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 {uj }∞j =1 and {vj }∞j =1 converge to u and v weakly in Lr(Ω) and Lr (Ω), respectively, where 1 < r <∞ with 1/r + 1/r = 1. Assume also that {div uj }∞j =1 is bounded in Lq(Ω) for q >max{1, 3r/(3+r)} and that {rot vj }∞j =1 is bounded in Ls(Ω) for s >max{1, 3r /(3+r )}, respectively. If either {uj · ν|∂Ω}∞j =1 is bounded in W1−1/q,q (∂Ω), or {vj × ν|∂Ω}∞j =1 is bounded in W1−1/s,s (∂Ω) (ν: unit outward normal to ∂Ω), then it holds that Ω uj ·vj dx→ Ω u·v dx. In particular, if either uj · ν = 0 or vj ×ν = 0 on ∂Ω for all j = 1, 2, . . . is satisfied, then we have that Ω uj · vj dx→ Ω u · v dx. 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. © 2009 Elsevier Inc. All rights reserved.