Distributional convergence of null Lagrangians under very mild conditions

We consider sequences Uε in W 1,m(Ω®;n), where Ω is a bounded connected open subset of ®n, 2 ≤ m ≤ n. The classical result of convergence in distribution of any null Lagrangian states, in particular, that if Uε converges weakly in W1,m(Ω) to U, then det(DUε) converges to det(DU in D′(Ω). We prove convergence in distribution under weaker assumptions. We assume that the gradient of one of the coordinates of Uε is bounded in the weighted space L2(Ω,Aε(x)dx;®n), where Aε is a non-equicoercive sequence of symmetric positive definite matrix-valued functions, while the other coordinates are bounded in W1,m(Ω.). Then, any momogeneous minor of the Jacobian matrix of Uε converges in distribution to a generalized minor provide that |-Aε-1 |n/2 converges to a Radon measure which does not load any point of Ω. A counter-example shows that this latter condition cannot be removed. As a by-product we derive improved div-curl results in any dimension n ≥ 2.