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Kazdan-Warner obstructions for a 4$th-$order boundary problem

Sergio Cruz-Blázquez, Azahara DelaTorre

Abstract

We derive Kazdan-Warner type identities for the boundary problem of prescribing nonconstant interior $Q$ curvature and boundary $T$ curvature on the upper hemisphere $\mathbb{S}^4_+$ by a conformal change of the standard metric. Using the natural variational formulation and conformal variations generated by boundary-preserving conformal vector fields, we obtain nontrivial integral obstructions to solvability.

Kazdan-Warner obstructions for a 4$th-$order boundary problem

Abstract

We derive Kazdan-Warner type identities for the boundary problem of prescribing nonconstant interior curvature and boundary curvature on the upper hemisphere by a conformal change of the standard metric. Using the natural variational formulation and conformal variations generated by boundary-preserving conformal vector fields, we obtain nontrivial integral obstructions to solvability.
Paper Structure (5 sections, 6 theorems, 79 equations)

This paper contains 5 sections, 6 theorems, 79 equations.

Table of Contents

  1. Introduction
  2. The variational setting
  3. Conformal variations: proof of Theorem \ref{['thm:KW']}
  4. Nonexistence criterions
  5. Weak Formulation

Key Result

Theorem 1.1

Let $u\in\mathcal{H}$ be a solution to eq:P. Then for every boundary-preserving conformal vector field $X$ on ${\mathbb{S}}^{4}_{+}$,

Theorems & Definitions (12)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Proposition 3.1
  • proof
  • Corollary 4.1
  • Corollary 4.2
  • ...and 2 more