Scalar problems in junctions of rods and a plate

II. Self-adjoint extensions and simulation models

¹ University of Lorraine, IECL, CNRS UMR 7502, 3 rue Augustin Fresnel, 57073, Metz, France
² Università del Sannio, Department of Engineering, Corso Garibaldi, 107, 82100, Benevento, Italy
³ Mathematics and Mechanics Faculty, St. Petersburg State University, 198504, Universitetsky pr., 28, St. Petersburg, 195251, Russia
⁴ Institute of Problems of Mechanical Engineering RAS, V.O., Bolshoj pr., 61, St. Petersburg, 199178, Russia

^* Corresponding author: giuseppe.cardone@unisannio.it

Received: 19 November 2016
Accepted: 11 September 2017

Abstract

In this work we deal with a scalar spectral mixed boundary value problem in a spacial junction of thin rods and a plate. Constructing asymptotics of the eigenvalues, we employ two equipollent asymptotic models posed on the skeleton of the junction, that is, a hybrid domain. We, first, use the technique of self-adjoint extensions and, second, we impose algebraic conditions at the junction points in order to compile a problem in a function space with detached asymptotics. The latter problem is involved into a symmetric generalized Green formula and, therefore, admits the variational formulation. In comparison with a primordial asymptotic procedure, these two models provide much better proximity of the spectra of the problems in the spacial junction and in its skeleton. However, they exhibit the negative spectrum of finite multiplicity and for these “parasitic” eigenvalues we derive asymptotic formulas to demonstrate that they do not belong to the service area of the developed asymptotic models.