Skip to main content

Eigenfrequencies of isotropic fractal drums

  • Conference paper
The Maz’ya Anniversary Collection

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 110))

Abstract

We study the behaviour of eigenvalues in problems which correspond to the vibrations of a drum, the whole mass of which is concentrated on a fractal subset of the drum.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. B. Carl and I. Stephani. Entropy, compactness and the approximation of operators. Cambridge Univ. Press. 1990.

    Google Scholar 

  2. D. E. Edmunds and H. Triebel. Function spaces, entropy numbers, differential operators. Cambridge Univ. Press, 1996.

    Google Scholar 

  3. K. J. Falconer. The geometry of fractal sets. Cambridge Univ. Press, 1985.

    Google Scholar 

  4. K. J. Falconer. Fractal geometry. Chichester, Wiley, 1990.

    MATH  Google Scholar 

  5. K. J. Falconer. Techniques in fractal geometry. Chichester, Wiley, 1997.

    MATH  Google Scholar 

  6. W. Farkas and H. Triebel. The distribution of eigenfrequencies of anisotropic fractal drums. J. London Math. Soc. (to appear).

    Google Scholar 

  7. T. Fujita. A fractal dimension, self-similarity and generalized diffusion operators. In: Probabilistic methods in math. physics. Boston, Academic Press, 1987, 83–90.

    Google Scholar 

  8. A. Jonsson and H. Wallin. Function spaces on subsets of ℝn. Math. reports 2, 1. London, Harwood acad. publ., 1984.

    Google Scholar 

  9. H.-G. Leopold. Limiting embeddings and entropy numbers. Preprint, Forschungsbericht, Uni. Jena, 1998.

    Google Scholar 

  10. P. Mattila. Geometry of sets and measures in euclidean spaces. Cambridge Univ. Press, 1995.

    Google Scholar 

  11. K. Naimark and M. Solomyak. On the eigenvalue behaviour for a class of operators related to self-similar measures on ℝd. C. R. Acad. Sci. Paris 319(I) (1994), 837–842.

    MathSciNet  MATH  Google Scholar 

  12. K. Naimark and M. Solomyak. The eigenvalue behaviour for the boundary value problems related to self-similar measures on ℝd. Math. Research Letters 2 (1995), 279–298.

    MathSciNet  MATH  Google Scholar 

  13. Th. Runst and W. Sickel. Sobolev spaces of fractional order, Nemytskij operators, and nonlinear partial differential equations. Berlin, de Gruyter, 1996.

    Book  MATH  Google Scholar 

  14. M. Solomyak and E. Verbitsky. On a spectral problem related to self-similar measures. Bull. London Math. Soc. 27 (1995), 242–248.

    Article  MathSciNet  MATH  Google Scholar 

  15. H. Triebel. Interpolation theory, function spaces, differential operators. Amsterdam, North-Holland, 1978 (sec. ed. Heidelberg, Barth, 1995).

    Google Scholar 

  16. H. Triebel. Theory of function spaces. Basel, Birkhäuser, 1983.

    Book  Google Scholar 

  17. H. Triebel. Theory of function spaces II. Basel, Birkhäuser, 1992.

    Book  MATH  Google Scholar 

  18. H. Triebel. Fractals and spectra. Basel, Birkhäuser, 1997.

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 1999 Springer Basel AG

About this paper

Cite this paper

Edmunds, D., Triebel, H. (1999). Eigenfrequencies of isotropic fractal drums. In: Rossmann, J., Takáč, P., Wildenhain, G. (eds) The Maz’ya Anniversary Collection. Operator Theory: Advances and Applications, vol 110. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8672-7_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8672-7_7

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9725-9

  • Online ISBN: 978-3-0348-8672-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics