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 {u_j}_{j = 1}^∞ and {v_j}_{j = 1}^∞ converge to u and v weakly in L^r (Ω) and L^r′ (Ω), respectively, where 1 < r < ∞ with 1 / r + 1 / r^′ = 1. Assume also that {div u_j}_{j = 1}^∞ is bounded in L^q (Ω) for q > max {1, 3 r / (3 + r)} and that {rot v_j}_{j = 1}^∞ is bounded in L^s (Ω) for s > max {1, 3 r^′ / (3 + r^′)}, respectively. If either {u_j ṡ ν |_{∂ Ω}}_{j = 1}^∞ is bounded in W^{1 - 1 / q, q} (∂ Ω), or {v_j × ν |_{∂ Ω}}_{j = 1}^∞ is bounded in W^{1 - 1 / s, s} (∂ Ω) (ν: unit outward normal to ∂Ω), then it holds that ∫_Ω u_j ṡ v_j d x → ∫_Ω u ṡ v d x. In particular, if either u_j ṡ ν = 0 or v_j × ν = 0 on ∂Ω for all j = 1, 2, ... is satisfied, then we have that ∫_Ω u_j ṡ v_j d x → ∫_Ω u ṡ v d x. As an immediate consequence, we prove the well-known Div-Curl lemma for any open set in R³. The Helmholtz-Weyl decomposition for L^r (Ω) plays an essential role for the proof.