Homogenization of a class of singular elliptic problems in perforated domains

Abstract

In this work we study the asymptotic behavior of a class of quasilinear elliptic problems posed in a domain perforated by ε-periodic holes of size ε. The quasilinear equations present a nonlinear singular lower order term fζ(uε), where uε is the solution of the problem at ε-level, ζ is a continuous function singular in zero and f a function whose summability depends on the growth of ζ near its singularity. We prescribe a nonlinear Robin condition on the boundary of the holes contained in Ω and a homogeneous Dirichlet condition on the exterior boundary. The particular case of a Neumann boundary condition on the holes is already new.The main tool in the homogenization process consists in proving a suitable convergence result, which shows that the gradient of uε behaves like that of the solution of a suitable linear problem associated with a weak cluster point of the sequence {uε}, as ε→0. This allows us not only to pass to the limit in the quasilinear term, but also to study the singular term near its singularity, via an accurate a priori estimate. We also get a corrector result for our problem.The main novelty of this work is that for the first time the unfolding method is used to treat a singular term as fζ(uε). This plays an essential role in order to get an almost everywhere convergence of the solution uε, needed in the study the asymptotic behavior of the problem.