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The first boundary value problem in domains with a complicated boundary for higher order equations
E. Ya. Khruslov
Abstract:
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.
Bibliography: 7 titles.
UDC:
517.946
MSC: Primary
35J40; Secondary
47B25 Received: 09.11.1976
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