This talk presents an asymptotic approach based on the method of
compound asymptotic expansions. Elliptic boundary value problems
are considered in domains whose geometry depends on a small parameter
in such a way that the limit set consists of subsets of different
limit dimensions. Examples include 3D bodies linked to 1D segments
and 3D bodies connected to 2D plates (or shells). Boundary value problems
for the Laplacian as well as the Navier operators are analysed.
The asymptotic approximations of solutions of these singularly perturbed
problems involve boundary layers in the vicinty of junction regions.
Analysis of spectral problems for multi-structures gives
accurate asymptotic formulae for the first eigenfrequencies. Applications
are given in problems of mechanics, electrostatics and heat conduction.