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Asymptotic analysis of problems on junctions of domains of different limit dimensions. A body pierced by a thin rod
I. I. Argatov,
S. A. Nazarov
Abstract:
We consider the junction problem on the union of two bodies: a thin cylinder
$Q_\varepsilon$ and a massive body
$\Omega(\varepsilon)$ with an opening into which this cylinder has been inserted. The equations on
$Q_\varepsilon$ and
$\Omega(\varepsilon)$ contain the operators
$\mu\Delta$ and
$\Delta$ (where
$\mu =\mu (\varepsilon)$ is a large parameter and
$\Delta$ is the Laplacian): Dirichlet conditions are imposed on the ends of
$Q_\varepsilon$ and Neumann conditions on the remainder of the exterior boundary. We study the asymptotic behaviour of a solution
$\{u_Q,u_\Omega\}$ as
$\varepsilon\to+0$. The principal asymptotic formulae are as follows:
$u_Q\sim w$ on
$Q_\varepsilon$ and
$u_\Omega\sim v$ on
$\Omega(\varepsilon)$, where
$v$ is a solution of the Neumann problem in
$\Omega$ and the Dirac function is distributed along the interval
$\Omega\setminus\Omega(0)$ with density
$\gamma$. The functions
$w$ and
$\gamma$, depending on the axis variable of the cylinder, are found as solutions of a so-called resulting problem, in which a second-order differential equation and an integral equation (principal symbol of the operator
$(2\pi)^{-1}\ln|\xi|$) are included. In the resulting problem the large parameter
$\lvert\ln\varepsilon\rvert$ remains. Various methods of constructing its asymptotic solutions are discussed. The most interesting turns out to be the case
$\mu(\varepsilon)=O(\varepsilon^{-2}\lvert\ln\varepsilon\rvert^{-1})$) (even the principal terms of the functions
$w$ and
$\gamma$ are not found separately). All the asymptotic formulae are justified; the remainders are estimated in the energy norm.
MSC: Primary
35J25,
35B40,
73C35; Secondary
35A35,
35C10 Received: 23.05.1994
DOI:
10.4213/im60
Fulltext:
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