Density in $W^{s,p}(\Omega ; N)$

TL;DR

This paper investigates the density of smooth maps in fractional Sobolev spaces with target manifolds, providing new approximation methods and extending known results to cases where the smoothness parameter is less than one.

Contribution

It introduces a novel approximation technique for $W^{s,p}$ maps with $s<1$, and characterizes when smooth maps are dense in these spaces for various target manifolds.

Findings

Smooth maps are dense in $W^{s,p}( abla; N)$ under certain topological conditions.

New approximation method for $W^{s,p}$-maps when $s<1$.

Extension of density results to fractional Sobolev spaces with $s<1$.

Abstract

Let $\Omega$ be a smooth bounded domain in ${\mathbb R}^n$, $0\textless{}s\textless{}\infty$ and $1\le p\textless{}\infty$. We prove that $C^\infty(\overline\Omega\, ; {\mathbb S}^1)$ is dense in $W^{s,p}(\Omega ; {\mathbb S}^1)$ except when $1\le sp\textless{}2$ and $n\ge 2$. The main ingredient is a new approximation method for $W^{s,p}$-maps when $s\textless{}1$. With $0\textless{}s\textless{}1$, $1\le p\textless{}\infty$ and $sp\textless{}n$, $\Omega$ a ball, and $N$ a general compact connected manifold, we prove that $C^\infty(\overline\Omega \, ; N)$ is dense in $W^{s,p}(\Omega \, ; N)$ if and only if $\pi\_{[sp]}(N)=0$. This supplements analogous results obtained by Bethuel when $s=1$, and by Bousquet, Ponce and Van Schaftingen when $s=2,3,\ldots$ [General domains $\Omega$ have been treated by Hang and Lin when $s=1$; our approach allows to extend their result to $s\textless{}1$.] The case where $s\textgreater{}1$, $s\not\in{\mathbb N}$, is still open.