Abstract
A mathematical description of free vibrations of a membrane leads to eigenvalue problems for elliptic differential operators containing a small positive parameter ɛ in the highest order part. The asymptotic behaviour (for ɛ → +0) of the eigenvalues is studied in second order problems that reduce to zero-th and first order for ɛ=0 and in a fourth order problem that reduces to an elliptic problem of second order. In the case of reduction to zero-thorder the density of the eigenvalues on a half-axis grows beyond bound and is proportional to ɛ−n/2 (in n dimensions). In the case of reduction to first order the relation between the asymptotic behaviour of the spectrum and the critical points of the reduced operator is shown. In the case of reduction to second order an asymptotic series expansion is constructed for every eigenvalue.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Ackerberg, R.C. & R.E. O'Malley, Boundary layer problems exhibiting resonance, Studies in Appl. Math. 49 (1970), p. 277–295.
Besjes, J.G., Singular perturbation problems for linear elliptic differential operators of arbitrary order, part I, Degeneration to elliptic operators, J. Math. Anal. Appl. 49 (1975), p. 24–46, part II, Degeneration to first order operators, J. Math. Anal. Appl. 49 (1975), p. 324–346.
Besjes, J.G., Singular perturbation problems for linear elliptic boundary value problems, part III, Counterexamples, Nieuw Archief voor Wiskunde, ser. 3, 25 (1977), p. 1–39.
Eckhaus, W. & E.M. de Jager, Asymptotic solutions of singular perturbation problems for linear differential equations of elliptic type, Arch. Rat. Mech. Anal. 23 (1966), p. 26–86.
Garabedian, P.R., Partial differential equations, John Wiley & Sons, inc., 1964.
Gårding, Lars, On the asymptotic distribution of the eigenvalues and eigenfunctions of elliptic differential operators, Math. Scand. 1 (1953), p. 237–255.
Greenlee, W.M., Singular perturbations of eigenvalues, Arch. Rat. Mech. Anal. 34 (1969), p. 143–164.
de Groen, P.P.N., Singularly perturbed differential operators of second order, Math. Centre Tract 68, Mathematisch Centrum, Amsterdam, 1976.
de Groen, P.P.N., Spectral properties of second order singularly perturbed boundary value problems with turning points, J. Math. Anal. Appl. 57 (1977), p. 119–149.
de Groen, P.P.N., The nature of resonance in a singular perturbation problem of turning point type, to appear in SIAM. J. Math. Anal.
de Groen, P.P.N., A singular perturbation problem of turning point type, in: New developments in differential equations, Proceedings of the 2nd Scheveningen conference on differential equations, ed. W. Eckhaus, North-Holland Mathematics Studies 21, North-Holland publ. Co., Amsterdam 1976, p. 117–124.
Kato, T., Perturbation theory for linear operators, Springer-Verlag, Berlin etc., 1966.
Kifer, Ju.I., On the spectral stability of invariant tori under small random perturbations of dynamical systems, Doklady akademia nauk 235 (1977), p. 512–515, Soviet Math. 18 (1977), p. 952–956.
Matkovsky, B.J., On boundary layer problems exhibiting resonance, SIAM Review 17 (1975), p. 82–100.
Moser, J., Singular perturbation of eigenvalue problems for linear differential equations of even order, Comm. Pure Appl. Math. 8 (1955), p. 649–675.
Timoshenko, S., Theory of plates and shells, McGraw-Hill Book Company, inc., New York, 1940.
Vishik, M.I. & L.A. Lyusternik, Regular degeneration and boundary layer for linear differential equations with small parameter, Uspehi Math. Nauk 12 (1957), p. 3–122, Am. Math. Soc. Trans. Ser. 2, 20 (1962), p. 239–364.
Wilkinson, J.H., The algebraic eigenvalue problem, Oxford University Press, Oxford, 1965.
Yosida, K., Functional analysis, Springer-Verlag, Berlin, etc., 1965.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1979 Springer-Verlag
About this paper
Cite this paper
de Groen, P.P.N. (1979). Singular perturbations of spectra. In: Verhulst, F. (eds) Asymptotic Analysis. Lecture Notes in Mathematics, vol 711. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0062945
Download citation
DOI: https://doi.org/10.1007/BFb0062945
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-09245-2
Online ISBN: 978-3-540-35332-4
eBook Packages: Springer Book Archive