Fractal differential equations on the Sierpinski gasket

  • Kyallee Dalrymple
  • Robert S. Strichartz
  • Jade P. Vinson
Article

Abstract

Let Δ denote the symmetric Laplacian on the Sierpinski gasket SG defined by Kigami [11] as a renormalized limit of graph Laplacians on the sequence of pregaskets Gm whose limit is SG. We study the analogs of some of the classical partial differential equations with Δ playing the role of the usual Laplacian. For harmonic functions, biharmonic functions, and Dirichlet eigenfunctions of Δ, we give efficient algorithms to compute the solutions exactly, we display the results of implementing these algorithms, and we prove various properties of the solutions that are suggested by the data. Completing the work of Fukushima and Shima [8] who computed the Dirichlet eigenvalues and their multiplicities, we show how to construct a basis (but not orthonormal) for the eigenspaces, so that we have the analog of Fourier sine series on the unit interval. We also show that certain eigenfunctions have the property that they are a nonzero constant along certain lines contained in SG. For the analogs of the heat and wave equation, we give algorithms for approximating the solution, and display the results of implementing these algorithms. We give strong evidence that the analog of finite propagation for the wave equation does not hold because of inconsistent scaling behavior in space and time.

Math subject classifications

Primary 31C45 42C99 

Keywords and phrases

Fractal differential equations analysis on fractals Sierpinski gasket eigenfunctions of the Laplacian wave propagation on fractals 

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References

  1. [1]
    Ayer, E. and Strichartz, R. Hausdorff measure and intervals of maximal density for Cantor sets,Trans. Amer. Math. Soc., (to appear).Google Scholar
  2. [2]
    Barlow, M.T. and Kigami, J. (1997). Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets,J. London Math. Soc. Google Scholar
  3. [3]
    Ben-Bassat, O., Strichartz, R., and Teplyaev, A. What is not in the domain of the Laplacian on a Sierpinski gasket type fractal,J. Functional Anal., (to appear).Google Scholar
  4. [4]
    Berry, M.V. (1980). Some geometric aspects of wave motion: wavefront dislocations, diffraction catastrophes, diffractals in Geometry of the Laplace Operator,Proc. Symp. Pure Math.,36, Amer. Math. Soc., Providence, 13–38.Google Scholar
  5. [5]
    Brossard, J. and Carmona, R. (1986). Can one hear the dimension of a fractal?Commun. Math. Phys.,104, 103–122.Google Scholar
  6. [6]
    Fleckinger, J., Levitin, M., and Vassiliev, D. (1995). Heat equation on the triadic von Koch snowflake,Proc. London Math. Soc.,3(sn71), 372–396.Google Scholar
  7. [7]
    Fleckinger, J. and Vassiliev, D. (1993). An example of a two-term asymptotics for the “counting function” of a fractal drum,Trans. Amer. Math. Soc.,337, 99–116.Google Scholar
  8. [8]
    Fukushima, M. and Shima, T. (1992). On a spectral analysis for the Sierpinski gasket,Potential Anal.,1, 1–35.Google Scholar
  9. [9]
    Jonsson, A. (1996). Brownian motion on fractals and function spaces,Math. Zeit.,222, 495–504.Google Scholar
  10. [10]
    Kigami, J. Personal communication.Google Scholar
  11. [11]
    Kigami, J. (1989). A harmonic calculus on the Sierpinski gasket,Japan J. Appl. Math.,6, 259–290.Google Scholar
  12. [12]
    Kigami, J. (1993). Harmonic calculus on p.c.f. self-similar sets,Trans. Amer. Math. Soc.,335, 721–755.Google Scholar
  13. [13]
    Kigami, J. (1993). Harmonic metric and Dirichlet form on the Sierpinski gasket, in Asymptotic problems in probability theory: stochastic models and diffusions on fractals, Elworthy, K.D. and Bceda, N., Eds.,Pitman Research Notes in Math.,283, 201–218.Google Scholar
  14. [14]
    Kigami, J. and Lapidus, M. (1993). Weyl's problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals,Comm. Math. Phys.,158, 93–125.Google Scholar
  15. [15]
    Kusuoka, S. and Zhou X.Y. Waves on fractal-like manifolds and effective energy propagation, preprint.Google Scholar
  16. [16]
    Levitin, M. and Vassiliev, D. (1996). Spectral asymptotics, renewal theorem, and the Berry conjecture for a class of fractals,Proc. London Math. Soc.,3(sn72), 188–214.Google Scholar
  17. [17]
    Lindstrom, T. (1990). Brownian motion on nested fractals,Memoir AMS,83(420).Google Scholar
  18. [18]
    Milnor, J. (1990).Dynamics in One Complex Variable: Introductory Lectures, SUNY Stony Brook, Institute for Mathematical Sciences.Google Scholar
  19. [19]
    Rammal, R. (1984). Spectrum of harmonic excitations on fractals,J. Physique,45, 191–206.Google Scholar
  20. [20]
    Rammal, R. and Toulouse, G. (1982). Random walks on fractal structures and percolation clusters,J. Physique Lett.,43, 13–22.Google Scholar
  21. [21]
    Shima, T. (1991). On eigenvalue problems for the random walks on the Sierpinski pre-gaskets,Japan J. Indust. Appl. Math.,8, 127–141.Google Scholar
  22. [22]
    Shima, T. (1996). On eigenvalue problems for Laplacians on p.c.f. self-similar sets,Japan J. Indust. Appl. Math.,13, 1–23.Google Scholar
  23. [23]
    Strichartz, R. (1996). Fractals in the large,Can. Math. J.,50, 638–657.Google Scholar
  24. [24]
    Strichartz, R. (1997). Piecewise linear wavelets on Sierpinski gasket type fractals,J. Four. Anal. Appl.,3, 387–416.Google Scholar
  25. [25]
    Strichartz, R. Isoperimetric estimates on Sierpinski gasket type fractalsTrans. Amer. Math. Soc., (to appear).Google Scholar
  26. [26]
    Strichartz, R. Some properties of Laplacians on fractals,J. Functional Anal., (to appear).Google Scholar
  27. [27]
    Teplyaev, A. (1998). Spectral analysis on infinite Sierpinski gaskets,J. Functional Anal.,159, 537–567.Google Scholar

Copyright information

© Birkhäuser Boston 1999

Authors and Affiliations

  • Kyallee Dalrymple
    • 1
  • Robert S. Strichartz
    • 2
  • Jade P. Vinson
    • 3
  1. 1.Polygon Network, Inc.Golden
  2. 2.Mathematics DepartmentCornell UniversityIthaca
  3. 3.Department of MathematicsPrinceton UniversityPrinceton

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