Some extensions of the div-curl lemma and applications
Juan Casado D´ıaz Universidad de Sevilla
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 Lp and Lq 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.