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

Abstract

In this paper a new div-curl result is established in an open set Ω of R N , N ≥ 2, for the product of two sequences of vector-valued functions which are bounded respectively in Lp (Ω)N and Lq (Ω)N , with 1/p + 1/q = 1 + 1/(N − 1), and whose respectively divergence and curl are compact in suitable spaces. We also assume that the product converges weakly in W−1,1 (Ω). The key ingredient of the proof is a compactness result for bounded sequences in W1,q(Ω), based on the imbedding of W1,q(SN−1) into Lp ′ (SN−1) (SN−1 the unit sphere of R N ) through a suitable selection of annuli on which the gradients are not too high, in the spirit of [26, 32]. The div-curl result is applied to the homogenization of equi-coercive systems whose coefficients are equi-bounded in Lρ (Ω) for some ρ > N−1 2 if N > 2, or in L1 (Ω) if N = 2. It also allows us to prove a weak continuity result for the Jacobian for bounded sequences in W1,N−1 (Ω) satisfying an alternative assumption to the L∞-strong estimate of H. Brezis & H. Nguyen: “The Jacobian determinant revisited”, Invent. Math., 185 (1) (2011), 17-54. Two examples show the sharpness of the results.