  We derive asymptotic formulas for the solutions of the mixed boundary value
problem for the Poisson equation on the union of a thin cylindrical plate and
several thin cylindrical rods. One of the ends of each rod is set into a hole
in the plate and the other one is supplied with the Dirichlet condition. The
Neumann conditions are imposed on the whole remaining part of the boundary.
Elements of the junction are assumed to have contrasting properties so that the
small parameter, i.e. the relative thickness, appears in the differential
equation, too, while the asymptotic structures crucially depend on the
contrastness ratio. Asymptotic error estimates are derived in anisotropic
weighted Sobolev norms.
