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 ${\mathbb{R}}^N$, $N\ge 2$, 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 $L^{p^{\prime }}(S\_N-1)$ ($S\_N-1$ 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 $ρ\textgreater{}{N-1\over 2}$ 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.