, Volume 40, Issue 4–6, pp 419–436 | Cite as

Analysis on Fractal Objects

  • U. R. Freiberg


Irregular objects are often modeled by fractals sets. In order to formulate partial differential equations on these nowhere differentiable sets the development of a “new analysis” is necessary. With the help of the model case of the Sierpinski gasket the definition of energy forms and Laplacians on self-similar finitely ramified fractals is explained. Moreover, some results for certain classes of non-self-similar fractals are presented.


Fractals Hausdorff dimension Self-similarity Dirichlet form Laplacian Lagrangian 


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  1. 1.
    Barlow, M., ‘Diffusions on fractals’. Lect. Notes Noth. 1690 (1998).Google Scholar
  2. 2.
    Barlow, M., Bass, R. 1999motion and harmonic analysis on Sierpinski carpets’Cand. J. Math.51673744MathSciNetMATHGoogle Scholar
  3. 3.
    Capitanelli, R., ‘Lagrangians on homogeneous spaces’, PhD Thesis, Univ. di Roma “La Sapienza”, 2001.Google Scholar
  4. 4.
    Falconer, K.J. 1985The geometry of fractal setsCambridge Univerity PressCambridgeMATHGoogle Scholar
  5. 5.
    Freiberg, U. 2003‘Analytic properties of measure geometric Krein–Feller-operators on the real line’Math. Nach.2603447MATHMathSciNetGoogle Scholar
  6. 6.
    Freiberg, U. 2004‘Dirichlet forms on fractal subsets of the real line’R. Anal. Exchange.30589604MathSciNetGoogle Scholar
  7. 7.
    Freiberg, U. 2005‘Spectral asymptotics of generalized measure geometric Laplacians on Cantor like sets’Forum Math.1787104MATHMathSciNetGoogle Scholar
  8. 8.
    Freiberg, U.R., Lancia, M.R. 2004‘Energy form on a closed fractal curve’Z. Anal. Anwendungen.23115137MathSciNetMATHGoogle Scholar
  9. 9.
    Freiberg, U.R. and Lancia, M.R. ‘Energy forms on conformal images of nested fractals’. preprint MeMoMat. 15, (2004)Google Scholar
  10. 10.
    Freiberg, U. R. and Lancia, M. R. Can one hear the curvature of a fractal? S asymptotics of fractal Laplace Beltrami-operators. in preparation.Google Scholar
  11. 11.
    Fukushima, M., Oshima, Y. and Takeda, M. Dirichlet forms and symmetric Markov processes, in: Bauer Kazdan, Zehnder (eds) de Gruyter Studies in Mathematics, Vol. 19, Berlin, 1994.Google Scholar
  12. 12.
    Goldstein, S. Random walks and diffusions on fractals. in Percolation theory and ergodic theory of infinite particle systems, Minneapolis, Minn. 1984/85, 121–129; IMA Vol. Math. Appl. 8, Springer, New York, Berlin, 1987.Google Scholar
  13. 13.
    Hino, M.(2005). ‘Singularity of energy measures on self similar sets’, preprintGoogle Scholar
  14. 14.
    Hutchinson, J.E. 1981‘Fractals and self similarity’Indiana Univ. Math. J.30713747CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Jonsson, A. 1996‘Brownian motion on fractals and function spaces’Math. Z222495504MATHMathSciNetGoogle Scholar
  16. 16.
    Jonnson, A. and Wallin, H.‘Function spaces on subsets of R n’, Math. Rep. Ser. 2 1 (1984).Google Scholar
  17. 17.
    Kato, T. Pertubation theory for linear operators. 2nd edn, Springer, 1977.Google Scholar
  18. 18.
    Kigami, J. 1989‘A harmonic calculus on the Sierpinski spaces’Jpn. J. Appl. Math.6259290MATHMathSciNetGoogle Scholar
  19. 19.
    Kigami, J. 1993‘Harmonic calculus on p.c.f. self–similar sets’Trans. Am. Math. Soc.335721755MATHMathSciNetGoogle Scholar
  20. 20.
    Kigami, J. 2001Analysis on fractalsCambridge University PressCambridgeMATHGoogle Scholar
  21. 21.
    Kigami, J., Lapidus, M.L. 1993Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self–similar fractals’Commun. Math. Phys.15893125CrossRefMathSciNetMATHADSGoogle Scholar
  22. 22.
    Kusuoka, S. Diffusion processes on nested fractals. Lec. Notes in Math. 1567, Springer, 1993.Google Scholar
  23. 23.
    Lancia, M.R. 2003‘Second order transmission problems across a fractal surface’Rend. Accad. Naz. Sci. XL Mem. Mat. Appl.27191213MathSciNetGoogle Scholar
  24. 24.
    Lindstrøm, T. Brownian Motion on Nested Fractals’, Memoirs Amer. Math. Soc. 420 (1990).Google Scholar
  25. 25.
    Löbus, J.-U. 1991‘Generalized second order differential operators’Math. Nachr.152229245MATHMathSciNetGoogle Scholar
  26. 26.
    Löbus, J.-U. 1993‘Constructions and generators of one-dimensional quasidiffusions with applications to selfaffine diffusions and Brownian motion on the Cantor set. Stochast’Stochast. Rep.4293114MATHGoogle Scholar
  27. 27.
    Mandelbrot, B.B. 1982The fractal geometry of natureFreemanSan FransiscoMATHGoogle Scholar
  28. 28.
    Moran, P.A.P. 1946‘Additive functions of intervals and Hausdorff measure’Proc. Camb. Phil. Soc.421523MATHCrossRefGoogle Scholar
  29. 29.
    Mosco, U. 1994‘Composite media and asymptotic Dirichlet forms’J. Funct. Anal.123368421CrossRefMATHMathSciNetGoogle Scholar
  30. 30.
    Mosco, U. ‘Lagrangian metrics on fractals’, in: R. Spigler and S. Venakides (eds) Proc. Symp. Appl. Math, Vol. 54, Amer. Math. Soc., (1998) pp. 301–323.Google Scholar
  31. 31.
    Mosco, U. 2002‘Energy functionals on certain fractal structures’J. Convex Anal.9581600MATHMathSciNetGoogle Scholar
  32. 32.
    Mosco, U. Highly conductive fractal layers., Proc. Conf. Whence the boundary conditions in modern physics? Acad. Lincei, Rome, (2002).Google Scholar
  33. 33.
    Mosco, U. ‘An elementary introduction to fractal analysis’, Nonlinear analysis and applications to physical sciences, Italia, Springer, Milan, 2004, pp. 51–90.Google Scholar
  34. 34.
    Triebel, H. ‘Fractals and spectra related to Fourier analysis and function spaces. Monographs in Mathematics, Vol. 91, Birkhäuser, Basel, 1997Google Scholar
  35. 35.
    Weyl, H. 1915‘Das asymptotische Verteilungsgesetz der Eigenschwingungen eines beliebig gestalteten elastischen Körpers’Rend. Cir. Mat. Palermo.39150MATHGoogle Scholar

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Mathematisches InstitutFriedrich-Schiller-Universität JenaJenaGermany

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