Eigenvalue asymptotics for a star-graph damped vibrations problem

Article type: Research Article

Authors: Pivovarchik, Vyacheslav | Woracek, Harald

Affiliations: Department of Applied Mathematics and Informatics, South-Ukrainian State Pedagogical University, Staroportofrankovskaya str. 26, UA-65020 Odessa, Ukraine. E-mail: v.pivovarchik@paco.net, vnp@dtp.odessa.ua | Institute for Analysis and Scientific Computing, Vienna University of Technology, Wiedner Hauptstr. 8–10/101, A-1040 Vienna, Austria. E-mail: harald.woracek@tuwien.ac.at

Abstract: We consider a boundary value problem generated by Sturm–Liouville equations on the edges of a star-shaped graph. Thereby a continuity condition and a condition depending on the spectral parameter is imposed at the interior vertex, corresponding to the case of damping in the problem of small transversal vibrations of a star graph of smooth inhomogeneous strings. At the pendant vertices Dirichlet boundary conditions are imposed. We describe the eigenvalue asymptotics of the problem under consideration.

Keywords: eigenvalue asymptotics, Sturm–Liouville theory, star-graph, Kirchhoff condition

DOI: 10.3233/ASY-2011-1040

Journal: Asymptotic Analysis, vol. 73, no. 3, pp. 169-185, 2011