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# Homogenization of singular quasilinear elliptic problems with natural growth in a domain with many small holes

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Research supported by MINECO-FEDER grant MTM2015-68210-P and Junta de Andalucía FQM-194 (first author) and FQM-116 (second author). Programa de Apoyo a la Investigación de la Fundación Séneca-Agencia de Ciencia y Tecnología de la Región de Murcia, reference 19461/PI/14 (second author)

• In this paper we consider the homogenization problem for quasilinear elliptic equations with singularities in the gradient, whose model is the following

$\begin{equation*}\begin{cases}\displaystyle -Δ u^\varepsilon + \frac{|\nabla u^\varepsilon|^2}{{(u^\varepsilon})^θ} = f (x)& \mbox{in} \; Ω^\varepsilon,\\u^\varepsilon = 0&\mbox{on} \; \partial Ω^\varepsilon,\\\end{cases}\end{equation*}$

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.

Mathematics Subject Classification: Primary:35B09, 35B25, 35B27, 35J25, 35J60, 35J75;Secondary:35A01, 35D30.

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