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Classification of solutions to the $Q$-flat and constant $T$-curvature equation on the half-space and ball

Xuezhang Chen, Shihong Zhang

TL;DR

The paper introduces a rigorous biharmonic Poisson kernel for conformal boundary operators associated with the Paneitz operator, yielding explicit kernel pairs $P_i^3 \oplus P_j^3$ and a complete representation framework for biharmonic boundary-value problems on $\mathbb{R}_+^{n+1}$ and $\mathbb{B}^{n+1}$. It then develops biharmonic Green functions and analyzes both noncritical and critical dimensions to obtain sharp integral representations. Using these tools, the authors prove Liouville-type classification theorems for nonnegative solutions to the $Q$-flat and constant $T$-curvature equations, showing that solutions either vanish or decompose into a geometric bubble plus lower-order terms, depending on the boundary operator pair and critical exponents. Applications include a detailed analysis of biharmonic boundary value problems on balls, boundary singularity behavior, and sharp geometric inequalities within $Q$-flat conformal classes. The work also discusses perspectives and conjectures on rigidity for geodesic balls in spheres and the uniqueness of radial solutions, highlighting the broader impact on prescribing boundary curvature in conformal geometry.

Abstract

For conformal boundary operators associated with the Paneitz operator, we introduce a rigorous definition of the biharmonic Poisson kernel consisting of a pair of kernel functions and derive its explicit representation formula. With this powerful tool, we establish classification theorems of nonnegative solutions to the $Q$-flat and constant $T$-curvature equations on $\mathbb{R}_+^{n+1}$ and $\mathbb{B}^{n+1}$.

Classification of solutions to the $Q$-flat and constant $T$-curvature equation on the half-space and ball

TL;DR

The paper introduces a rigorous biharmonic Poisson kernel for conformal boundary operators associated with the Paneitz operator, yielding explicit kernel pairs and a complete representation framework for biharmonic boundary-value problems on and . It then develops biharmonic Green functions and analyzes both noncritical and critical dimensions to obtain sharp integral representations. Using these tools, the authors prove Liouville-type classification theorems for nonnegative solutions to the -flat and constant -curvature equations, showing that solutions either vanish or decompose into a geometric bubble plus lower-order terms, depending on the boundary operator pair and critical exponents. Applications include a detailed analysis of biharmonic boundary value problems on balls, boundary singularity behavior, and sharp geometric inequalities within -flat conformal classes. The work also discusses perspectives and conjectures on rigidity for geodesic balls in spheres and the uniqueness of radial solutions, highlighting the broader impact on prescribing boundary curvature in conformal geometry.

Abstract

For conformal boundary operators associated with the Paneitz operator, we introduce a rigorous definition of the biharmonic Poisson kernel consisting of a pair of kernel functions and derive its explicit representation formula. With this powerful tool, we establish classification theorems of nonnegative solutions to the -flat and constant -curvature equations on and .
Paper Structure (18 sections, 25 theorems, 371 equations)

This paper contains 18 sections, 25 theorems, 371 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Biharmonic Poisson kernel
  4. The half-space
  5. The unit ball
  6. Biharmonic Green function
  7. Noncritical dimensions
  8. Critical dimension
  9. Classification theorems
  10. Proof of Theorem \ref{['Main thm 1']}
  11. Proof of Theorem \ref{['Main thm 2']}
  12. Proof of Theorem \ref{['Main thm 3']}
  13. Applications to biharmonic boundary value problems on balls
  14. Solutions with single boundary point singularity
  15. Sharp geometric inequalities
  16. ...and 3 more sections

Key Result

Theorem 1.1

For $n \geq 2$ and $X=(x,t) \in \mathbb{R}_+^{n+1}$, we letA heuristic observation allows us to find these explicit kernel functions $P_k^3, k \in \{0,1,2,3\}$. and Suppose (i) $f_0 \in C^2(\mathbb{R}^n)\cap L^\infty(\mathbb{R}^n)$ ; (ii) $k \in \{1,2,3\}$, $f_k \in C^2(\mathbb{R}^n)$ and $f_k=O(|x|^{-k-\delta_k})$ as $|x| \to \infty$ for some $\delta_k \in \mathbb{R}_+$. Then for $i,j \in \{0,1,

Theorems & Definitions (36)

  • Definition 1.1: Biharmonic Poisson kernel
  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Definition 1.2: Biharmonic Green function
  • Remark 1.3
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • ...and 26 more