Extension and Representation of Divergence-free Vector Fields on Bounded Domains

Let Ω ⊂ Rn be a bounded, connected domain, with b + 1 boundary components, ∂Ω = Γ0∪...∪Γb. Say O0,...,Ob are the connected components of Rn\nΩ, O0 being the unbounded component, and Γj = ∂Oj . If b > 0, pick yj ∊ Oj ; 1 ≤ j ≤ b, and set (1.1) gj(x) :=x - yj/|x - y|n, x 2 Rn \ {yj}