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On the Neumann boundary problem in a domain with complicated boundary
E. Ya. Khruslov
Abstract:
The second boundary value problem is studied for a Helmholtz equation in a domain
$G^{(n)}$, which is the complement of a strongly disconnected set
$F^{(n)}$, contained in a neighborhood of a fixed surface
$\Gamma$.
An approximate description of a solution
$u^{(n)}(x)$ of this problem is based on the study of the sequence
$\{u^{(n)}(x),n=1,2,\dots\}$ of solutions corresponding to a sequence
$\{F^{(n)}\}$ such that for
$n\to\infty$ the set
$F^{(n)}$ becomes infinitely close to
$\Gamma$ and becomes increasingly disconnected.
The sets
$F^{(n)}$ are characterized by the notion of conductivity, introduced in this paper. Necessary and sufficient conditions are given (in terms of conductivity) for the existence of a function
$v(x)$ as a limit of the sequence
$\{u^{(n)}(x)\}$ for
$n\to\infty$ such that it satisfies the same conditions outside
$\Gamma$, and on
$\Gamma$ the conjugacy conditions of the form
$$
\biggl(\frac{\partial v}{\partial\nu}\biggr)_+=\biggl(\frac{\partial v}{\partial\nu}\biggr)_-=p(x)[v_+-v_-],
$$
where the limits of functions from different sides of
$\Gamma$ are indicated by the signs
$+$ and
$-$;
$\nu$ is the normal to
$\Gamma$.
Figure: 1.
Bibliography: 7 titles.
UDC:
517.944
MSC: 35N15,
35J05,
35J25 Received: 27.03.1970
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