Density of smooth maps for fractional Sobolev spaces $W^{s, p}$ into $\ell$ simply connected manifolds when $s \ge 1$

TL;DR

This paper establishes density results for smooth maps into simply connected manifolds within fractional Sobolev spaces, depending on the connectivity level related to the fractional order and integrability parameters.

Contribution

It proves strong density of smooth maps for certain connectivity conditions and weak density for integer cases, extending the understanding of approximation in fractional Sobolev spaces.

Findings

Strong density of smooth maps when $N^n$ is $loor{sp}$ simply connected.

Weak density of smooth maps when $sp$ is integer and $N^n$ is $(sp-1)$ simply connected.

Utilizes retraction techniques and pointwise fractional derivative estimates.

Abstract

Given a compact manifold $N^n \subset \mathbb{R}^\nu$, $s \ge 1$ and $1 \le p < \infty$, we prove that the class of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s, p}(Q^m; N^n)$ when $N^n$ is $\lfloor sp \rfloor$ simply connected. For $sp$ integer, we prove weak density of smooth maps with values into $N^n$ when $N^n$ is $sp - 1$ simply connected. The proofs are based on the existence of a retraction of $\mathbb{R}^\nu$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s, p} \cap W^{1, sp}$.