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


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.

If the inline PDF is not rendering correctly, you can download the PDF file here.

  • [1] M. S. Ashbaugh and R. D. Benguria. Eigenvalue ratios for Sturm-Liouville operators. Journal of Differential Equations, 103 (1):205-219,1993.

  • [2] M. S. Ashbaugh and R. D. Benguria. Isoperimetric bounds for higher eigenvalue ratios for the n-dimensional fixed membrane problem. Proceedings of the Royal Society of Edinburgh, Section: A Mathematics, 123(06):977-985,1993.

  • [3] M. S. Ashbaugh, E. M. Harrell, and R. Svirsky. On minimal and maximal eigenvalue gaps and their causes. Pacific Journal of Mathematics, 147(1):1–24,1991.

  • [4] D. Barnes. Extremal problems for eigenvalues with applications to buckling, vibration and sloshing. SIAM Journal on Mathematical Analysis, 16(2):341-357,1985.

  • [5] S. Chanillo, D. Grieser, M. Imai, K. Kurata, and I. Ohnishi. Symmetry breaking and other phenomena in the optimization of eigenvalues for composite membranes. Communications in Mathematical Physics, 214(2):315-337, 2000.

  • [6] Shiu-Yuen Cheng and Kevin Oden. Isoperimetric inequalities and the gap between the first and second eigenvalues of an Euclidean domain. The Journal of Geometric Analysis, 7(2):217-239,1997.

  • [7] Eduardo Colorado and Jorge García-Melián. The behavior of the principal eigenvalue of a mixed elliptic problem with respect to a parameter. Journal of Mathematical Analysis and Applications, 377(1):53-69, 2011.

  • [8] Lennart Edsberg. Introduction to Computation and Modeling for Differential Equations. John Wiley & Sons, Inc., 2008.

  • [9] Behrouz Emamizadeh and Ryan I. Fernandes. Optimization of the principal eigenvalue of the one-dimensional Schrödinger operator. Electronic Journal of Differential Equations, 65, 2008. 11 pp.

  • [10] Julián Fernández Bonder, Julio D. Rossi, and Carola-Bibiane Schönlieb. The best constant and extremals of the Sobolev embeddings in domains with holes: the L case. Illinois J. Math., 52(4):1111–1121, 2008.

  • [11] Evans M. Harrell Li, Pawel Kröger, and Kazuhiro Kurata. On the placement of an obstacle or a well so as to optimize the fundamental eigenvalue. SIAM J. Math. Anal., 33(1):240-259, 2001.

  • [12] Antoine Henrot, El-Haj Laamri, and Didier Schmitt. On some spectral problems arising in dynamic populations. Commun. Pure Appl. Anal., 11(6):2429-2443, 2012.

  • [13] S. Karaa. Extremal eigenvalue gaps for the Schrödinger operator with Dirichlet boundary conditions. J. Math. Phys., 39: 2325-2332,1998.

  • [14] Bernhard Kawohl. Rearrangements and Convexity of Level Sets inPDE. Number 1150 in Lecture Notes in Mathematics. Springer-Verlag, 1985.

  • [15] M. Levitin and L. Parnovski. On the principal eigenvalue of a Robin problem with a large parameter. Math. Nachr., 281: 272-281, 2008.

  • [16] P. Li and S.-T. Yau. On the Schrödinger equation and the eigenvalue problem. Comm. Math. Phys., 88:309-318,1983.

  • [17] U. Lumiste and J. Peetre, editors. Edgar Krahn, 1894-1961, A Centenary Volume. IOS Press, Amsterdam, 1994.

  • [18] P. Marcellini. Bounds for the third membrane eigenvalue. J. Differential Equations, 37:438-443,1980.

  • [19] N. S. Nadirashvili. Rayleigh's conjecture on the principal frequency of the clamped plate. Arch. Rational Mech. Anal., 129: 1-10,1995.

  • [20] R. D. Nussbaum and Y. Pinchover. On variational principles for the generalized principal eigenvalue of second order elliptic operators and some applications. J. Anal. Math., 59:161-177,1992.

  • [21] R. Osserman. The isoperimetric inequality. Bull. Amer. Math. Soc, 84:1182-1238,1978.

  • [22] L. E. Payne, G. Pólya, and H. F. Weinberger. On the ratio of consecutive eigenvalues. J. Math. and Phys., 35:289-298, 1956.

  • [23] Leandro Del Pezzo, Julián Fernández Bonder, and Wladimir Neves. Optimal boundary holes for the Sobolev trace constant. Journal of Differential Equations, 251(8):2327-2351, 2011.

  • [24] G. Pólya. On the characteristic frequencies of a symmetric membrane. Math. Z., 63:331-337,1955.

  • [25] G. Pólya. On the eigenvalues of vibrating membranes. Proc. London Math. Soc, 11:419-433,1961.


Journal + Issues

Nonautonomous Dynamical Systems (NDS) covers all areas and subareas of nonautonomous dynamical systems. In particular, it encourages interdisciplinary papers that cut across sub-disciplines of nonautonomous dynamical systems to neighboring fields.