Scalar problems in junctions of rods and a plate. II. Self-adjoint extensions and simulation models

TL;DR

This paper investigates scalar spectral problems in junctions of rods and a plate, developing asymptotic models using self-adjoint extensions and algebraic conditions to improve spectral approximation accuracy.

Contribution

It introduces two novel asymptotic models based on self-adjoint extensions and algebraic junction conditions, enhancing spectral analysis in complex junction geometries.

Findings

Models closely approximate spectra of the junction problem.

Negative eigenvalues are identified and asymptotically characterized.

Asymptotic models outperform primordial procedures in spectral proximity.

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.