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

* Corresponding author 1 IRMAR - Institut de Recherche Mathématique de Rennes 2 EDAN US - Departamento de Ecuaciones Diferenciales y Analisis Numerico

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 $L^ ho\Omega$ for some $ho>{N-1\over 2}$ if $N>2$, 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.

Keywords : Jacobian weak continuity H-convergence Γ-convergence non equi-bounded coefficients homogenization elliptic systems div-curl

Author: Marc Briane - Juan Casado-Diaz -

Source: https://hal.archives-ouvertes.fr/