Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Asymptotics of resonances in a thermoelastic model with light local mass perturbations

Authors: Natalia Babych and Johannes Zimmer
Journal: Quart. Appl. Math. 67 (2009), 311-326
MSC (2000): Primary 35P15; Secondary 34E10, 74F05
DOI: https://doi.org/10.1090/S0033-569X-09-01112-8
Published electronically: March 20, 2009
MathSciNet review: 2514637
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: The limit behaviour of a linear one-dimensional thermoelastic system with local mass perturbations is studied. The mass density is supposed to be nearly homogeneous everywhere except in an $ \varepsilon$-vicinity of a given point, where it is of order $ \varepsilon^{-m}$, with $ m \in \mathbb{R}$. The resonance vibrations of the string are investigated as $ \varepsilon\to0$. An important ingredient of the analysis is the construction of an operator in a space of higher regularity such that its spectrum coincides with that of the classical operator in linearised thermoelasticity, with a correspondence of generalised eigenspaces. The convergence of eigenvalues and eigenprojectors is established along with error bounds for two classes of relatively light mass perturbations, $ m<1$ and $ m=1$, which exhibit contrasting limit behaviour.

References [Enhancements On Off] (What's this?)

  • 1. Christer Bennewitz and Yoshimi Saitō.
    Approximation numbers of Sobolev embedding operators on an interval.
    J. London Math. Soc. (2), 70(1):244-260, 2004. MR 2064761 (2005c:46037)
  • 2. I. C. Gohberg and M. G. Kreĭn.
    Vvedenie v teoriyu lineinykh nesamosopryazhennykh operatorov v gilbertovom prostranstve.
    Izdat. ``Nauka'', Moscow, 1965. MR 0220070 (36:3137)
  • 3. Yu. D. Golovaty.
    Spectral properties of oscillatory systems with added masses.
    Trudy Moskov. Mat. Obshch., 54:29-72, 278, 1992; translation in Trans. Moscow Math. Soc., 1993:23-59, 1993. MR 1256922 (95b:73012)
  • 4. Yu. D. Golovaty, S. A. Nazarov, O. A. Oleinik, and T. S. Soboleva.
    Natural oscillations of a string with an additional mass.
    Sibirsk. Mat. Zh., 29(5):71-91, 237, 1988. MR 971229 (90e:34044)
  • 5. Song Jiang and Reinhard Racke.
    Evolution equations in thermoelasticity, volume 112 of Chapman & Hall/CRC Monographs and Surveys in Pure and Applied Mathematics.
    Chapman & Hall/CRC, Boca Raton, FL, 2000. MR 1774100 (2001g:74013)
  • 6. Tosio Kato.
    Perturbation theory for linear operators.
    Classics in Mathematics. Springer-Verlag, Berlin, 1995.
    Reprint of the 1980 edition. MR 1335452 (96a:47025)
  • 7. O. A. Oleinik, J. Sanchez-Hubert, and G. A. Yosifian.
    On vibrations of a membrane with concentrated masses.
    Bull. Sci. Math., 115(1):1-27, 1991. MR 1086936 (92a:73021)
  • 8. J. Sanchez Hubert and E. Sánchez-Palencia.
    Vibration and coupling of continuous systems.
    Springer-Verlag, Berlin, 1989.
    Asymptotic methods. MR 996423 (91c:00018)
  • 9. E. Sánchez-Palencia.
    Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses.
    In Trends and applications of pure mathematics to mechanics (Palaiseau, 1983), volume 195 of Lecture Notes in Phys., pages 346-368. Springer, Berlin, 1984. MR 755735 (85m:73010)
  • 10. E. Sánchez-Palencia and H. Tchatat.
    Vibration de systèmes élastiques avec des masses concentrées.
    Rend. Sem. Mat. Univ. Politec. Torino, 42(3):43-63, 1984. MR 834781 (87i:73039)
  • 11. Enrique Sánchez-Palencia.
    Nonhomogeneous media and vibration theory, volume 127 of Lecture Notes in Physics.
    Springer-Verlag, Berlin, 1980. MR 578345 (82j:35010)
  • 12. Yakov Yakubov.
    Completeness of root functions and elementary solutions of the thermoelasticity system.
    Math. Models Methods Appl. Sci., 5(5):587-598, 1995. MR 1347149 (96c:35137)

Similar Articles

Retrieve articles in Quarterly of Applied Mathematics with MSC (2000): 35P15, 34E10, 74F05

Retrieve articles in all journals with MSC (2000): 35P15, 34E10, 74F05

Additional Information

Natalia Babych
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

Johannes Zimmer
Affiliation: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom

DOI: https://doi.org/10.1090/S0033-569X-09-01112-8
Received by editor(s): November 3, 2007
Published electronically: March 20, 2009
Article copyright: © Copyright 2008 by the authors
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society