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

José Carmona; Pedro J. Martínez-Aparicio

doi:10.3934/dcds.2017002

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2017, Volume 37, Issue 1: 15-31. Doi: 10.3934/dcds.2017002

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

José Carmona^1, , and
Pedro J. Martínez-Aparicio^2,

1.
Departamento de Matemáticas, Universidad de Almería, Ctra. Sacramento s/n, La Cañada de San Urbano 04120, Almería, Spain
2.
Departamento de Matemática Aplicada y Estadística, Campus Alfonso XIII, Universidad Politécnica de Cartagena, 30203, Murcia, Spain

* Corresponding author
* Corresponding author

Received: February 29, 2016

Revised: July 31, 2016

Published: September 2017

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)

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Abstract

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.

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

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