Homogenization of a class of singular elliptic problems in perforated domains (Q1750271)

In this paper the authors study the homogenization of a family of elliptic equations like \[ - \text{div} \, \big( A^{\varepsilon} (x, u_{\varepsilon}) \cdot \nabla u_{\varepsilon}) = f(x) \zeta (u_{\varepsilon}) \] where $\zeta$ is a continuous function, singular at zero, which satisfies \[ \zeta : [0, + \infty) \to \mathbb{R} , \qquad 0 \leqslant \zeta (s) \leqslant \frac{1}{s^{\theta}} \quad \text{for some } \theta \in (0,1] , \] and $f$ is a non-negative function whose summability depends on $\theta$, $f \in L^p$ with $p \geqslant 2 / (1 + \theta)$. The equations are studied in some perforated domain with zero Dirichlet conditions in the external boundary and a non-linear Robin conditions at the boundary of the holes. The authors study the limit behaviour as $\varepsilon$ goes to zero of the solutions (whose existence is proved by the same authors in a previous paper) via the unfolding method by which they get also the pointwise (almost everywhere) convergence of the solutions, tool needed in the characterization of the limit problem. They also give a corrector result.