Degree and Sobolev spaces - Brezis, Haïm; Li, YanYan; Mironescu, Petru; Nirenberg, Louis (Topological Methods in Nonlinear Analysis, 1999)
Let $u$ belong (for example) to $W^{1,n+1}(S^n× Λ, S^n)_{λ\inΛ}$where $Λ$ is a connected open set in ${\mathbb R}^k$. For a.e. the map$x\mapsto u(x,λ)$ is continuous from $S^n$ into $S^n$ and therefore its(Brouwer) degree is well defined. We prove that this degree is independent of$λ$ a.e. in $Λ$. This result is extended to a more general setting, aswell to fractional Sobolev spaces $W^{s,p}$ with $sp≥ n+1$.