American Institute of Mathematical Sciences

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