The div–curl lemma for sequences whose divergence and curl are compact in [formula omitted

Abstract

It is shown that u k ⋅ v k converges weakly to u ⋅ v if u k ⇀ u weakly in L p and v k ⇀ v weakly in L q with p , q ∈ ( 1 , ∞ ) , 1 / p + 1 / q = 1 , under the additional assumptions that the sequences div u k and curl v k are compact in the dual space of W 0 1 , ∞ and that u k ⋅ v k is equi-integrable. The main point is that we only require equi-integrability of the scalar product u k ⋅ v k and not of the individual sequences.