Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the sharp constants in the regional fractional Sobolev inequalities

Rupert L. Frank, Tianling Jin, Wei Wang

Abstract

In this paper, we study the sharp constants in fractional Sobolev inequalities associated with the regional fractional Laplacian in domains.

On the sharp constants in the regional fractional Sobolev inequalities

Abstract

In this paper, we study the sharp constants in fractional Sobolev inequalities associated with the regional fractional Laplacian in domains.
Paper Structure (4 sections, 7 theorems, 117 equations)

This paper contains 4 sections, 7 theorems, 117 equations.

Table of Contents

  1. Introduction
  2. Radial symmetry
  3. Sharp constants
  4. Proof of Proposition \ref{['prop:integrability']}

Key Result

Theorem 1.1

Let $n\geq 4$, $\frac{1}{2}<\sigma<1$ and $\Omega\subset\mathbb{R}^n$ be an open set. Suppose there exists a point $a\in \partial\Omega$ such that $\partial\Omega$ is $C^3$ near the point $a$. Then there exist two positive constants $c$ and $C$, both of which depend only on $n,\sigma$ and $\Omega$, for all large $\lambda$, where $H(a)$ is the mean curvature of $\partial \Omega$ at $a$, and with

Theorems & Definitions (15)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Proposition 2.1
  • proof : Proof of Theorem \ref{['thm:minimizersymmetry']}
  • proof : Proof of Proposition \ref{['prop:rearrangement']}
  • Proposition 2.2
  • proof
  • proof : Proof of Theorem \ref{['thm:symmetry']}
  • ...and 5 more