Two-Dimensional Div-Curl Results: Application to the Lack of Nonlocal Effects in Homogenization
Marc Briane, Juan Casado‐Díaz · Communications in Partial Differential Equations · 2007
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.