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A sharp Sobolev trace inequality of order four on three-balls

Xuezhang Chen, Shihong Zhang

Abstract

We establish a fourth order sharp Sobolev trace inequality on three-balls, and its equivalence to a third order sharp Sobolev inequality on two-spheres.

A sharp Sobolev trace inequality of order four on three-balls

Abstract

We establish a fourth order sharp Sobolev trace inequality on three-balls, and its equivalence to a third order sharp Sobolev inequality on two-spheres.
Paper Structure (12 sections, 15 theorems, 219 equations)

This paper contains 12 sections, 15 theorems, 219 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Conformally covariant boundary operators
  4. The equivalence of $P_3$ and $\mathscr{B}_3^3$ in class $\mathcal{N}$
  5. Fourth order sharp Sobolev trace inequality on three-balls
  6. Outline of the proof
  7. A bridge between $\mathscr{B}_3^3$ and $(-\Delta)^{3/2}$
  8. An extension problem of $\mathscr{B}_3^3$ in the upper half-space
  9. An integral equation
  10. Proof of Theorem \ref{['Dim 3 integral equ thm']}
  11. Third order sharp Sobolev inequality on two-spheres
  12. Extremal functions on balls for Sobolev trace inequalities

Key Result

Theorem 1.1

Given $0<u\in C^{\infty}(\mathbb{S}^{2})$, let $U$ be a smooth extension of $u$ to $\mathbb{B}^3$ satisfying then with equality if and only if modulo a positive constant, is biharmonic in $\overline{\mathbb{B}^3}$, where $a \in \mathbb{B}^3$.

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • Proposition 2.1
  • Proposition 2.2
  • Proposition 4.1
  • Theorem 4.1
  • Lemma 4.1
  • Theorem 4.2
  • Lemma 4.2
  • ...and 6 more