The first boundary value problem in domains with a complicated boundary for higher order equations

The first boundary value problem is considered for an elliptic selfadjoint operator $L$ of order $2m$ in a domain $\Omega^{(s)}$ of complicated structure of the form $\Omega^{(s)}=\Omega\setminus F^{(s)}$, where $\Omega$ is a comparatively simple domain in $\mathbf R_n$ ($n\geqslant2$) and $F^{(s)}$ is a closed, connected, highly fragmented set in $\Omega$. The asymptotic behavior of the resolvent $R^{(s)}$ of this problem is studied for $s\to\infty$ when the set $F^{(s)}$ becomes ever more fragmented and is disposed volumewise in $\Omega$ so that the distance from $F^{(s)}$ to any point $x\in\Omega$ tends to zero.
It is shown that $R^{(s)}$ converges in norm to the resolvent $R^c$ of an operator $L+c(x)$, which is considered in the simple domain $\Omega$ under null conditions in $\partial\Omega$. A massivity characteristic of the sets $F^{(s)}$ (of capacity type) is introduced, which is used to formulate necessary and sufficient conditions for convergence, and the function $c(x)$ is described.
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