A new div–curl result. Applications to the homogenization of elliptic systems and to the weak continuity of the Jacobian

In this paper a new div–curl result is established in an open set Ω of RNRN, N≥2N≥2, for the product σn⋅ηnσn⋅ηn of two sequences of vector-valued functions σnσn, ηnηn such that σnσn is bounded in Lp(Ω)NLp(Ω)N, ηnηn is bounded in Lq(Ω)NLq(Ω)N, with 1/p+1/q=1+1/(N−1)1/p+1/q=1+1/(N−1), and such that divσn, curlηn are compact in suitable spaces. The new assumption is that the product converges weakly in W−1,1(Ω)W−1,1(Ω). The approach is also new in the topic, and is based on a compactness result for bounded sequences in W1,q(Ω)W1,q(Ω) through a suitable selection of annuli on which the gradients are not too high, in the spirit of [26] and [32] and using the imbedding of W1,qW1,q into Lp′Lp′ for the unit sphere of RNRN. The div–curl result is applied to the homogenization of equi-coercive systems whose coefficients are equi-bounded in Lρ(Ω)Lρ(Ω) for some ρ>N−12 if N>2N>2, or in L1(Ω)L1(Ω) if N=2N=2. It also allows us to prove a weak continuity result for the Jacobian for bounded sequences in W1,N−1(Ω)W1,N−1(Ω) satisfying an alternative assumption to the L∞L∞-strong estimate of [8]. Two examples show the sharpness of the results.