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

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 $W^{s,p}\cap W^{1,sp}$.

20 pages, abstract reformatted for MathJax

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