positive solutions in the tangent cones at singular boundary points. This extends the results of [10, 12, 26].
Paper Structure (8 sections, 9 theorems, 152 equations)

This paper contains 8 sections, 9 theorems, 152 equations.

Table of Contents

  1. Introduction
  2. Assumption
  3. Main results
  4. Preliminaries
  5. Proof of Theorem \ref{['thm-main-1']}
  6. Proof of Theorem \ref{['thm-main-2']}
  7. Examples with no tangential decay
  8. Existence of Solutions to Degenerate/Singular Elliptic Equations

Key Result

Theorem 1.2

Let $n\geq 3$, and let $\Omega$, $V$, and $T$ be as in assumption assum. Suppose $u\in C^\infty(\Omega\cap B_1(0))$ is a local positive solution to equation-Yamabe-main-boundary-condition-main. Then, there is a constant $r_0>0$ small enough, such that where $d_g(x, 0)$ is the distance from $x$ to $0$ with respect to the metric $g$, and $C$ is a positive constant depending only on $n$, $g$, $\|T\|

Theorems & Definitions (21)

  • Example 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Remark 1.4
  • Remark 1.5
  • Remark 1.6
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • ...and 11 more