The asymptotic behavior of solutions of the second boundary value problem under fragmentation of the boundary of the domain

The second boundary value problem is considered for the equation $\Delta u-cu=f$ in a domain $G^{(s)}$ of complicated structure of the form $G^{(s)}=\mathbf R_n\setminus F^{(s)}$, where $F^{(s)}$ is a closed finely partitioned set lying in a domain $\Omega\subset\mathbf R_n$ ($n\geqslant 2$) for all $s=1,2,\dots$. The asymptotic behavior of a solution $u^{(s)}(x)$ of this problem is studied as $s\to\infty$, when $F^{(s)}$ becomes more and more finely divided and is situated in $\Omega$ so that the distance from $F^{(s)}$ to any point $x\in\Omega$ tends to zero. It is proved that under specific conditions $u^{(s)}(x)$ converges in $\mathbf R_n\setminus\overline\Omega$ to a function $u(x)$ that is a solution of a conjugation problem. Sufficient conditions for convergence are formulated.
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