Global Div-Curl lemma on bounded domains in R3

We consider a global version of the Div-Curl lemma for vector fields in a bounded domain Ω⊂R3 with the smooth boundary ∂Ω. Suppose that and converge to u and v weakly in Lr(Ω) and Lr′(Ω), respectively, where 1max{1,3r/(3+r)} and that is bounded in Ls(Ω) for s>max{1,3r′/(3+r′)}, respectively. If either is bounded in W1−1/q,q(∂Ω), or is bounded in W1−1/s,s(∂Ω) (ν: unit outward normal to ∂Ω), then it holds that . In particular, if either uj⋅ν=0 or vj×ν=0 on ∂Ω for all j=1,2,… is satisfied, then we have that . As an immediate consequence, we prove the well-known Div-Curl lemma for any open set in R3. The Helmholtz–Weyl decomposition for Lr(Ω) plays an essential role for the proof.