### Some extensions of the div-curl lemma and applications

Juan Casado D´ıaz Universidad de Sevilla

jcasadod@us.es Abstract.

Abstract. The classical div-curl lemma by F. Murat and L. Tartar asserts
that the limit of the scalar product of two vectorial sequences converging weakly
in L^{p} and L^{q} respectively (p and q conjugated exponents) converges weakly
to the product of the limits if we also know that the divergence of one of the
sequences and the curl of the other one are compact in a suitable topology. This
result was extended in a work in collaboration with M. Briane and F. Murat to
the case where p and q satisfy 1/p+1/q ≤ 1+1/N (N the dimension of the space).

Recently, in a work with M. Briane we have extended the result to 1/p + 1/q <

1 + 1/(N − 1) or even 1/p + 1/q ≤ 1 + 1/(N − 1) with some equiintegrability condition. The result applies to the convergence of the Jacobian of a sequence converging weakly in a Sobolev space and to the homogenization of partial differential systems with unbounded coefficients.

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