Monotonicity of the principal eigenvalue related to a non-isotropic vibrating string

Behrouz Emamizadeh 1  and Amin Farjudian 2
  • 1 Department of Mathematical Sciences, University of Nottingham Ningbo China, 199 Taikang East Road, Ningbo, 315100, China
  • 2 School of Computer Science, University of Nottingham Ningbo China, 199 Taikang East Road, Ningbo, 315100, China

Abstract

In this paper we consider a parametric eigenvalue problem related to a vibrating string which is constructed out of two different materials. Using elementary analysis we show that the corresponding principal eigenvalue is increasing with respect to the parameter. Using a rearrangement technique we recapture a part of our main result, in case the difference between the densities of the two materials is sufficiently small. Finally, a simple numerical algorithm will be presented which will also provide further insight into the dynamics of the non-principal eigenvalues of the system.

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