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)_λΛ where Λ is a connected open set in R^k. For a.e. the map xu(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, as well to fractional Sobolev spaces W^s,p with sp≥ n+1.