Existence of a sequence satisfying Cioranescu-Murat conditions in homogenization of Dirichlet problems in perforated domains

Abstract

In a paper of 1982, D. Cioranescu and F. Murat considered the problem satisfied by the limit u of the sequence un solution of 0 −∆un = f in Ωn, un = 0 on ∂Ωn, where Ωn is a sequence of open sets which are contained in a fixed bounded open set Ω. In order to make this, they imposed several hypotheses about the sequence Ωn. Their results were later extended to the p-Laplacian operator by N. Labani and C. Picard. In the present paper, we prove that these hypotheses may be reduced to the following one: There exists a sequence zn ∈ W1,p(Ω) which is zero in Ω \ Ωn and which converges weakly to 1 in W1,p(Ω). Indeed, G. Dal Maso and U. Mosco have solved the above homogenization problem in the general case in which we do not make any hypothesis about Ωn using Γconvergence methods and recently, G. Dal Maso and A. Garroni have also solved this general problem by a method close to the one used by D. Cioranescu and F. Murat.

Keywords

• homogenization
• perforated domains