Homogenization of quasilinear elliptic problems with nonlinear Robin conditions and L1 data

Abstract

In the present paper, we study the homogenization of a class of quasilinear elliptic problems in a periodically perforated domain Ωε, with L1 data and nonlinear Robin conditions on the boundary of the holes. Considering that we deal with L1 data, we cannot have solutions in H1(Ωε). Therefore, we use here the convenient notion of renormalized solutions. For the homogenization, we use the periodic unfolding method but we can only apply it to the truncated solutions, which are in H1(Ωε). Hence, as a main difficulty, we have to carefully describe the limits of the truncated unfolded solutions and of their gradients. This allow us to prove that we obtain at the limit an unfolded renormalized problem, as well as a (renormalized) homogenized problem in Ω.