On the basis of an asymptotic analysis of elliptic problems on thin domains and their junc tions, a model of a mixed boundary value problem for a secondorder scalar differential equation on the 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 bound ary, the Neumann boundary conditions are set. An asymptotic expansion of the solution to such a problem has certain distinguishing features; namely, the expansion coefficients turn out to be rational functions of the large parameter |lnh| (where h ∈ (0, 1] is a small geometric parameter), and the solu tion to the limit problem in the longitudinal section of the plate has logarithmic singularities at the junction points with the beams. Thus, the classical settings of boundary value problems are inadequate to describe the asymptotics, and the technique of selfadjoint extensions and function spaces with sep arated asymptotics must be used.

### Modeling Junctions of Plates and Beams by Means of Self Adjoint Extensions

#### Abstract

On the basis of an asymptotic analysis of elliptic problems on thin domains and their junc tions, a model of a mixed boundary value problem for a secondorder scalar differential equation on the 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 bound ary, the Neumann boundary conditions are set. An asymptotic expansion of the solution to such a problem has certain distinguishing features; namely, the expansion coefficients turn out to be rational functions of the large parameter |lnh| (where h ∈ (0, 1] is a small geometric parameter), and the solu tion to the limit problem in the longitudinal section of the plate has logarithmic singularities at the junction points with the beams. Thus, the classical settings of boundary value problems are inadequate to describe the asymptotics, and the technique of selfadjoint extensions and function spaces with sep arated asymptotics must be used.
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2009
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Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11386/3797278`
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