Geodesic
Averaging of boundary-value problems for the Laplace operator in perforated domains with a nonlinear boundary condition of the third type on the boundary of cavities
Contemporary Mathematics. Fundamental Directions, Partial differential equations, Tome 39 (2011), pp. 173-184 
M. N. Zubova; T. A. Shaposhnikova. Averaging of boundary-value problems for the Laplace operator in perforated domains with a nonlinear boundary condition of the third type on the boundary of cavities. Contemporary Mathematics. Fundamental Directions, Partial differential equations, Tome 39 (2011), pp. 173-184. http://geodesic.mathdoc.fr/item/CMFD_2011_39_a10/
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     author = {M. N. Zubova and T. A. Shaposhnikova},
     title = {Averaging of boundary-value problems for the {Laplace} operator in perforated domains with a~nonlinear boundary condition of the third type on the boundary of cavities},
     journal = {Contemporary Mathematics. Fundamental Directions},
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     url = {http://geodesic.mathdoc.fr/item/CMFD_2011_39_a10/}
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Voir la notice du chapitre de livre provenant de la source Math-Net.Ru

In this paper, the asymptotic behavior of solutions $u_\varepsilon$ of the Poisson equation in the $\varepsilon$-periodically perforated domain $\Omega_\varepsilon\subset\mathbb R^n$, $n\ge3$, with the third nonlinear boundary condition of the form $\partial_\nu u_\varepsilon+\varepsilon^{-\gamma}\sigma(x,u_\varepsilon)=\varepsilon^{-\gamma}g(x)$ on a boundary of cavities, is studied. It is supposed that the diameter of cavities has the order $\varepsilon^\alpha$ with $\alpha>1$ and any $\gamma$. Here, all types of asymptotic behavior of solutions $u_\varepsilon$, corresponding to different relations between parameters $\alpha$ and $\gamma$, are studied.

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