Integral conditions for convergence of solutions of non-linear Robin's problem in strongly perforated domain

E. Ya. Khruslov   ; L. O. Khilkova   ; M. V. Goncharenko

Žurnal matematičeskoj fiziki, analiza, geometrii, Tome 13 (2017) no. 3, pp. 283-313

Résumé

We consider a boundary-value problem for the Poisson equation in a strongly perforated domain $\Omega^\varepsilon =\Omega\setminus F^\varepsilon \subset R^n$ ($n\geqslant 2$) with non-linear Robin's condition on the boundary of the perforating set $F^\varepsilon$. The domain $\Omega^\varepsilon$ depends on the small parameter $\varepsilon>0$ such that the set $F^\varepsilon$ becomes more and more loosened and distributes more densely in the domain $\Omega$ as $\varepsilon\to0$. We study the asymptotic behavior of the solution $u^\varepsilon(x)$ of the problem as $\varepsilon\to0$. A homogenized equation for the main term $u(x)$ of the asymptotics of $u^\varepsilon(x)$ is constructed and the integral conditions for the convergence of $u^\varepsilon(x)$ to $u(x)$ are formulated.

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