Asymptotic analysis of problems on junctions of domains of different limit dimensions. A body pierced by a thin rod

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.