Degree and Sobolev spaces

Abstract

Let $u$ belong (for example) to $W^{1,n+1}(S^n\times \Lambda, S^n)_{\lambda\in\Lambda}$
where $\Lambda$ is a connected open set in ${\mathbb R}^k$. For a.e. the map
$x\mapsto u(x,\lambda)$ is continuous from $S^n$ into $S^n$ and therefore its
(Brouwer) degree is well defined. We prove that this degree is independent of
$\lambda$ a.e. in $\Lambda$. This result is extended to a more general setting, as
well to fractional Sobolev spaces $W^{s,p}$ with $sp\geq n+1$.