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 $\Omega$ of $\mathbb{R}^N$, $N\geq 2$, for the product of two sequences of vector-valued functions which are bounded respectively in $L^p(\Omega)^N$ and $L^q(\Omega)^N$, 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 $W^{-1,1}(\Omega)$. The key ingredient of the proof is a compactness result for bounded sequences in $W^{1,q}(\Omega)$, based on the imbedding of $W^{1,q}(S_{N-1})$ into $L^{p'}(S_{N-1})$ ($S_{N-1}$ the unit sphere of $\mathbb{R}^N$) through a suitable selection of annuli on which the gradients are not too high, in the spirit of De Giorgi and Manfredi. The div-curl result is applied to the homogenization of equi-coercive systems whose coefficients are equi-bounded in $L^\rho(\Omega)$ for some $\rho\textgreater{}{N-1\over 2}$ if $N\textgreater{}2$, or in $L^1(\Omega)$ if $N=2$. It also allows us to prove a weak continuity result for the Jacobian for bounded sequences in $W^{1,N-1}(\Omega)$ satisfying an alternative assumption to the $L^\infty$-strong estimate of Brezis and Nguyen. Two examples show the sharpness of the results.