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

M. N. Zubova ; T. A. Shaposhnikova

Contemporary Mathematics. Fundamental Directions, Partial differential equations, Tome 39 (2011), pp. 173-184

Résumé

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.