June  2010, 5(2): 361-384. doi: 10.3934/nhm.2010.5.361

Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain

1. 

School of Mathematics, Institute for Advanced Study, 1, Einstein Drive, Princeton NJ 08540, United States

Received  July 2009 Revised  February 2010 Published  May 2010

We investigate the Neumann problem for a nonlinear elliptic operator $Au^{( s) }=-\sum_{i=1}^{n}\frac{\partial }{ \partial x_{i}}( a_{i}( x,\frac{\partial u^{( s) }}{ \partial x})) $ of Leray-Lions type in the domain $\Omega ^{( s) }=\Omega \backslash F^{( s) }$, where $\Omega $ is a domain in $\mathbf{R}^{n}$($n\geq 3$), $F^{( s) }$ is a closed set located in the neighbourhood of a $(n-1)$-dimensional manifold $ \Gamma $ lying inside $\Omega $. We study the asymptotic behaviour of $ u^{( s) }$ as $s\rightarrow \infty $, when the set $F^{( s) }$ tends to $\Gamma $. Under appropriate conditions, we prove that $ u^{( s) }$ converges in suitable topologies to a solution of a limit boundary value problem of transmission type, where the transmission conditions contain an additional term.
Citation: Mamadou Sango. Homogenization of the Neumann problem for a quasilinear elliptic equation in a perforated domain. Networks & Heterogeneous Media, 2010, 5 (2) : 361-384. doi: 10.3934/nhm.2010.5.361
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