Estimates for the Counting Function of the Laplace Operator on Domains with Rough Boundaries

Chapter
Part of the International Mathematical Series book series (IMAT, volume 13)

Abstract

We present explicit estimates for the remainder in the Weyl formula for the Laplace operator on a domain Ω, which involve only the most basic characteristics of Ω and hold under minimal assumptions about the boundary \(\partial \Omega\).

Keywords

Stein Tral Seco 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Berry, M. V.: Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals. Geometry of the Laplace Operator. Proc. Sympos. Pure Math. 36, 13–38 (1980)Google Scholar
  2. 2.
    Brossard, J., Carmona, R.: Can one hear the dimension of a fractal? Commun. Math. Phys. 104, 103–122 (1986)MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    van den Berg, M., Lianantonakis, M.: Asymptotics for the spectrum of the Dirichlet Laplacian on horn-shaped regions. Indiana Univ. Math. J. 50, 299–333 (2001)MATHMathSciNetGoogle Scholar
  4. 4.
    Birman, M.S., Solomyak, M.Z.: The principal term of spectral asymptotics for “non-smooth” elliptic problems (Russian). Funkt. Anal. Pril. 4:4, 1–13 (1970); English transl.: Funct. Anal. Appl. 4 (1971)Google Scholar
  5. 5.
    Fleckinger-Pellé, J., Vassiliev, D.: An example of a two-term asymptotics for the “counting function” of a fractal drum. Trans. Am. Math. Soc. 337:1, 99–116 (1993)MATHCrossRefGoogle Scholar
  6. 6.
    Hempel, R., Seco, L., Simon, B.: The essential spectrum of Neumann Laplacians on some bounded singular domains. J. Funct. Anal. 102, 448–483 (1991)MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Ivrii, V.: Sharp spectral asymptotics for operators with irregular coefficients. II. Domains with boundaries and degenerations. Commun. Partial Differ. Equ. 28, 103–128 (2003)MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Maz'ya, V.G.: On Neumann's problem for domains with irregular boundaries (Russian). Sib. Mat. Zh. 9, 1322–1350 (1968); English transl.: Sib. Math. J. 9, 990–1012 (1968)Google Scholar
  9. 9.
    Maz'ya, V.G.: Sobolev Spaces. Springer, Berlin etc. (1985)Google Scholar
  10. 10.
    Netrusov, Y.: Sharp remainder estimates in the Weyl formula for the Neumann Laplacian on a class of planar regions. J. Funct. Anal. 250, 21–41 (2007)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Netrusov, Y., Safarov, Y.: Weyl asymptotic formula for the Laplacian on domains with rough boundaries. Commun. Math. Phys. 253, 481–509 (2005)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Safarov, Y.: Fourier Tauberian Theorems and applications. J. Funct. Anal. 185, 111–128 (2001)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Stein, E.: Singular Integrals and Differentiability Properties of Functions. Princeton Univ. Press, Princeton (1970)MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of BristolBristolUK
  2. 2.Department of MathematicsKing’s College London, StrandLondonUK

Personalised recommendations