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

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  • 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.
    Mathematics Subject Classification: Primary: 35B27; Secondary: 35J25.

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