Two-Dimensional Div-Curl Results: Application to the Lack of Nonlocal Effects in Homogenization

In this paper, we study the asymptotic behaviour of sequences of conduction problems and sequences of the associated diffusion energies. We prove that, contrary to the three-dimensional case, the boundedness of the conductivity sequence in L¹ combined with its equi-coerciveness prevents from the appearance of nonlocal effects in dimension two. More precisely, in the two-dimensional case we extend the Murat–Tartar H-convergence which holds for uniformly bounded and equi-coercive conductivity sequences, as well as the compactness result which holds for bounded and equi-integrable conductivity sequences in L¹. Our homogenization results are based on extensions of the classical div-curl lemma, which are also specific to the dimension two.