Skip to main content
SpringerLink
Log in

Heat kernel bounds for the Laplacian on metric graphs of polygonal tilings

  • RESEARCH ARTICLE
  • Published:
Semigroup Forum Aims and scope Submit manuscript

Abstract

We obtain an upper heat kernel bound for the Laplacian on metric graphs arising as one skeletons of certain polygonal tilings of the plane, which reflects the one dimensional as well as the two dimensional nature of these graphs.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. Barlow, M.T., Bass, R.F., Kumagai, T.: Stability of parabolic Harnack inequalities on metric measure spaces. J. Math. Soc. Jpn. 58, 485–519 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  2. Carlen, E.A., Loss, M.: Sharp constants in Nash’s inequality. Int. Math. Res. Not. 7, 213–215 (1998)

    Google Scholar 

  3. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  4. Davies, E.B., Pang, M.M.H.: Sharp heat kernel bounds for some Laplace operators. Q. J. Math. Oxf. 40, 281–290 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  5. Diestel, R.: Graph Theory. Springer, New York (2005)

    MATH  Google Scholar 

  6. Grünbaum, B., Shephard, G.C.: Tilings and Patterns. W.H. Freeman Company, New York (1987)

    MATH  Google Scholar 

  7. Haeseler, S.: Heat kernel estimates and related inequalities on metric graphs. arXiv:1101.3010 (2011)

  8. Haeseler, S., Lenz, D., Pogorzelski, F., Pröpper, R.: Note on heat kernel estimates on metric graphs via rough isometries (in preparation)

  9. Mugnolo, D.: Gaussian estimates for a heat equation on a network. Netw. Heterog. Media 2, 55–79 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  10. Pang, M.M.H.: The heat kernel of the Laplacian defined on a uniform grid. Semigroup Forum 78, 238–252 (2008)

    Article  Google Scholar 

  11. Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Institut für Analysis, Universität Ulm, Helmholtzstrasse 18, 89081, Ulm, Germany

    René Pröpper

Authors
  1. René Pröpper
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to René Pröpper.

Additional information

Communicated by Jerome A. Goldstein.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Pröpper, R. Heat kernel bounds for the Laplacian on metric graphs of polygonal tilings. Semigroup Forum 86, 262–271 (2013). https://doi.org/10.1007/s00233-012-9435-x

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00233-012-9435-x

Keywords

Search

Navigation