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Harmonization and homogenization on fractals

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Abstract

This paper suggests a direct approach to define the Laplacian, the spectral dimension of nested fractals and the pre-Sierpinski carpet conductivity. We find a geometric construction of the harmonic functions on the gasket and therefore can describe effectively the dense set of functions having finite energy. The paper is mostly aimed at the homogenization on the pre-Sierpinski gasket, whose horizontal and nonhorizontal bonds have different conductivities:a andb respectively. We prove the Γ-convergence of the rescaled energies on the pre-Sierpinski gasket to σ(a, b)ε, where ε is the standard energy on the gasket with uniform conductivities. We also find an explicit expression for the effective conductivity σ(a, b) and deduce that its set of singularities turns out to be the Julia set of a certain rational function. A special section is devoted to the problem of the pre-Sierpinski carpet conductivity asymptotic behavior; for this problem a new proof of Barlow-Bass inequalities with sharper constants is given.

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References

  1. Sierpinski, W.: Sur une courbe dont tout point est un point de ramification. C.R. Acad. Sci. Paris160, 302 (1915)

    Google Scholar 

  2. Mandelbrot, B.B.: The fractal geometry of nature. New York: Freeman 1977

    Google Scholar 

  3. Rammal, R., Touluse, G.: Random walks on fractal structures and percolation clusters. J. Physique Lett.44, L13-L22 (1983)

    Google Scholar 

  4. Halvin, S., Ben Avraham, D.: Diffusion in disordered media. Adv. Physics36, 695–798 (1987)

    Google Scholar 

  5. Goldstein, S.: Random walks and diffusion on fractals. In: Kesten, H. (ed.): Percolation theory and ergodic theory of finite practical systems. IMA v. Math. Appl.8. Berlin, Heidelberg, New York: Springer, 1987, pp. 121–128

    Google Scholar 

  6. Kusuoka, S.: A diffusion process on a fractal. In: Itô, K., Ikeda, N. (eds.): Probabilistic methods in mathematical physics. Proc. Tanriguche Symp. Katata 1985. Amsterdam: Kunyo-North-Holland 1987, pp. 251–274

    Google Scholar 

  7. Barlow, M.T., Perkins, E.A.: Brownian motion on the Sierpinski gasket. Prob. Th. Rel. Fields79, 543–623 (1988)

    Article  Google Scholar 

  8. Lindstrøm, T.: Brownian Motion on nested fractals. Memoirs of the AMS83, N 420 (1990)

  9. Barlow, M.T., Bass, R.F.: The construction of Brownian motion on the Sierpinski carpet. Ann. Inst. H. Poincaré25, 225–257 (1989)

    Google Scholar 

  10. Matheron, G.: Eléments pour une théorie des milieux poreux. Masson et cie, Editeurs 120 Boulevard Saint-Germain, Paris, VIe (1967)

    Google Scholar 

  11. Zhikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of differential operators: Berlin, Heidelberg, New York: Springer 1993

    Google Scholar 

  12. Kozlov, S.M.: Homogenization of random structures. Soviet Mathematics Doklady19, N4, 950–954 (1987)

    Google Scholar 

  13. Kozlov, S.M.: On the duality of one type variational problems: Funkz. Anal. i ego Prilozhenija,17/3, 9–15 (1983)

    Google Scholar 

  14. Bergman, D.J.: Bulk physical properties of composite media. In: Bergmann et al. (eds.): Les méthodes de l'homogénéisation: théorie et application en physique. Paris: Eyrolles 1985, pp. 1–128

    Google Scholar 

  15. Golden, K., Papanicolaou, G.C.: Bounds for effective parameters of heterogeneous media by analytic continuation. Commun. Math. Phys.90, 473–491 (1983)

    Article  Google Scholar 

  16. Belyaev, A.Yu., Kozlov, S.M.: Hierarchical structures and estimates for homogenised coefficients. Russ. J. Math. Phys.1, N 1, 3–15 (1992)

    Google Scholar 

  17. Julia, G.: Sur l'iteration des fonctions rationnelles. J. de Math. Pure et Appl.8, 47–245 (1918)

    Google Scholar 

  18. Barlow, M.T., Bass, R.F.: On the resistance of the Sierpinski carpet. Proc. Roy. Soc. London Ser. A431, N 1882, 345–360 (1990)

    Google Scholar 

  19. Kozlov, S.M., Vucan, Ya.: Explicit formula for thermoconductivity of quadratic lattice structure. C.R. Acad. Sci. Paris Ser. 1,314, 281–286 (1992)

    Google Scholar 

  20. Poliya, G., Szego, G.: Aufgaben und Lehrsätze aus der Analysis. Berlin, Heidelberg, New York: Springer Vol. 1, 1964

    Google Scholar 

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Authors and Affiliations

  1. Université de Provence, 3 place V. Hugo, F-13331, Marseille Cedex 3, France

    Serguei M. Kozlov

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  1. Serguei M. Kozlov
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Communicated by T. Spencer

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Kozlov, S.M. Harmonization and homogenization on fractals. Commun.Math. Phys. 153, 339–357 (1993). https://doi.org/10.1007/BF02096647

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  • DOI: https://doi.org/10.1007/BF02096647

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