Modeling junctions of plates and beams by means of self-adjoint extensions

On the basis of an asymptotic analysis of elliptic problems on thin domains and their junctions, a model of a mixed boundary value problem for a secondorder scalar differential equation onthe union of 3D thin beams and a plate is constructed. One end of each beam is attached to the plate,and on the other end, the Dirichlet conditions are imposed; on the remaining part of the joint boundary, the Neumann boundary conditions are set. An asymptotic expansion of the solution to such aproblem has certain distinguishing features; namely, the expansion coefficients turn out to be rationalfunctions of the large parameter |lnh| (where h ∈ (0, 1] is a small geometric parameter), and the solution to the limit problem in the longitudinal section of the plate has logarithmic singularities at thejunction points with the beams. Thus, the classical settings of boundary value problems are inadequateto describe the asymptotics, and the technique of selfadjoint extensions and function spaces with separated asymptotics must be used.