Abstract
Some recent results on the construction of energy forms on certain classes of non self-similar fractal sets are presented. In order to overcome the lack of self-similarity, the energy for these sets is obtained by integrating a Lagrangian.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Bassat, B., Strichartz R. and Teplyaev, What is not in the domain of the Laplacian on a Sierpinski gasket type fractal. J. Funct. Anal., 166, 192–217, (1999)
Capitanelli, R., Lagrangians on homogeneous spaces. PhD Thesis Univ. di Roma “La Sapienza”, 2001
Falconer, K. J., The geometry of fractal sets. Cambridge Univ. Press., Cambridge, 1985
Freiberg, U.R. and Lancia, M.R., Energy form on a closed fractal curve. Z. Anal. Anwendungen, 23 no. 1, 115–137, (2004)
Freiberg, U.R. and Lancia, M.R., Energy forms on conformal images of nested fractals. preprint MeMoMat, 15, 2004
Freiberg, U.R. and Lancia, M.R., Can one hear the curvature of a fractal? Spectral asymptotics of fractal Laplace-Beltrami-operators. in preparation
Fukushima, M., Oshima, Y. and Takeda, M., Dirichlet forms and symmetric Markov processes, de Gruyter Studies in Mathematics, vol. 19, Berlin Eds. Bauer Kazdan, Zehnder 1994
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
Hambly, B. and Kumagai, T., Diffusion processes on fractal fields: heat kernel estimates and large deviations. Probab. Theory Relat. Fields, 127(3), 305–352, (2003)
Hutchinson, J.E., Fractals and self-similarity. Indiana Univ. Math. J. 30, 713–747, (1981)
Jonnson, A. and Wallin, H., Function spaces on subsets of\(\mathbb{R}^n \). Math. Rep. Ser. 2 1, (1984)
Kato, T., Pertubation theory for linear operators. 2nd edit., Springer, 1977
Kigami, J., Harmonic calculus on p.c.f. self-similar sets. Trans. Am. Math. Soc. 335, 721–755, (1993)
Kigami, J., Analysis on fractals. Cambridge Univ. Press., Cambridge, 2001
Kusuoka, S., Diffusion processes on nested fractals. Lecture Notes in Math. 1567, Springer, 1993
Lancia, M.R. and Vivaldi, M.A., Lipschitz spaces and Besov traces on self-similar fractals. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl.(5) 23, 101–116, (1999)
Lancia, M.R., Second-order transmission problems across a fractal surface. Rend. Accad. Naz. Sci. XL Mem. Mat. Appl.(1) 27, 191–213, (2003)
Lindstrøm, T., Brownian Motion on Nested Fractals. Memoirs Amer. Math. Soc. 420, (1990)
Mosco, U., Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123 no. 2, 368–421, (1994)
Mosco, U., Lagrangian metrics on fractals. Proc. Symp. Appl. Math, 54, Amer. Math. Soc., R. Spigler and S. Venakides eds., 301–323, (1998)
Mosco, U., Energy functionals on certain fractal structures. J. Convex Anal. 9, 581–600, (2002)
Mosco, U., Highly conductive fractal layers. Proc. Conf. “Whence the boundary conditions in modern physics?” Acad. Lincei, Rome, (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Birkhäuser Verlag Basel/Switzerland
About this chapter
Cite this chapter
Freiberg, U.R., Lancia, M.R. (2005). Energy Forms on Non Self-similar Fractals. In: Bandle, C., et al. Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, vol 63. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7384-9_27
Download citation
DOI: https://doi.org/10.1007/3-7643-7384-9_27
Publisher Name: Birkhäuser Basel
Print ISBN: 978-3-7643-7249-1
Online ISBN: 978-3-7643-7384-9
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)