Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Trace conjunction inequalities

Jean Van Schaftingen

TL;DR

The paper introduces trace conjunction integrals as a unified tool linking a function to its trace, connecting Hardy and Gagliardo trace inequalities. It develops first-order and fractional versions, proves that the conjunction energy controls the Gagliardo energy and Hardy terms, and establishes a trace characterization via finiteness of the mixed energy. It also derives BBM-type limits for smooth maps and discusses extensions to weighted, fractional, and BV settings, posing open problems about normal derivatives and optimal spaces. Overall, the results provide a new, diffeomorphism-invariant framework for trace regularity and boundary behavior in Sobolev spaces.

Abstract

Trace conjunction integrals are introduced and studied. They appear in trace conjunction inequalities which unify the Hardy inequality on a halfspace and the classical Gagliardo trace inequality. At the endpoint they satisfy a Bourgain-Brezis-Mironescu formula for smooth maps, which raises some new open problems.

Trace conjunction inequalities

TL;DR

The paper introduces trace conjunction integrals as a unified tool linking a function to its trace, connecting Hardy and Gagliardo trace inequalities. It develops first-order and fractional versions, proves that the conjunction energy controls the Gagliardo energy and Hardy terms, and establishes a trace characterization via finiteness of the mixed energy. It also derives BBM-type limits for smooth maps and discusses extensions to weighted, fractional, and BV settings, posing open problems about normal derivatives and optimal spaces. Overall, the results provide a new, diffeomorphism-invariant framework for trace regularity and boundary behavior in Sobolev spaces.

Abstract

Trace conjunction integrals are introduced and studied. They appear in trace conjunction inequalities which unify the Hardy inequality on a halfspace and the classical Gagliardo trace inequality. At the endpoint they satisfy a Bourgain-Brezis-Mironescu formula for smooth maps, which raises some new open problems.

Paper Structure

This paper contains 8 sections, 14 theorems, 87 equations.

Table of Contents

  1. Introduction
  2. First-order trace conjunction inequality
  3. Fractional trace conjunction inequality
  4. Controling the Gagliardo integral by the conjunction integral
  5. Characterising the trace
  6. Controlling the Hardy integral by the conjunction integral
  7. From Hardy and Gagliardo to conjunction
  8. Bourgain-Brezis-Mironescu formulae

Key Result

Theorem 2.1

If $N \in \mathbb{N} \setminus \set{0, 1}$, if $s \in \intvo{0}{1}$ and if $p \in \intvo{1}{\infty}$, then every weakly differentiable function $u \colon \mathbb{R}^{N}_+ \to \mathbb{R}$ satisfying has a trace $v = \mathop{\mathrm{tr}}\limits_{\partial \mathbb{R}^{N}_+} u$ satisfying

Theorems & Definitions (27)

  • Theorem 2.1
  • Proposition 2.2
  • proof : Proof of \ref{['proposition_Hardy']}
  • Lemma 2.3
  • proof
  • proof : Proof of \ref{['theorem_first_order_trace_conjunction']}
  • Theorem 3.1
  • proof : Proof of \ref{['theorem_fractional_trace_conjunction']}
  • Theorem 4.1
  • Lemma 4.2
  • ...and 17 more