Laboratoire de Mathématiques Raphaël Salem

UMR 6085 CNRS - Université de Rouen Normandie

We present some existence, uniqueness and homogenization results for the solution of a quasilinear elliptic problem with a singular nonlinearity  posed in a periodically perforated domain. The quasilinear equation presents a term singular in the $u$-variable ($u$ being the solution), which is the product of a continuous singular function $\zeta$ and a fuction $f$ whose summability depends on the growth of $\zeta$ near its singularity. The domain is perforated by $\varepsilon$-holes of $\varepsilon$-size: on the boundary of the holes we impose a homogeneuos Neumann condition, while on the exterior boundary we prescribe a Dirichlet condition.

First of all, we introduce the homogenization theory and the natural framework for our problem. As far as it concerns the existence of a solution, we recall some a priori estimates, the existence and uniqueness results proved in \cite{DoMoRa}.

Then, we study the asymptotic behaviour, as $\varepsilon$ goes to zero, of our problem. In order to study the quasilinear term, which affects also  the study of the singular term near its singularity, we prove a suitable convergence result following the techniques in \cite{ChDo} and \cite{DoGi}. 

The main tool for the homogenization proof consists in proving that   the gradient of the solution behaves like that of a suitable linear problem associated with a weak cluster point, as $\varepsilon\to 0$. This idea was originally introduced in the literature for the homogenization of nonlinear problems with quadratic growth with respect to the gradient.  Here the additional difficulties due to the singular term are treated by means of the periodic unfolding method, introduced by Cioranescu-Damlamian-Griso in \cite{CiDaGr} and successively adapted in literature to perforated domains.