Asymptotic analysis of problems on junctions of domains of different limit dimensions. A body pierced by a thin rod
Ivan Ivanovich Argatov, С. А. Назаров · Izvestiya Mathematics · 1996
We consider the junction problem on the union of two bodies: a thin cylinder and a massive body with an opening into which this cylinder has been inserted. The equations on and contain the operators and (where is a large parameter and is the Laplacian): Dirichlet conditions are imposed on the ends of and Neumann conditions on the remainder of the exterior boundary. We study the asymptotic behaviour of a solution as . The principal asymptotic formulae are as follows: on and on , where is a solution of the Neumann problem in and the Dirac function is distributed along the interval with density . The functions and , 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 ) are included. In the resulting problem the large parameter remains. Various methods of constructing its asymptotic solutions are discussed. The most interesting turns out to be the case ) (even the principal terms of the functions and are not found separately). All the asymptotic formulae are justified; the remainders are estimated in the energy norm.