Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Existence of extremal functions in higher-order affine Sobolev inequalities

Tristan Bullion-Gauthier

Abstract

In this article, we prove the existence of extremal functions in higher-order affine Sobolev inequalities. Proofs rely on concentration-compactness methods in spaces of integer or fractional regularity. The tools we use, available in spaces of arbitrary regularity, might be of independent interest.

Existence of extremal functions in higher-order affine Sobolev inequalities

Abstract

In this article, we prove the existence of extremal functions in higher-order affine Sobolev inequalities. Proofs rely on concentration-compactness methods in spaces of integer or fractional regularity. The tools we use, available in spaces of arbitrary regularity, might be of independent interest.

Paper Structure

This paper contains 5 sections, 26 theorems, 150 equations.

Table of Contents

  1. Introduction
  2. Basic facts about affine energies and weak convergence
  3. Proof of Theorem \ref{['exist sob']} in the fractional case
  4. Proof of Theorem \ref{['exist sob']} in the integer case
  5. Proof of Lemma \ref{['non zero ex']} and application to the existence of extremal functions in higher-order fractional Sobolev inequalities

Key Result

Theorem 1.1

Let $s>0$ and $1\leq p<\infty$ be such that $sp<N$. When $s$ is an integer, we assume that $p>1$. There exists $f \in \dot{W}^{s,p}({\mathbb R}^N)$ such that ${\left\vert\left\vert f \right\vert\right\vert}_{L^{q}({\mathbb R}^N)}=1$ and $\mathscr{E}_{s,p}(f)=S$. Moreover, when $p>1$, minimizing sequ

Theorems & Definitions (45)

  • Theorem 1.1
  • Proposition 2.1
  • Lemma 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • Definition 2.5
  • Lemma 2.6
  • proof
  • Lemma 2.7
  • ...and 35 more