Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes
José Carmona, Pedro J. Martínez−Aparicio · Discrete and Continuous Dynamical Systems · 2016
In this paper we consider the homogenization problem for quasilinear elliptic equations with singularities in the gradient, whose model is the following \begin{document}$\begin{equation*}\begin{cases}\displaystyle -Δ u^\varepsilon + \frac{| abla u^\varepsilon|^2}{{(u^\varepsilon})^θ} = f (x)& \mbox{in} \; Ω^\varepsilon,\\u^\varepsilon = 0&\mbox{on} \; \partial Ω^\varepsilon,\\\end{cases}\end{equation*}$ \end{document} where Ω is an open bounded set of $\mathbb{R}^N$, $θ ∈ (0,1)$ and $f$ is positive function that belongs to a certain Lebesgue's space. The homogenization of these equations is posed in a sequence of domains $Ω^\varepsilon$ obtained by removing many small holes from a fixed domain Ω. We also give a corrector result.