Variational fractals

Mosco, Umberto 

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 4, Tome 25 (1997) no. 3-4, pp. 683-712.

Bibliographie

[1] S. Alexander - R. Orbach, Densities of states on fractals: "fractons", J. Physique Lett. 43 (1982) L-625.

[2] M.T. Barlow - R.F. Bass, The construction of Brownian motion on the Sierpinski carpet, Ann. Inst. Henri Poincaré 25, 3 (1989), 225-257. | Zbl | MR | mathdoc-id

[3] M.T. Barlow - E.A. Perkins, Brownian motion on the Sierpinski gasket, Prob. Theo. Rel. Fields 79 (1988), 543-624. | Zbl | MR

[4] M. Biroli - U. Mosco, Formes de Dirichlet et estimations structurelles dans les milieux discontinus, C. R. Acad. Sci. Paris Série I, t. 313 (1991), 593-598. | Zbl | MR

[5] M. Biroli - U. Mosco, A Saint-Venant type principle for Dirichlet forms on discontinuous media, Ann. Mat. Pura Appl. (IV) CLXX (1995), 125-181. | Zbl | MR

[6] M. Biroli - U. Mosco, Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces, Rend. Mat. Acc. Lincei 9, 6 (1995), 37-44. | Zbl | MR

[7] M. Biroli - U. Mosco, Sobolev inequalities for Dirichlet forms on homogeneous spaces, in: "Boundary value problems for partial differential equations and applications ", C. Baiocchi and J. L. Lions (eds.), Research Notes in Appl. Math., Masson, 1993; Sobolev inequalities on homogeneous spaces, Potential Anal. 4 (1995), 311-324. | Zbl | MR

[8] A. Bunde - S. Havlin, Fractals and Disordered Systems, Springer-Verlag, Berlin-Heidelberg, 1991. | Zbl | MR

[9] E.A. Carlen - S. Kusuoka - D.W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré 2 (1987), 245-287. | Zbl | MR | mathdoc-id

[10] R.R. Coifman - G. Weiss, Analyse harmonique sur certaines éspaces homogenes, Lect. Notes in Math. 242, Springer V., Berlin- Heidelberg-New York, 1971. | Zbl | MR

[11] M. Fukushima, Dirichlet forms, diffusion processes and spectral dimension for nested fractals, in "Ideas and Methods in Mathematical Analysis, Stochastics and Applications", S. Albeverio et al. eds., Cambridge Univ. Press, 1992, 151-161. | Zbl | MR

[12] M. Fukushima - Y. Oshima - M. Takeda, Dirichlet forms and Symmetric Markov Processes, Walter De Gruyter Co., 1995. | Zbl | MR

[13] M. Fukushima - T. Shima, On a spectral analysis for the Sierpinski gasket, Potential Anal. 1 (1992), 1-35. | Zbl | MR

[14] S. Goldstein, Random walks and diffusions on fractals, in "Percolation theory and ergodic theory of infinite particle systems", Minneapolis, Minn. 1984-85, pp. 121-129, IMA Vol. Math. Appl. 8, Springer, New York-Berlin- Heidelberg, 1987. | Zbl | MR

[15] J.E. Hutchinson, Fractals and selfsimilarity, Indiana Univ. Math. J. 30 (1981), 713-747. | Zbl | MR

[16] J. Kigami, A harmonic calculus on the Sierpinski spaces, Japan J. Appl. Math. 6 (1989), 259-290. | Zbl | MR

[17] S.M. Kozlov, Harmonization and homogenization on fractals, Commun. Math. Phys. 153 (1993), 159-339. | Zbl | MR

[18] S. Kusuoka, A diffusion process on a fractal, in "Probabilistic methods in Mathematical Physics", Proc. of Taniguchi Int. Symp., Katata and Kyoto, 1985, K. Ito and N. Ikeda eds., Kinokuniya, Tokio, 1987, pp. 251-274. | Zbl | MR

[19] S. Kusuoka, Diffusion processes in nested fractals, in "Statistical Mechanics and Fractals" , Lect. Notes in Math. 1567, Springer V., 1993. | Zbl

[20] S. Kusuoka - X.Y. Zhou, Dirichlet forms on fractals: Poincaré constant and resistance, Prob. Th. Rel. Fields 93 (1992) 169-196. | Zbl | MR

[21] T. Lindstrøm, Brownian motion on nested fractals, Memoirs AMS, N.420, 83 (1990). | Zbl | MR

[22] U. Mosco, Composite media and asymptotic Dirichlet forms, J. Funct. Anal. 123, No.2 (1994), 368-421. | Zbl | MR

[23] U. Mosco, Variational metrics on self-similar fractals, C. R. Acad. Sci. Paris, t. 321, Série I (1995), 715-720. | Zbl | MR

[24] U. Mosco, Variations and Irregularities, in "Second Topological Analysis Workshop on Degree, Singularities and Variations: Developments of the last 25 Years", M. Matzeu and A. Vignoli eds., Progress in Nonlinear Differential Equations and Their Applications, Vol. 27, Birkhäuser, 1997. | Zbl | MR

[25] U. Mosco, Lagrangian metrics on fractals, Proc. Conf. on "Recent Advances in Partial Differential Equations" Marking the 70th Birthdays of Peter Lax and Louis Nirenberg, Venezia, 1996; Amer. Math. Soc., Proc. Symp. Appl. Math., Spigler R. and Venakides S. eds., 54 (1998), 301-323. | Zbl | MR

[26] U. Mosco - L. Notarantonio, Homogeneous fractal spaces, Proc. Conf. on "Irregular variational problems", Como, 1994; Serapioni R. and Tomarelli F. eds, Progress in Nonlinear Differential Equations and Their Applications, Vol. 25, Birkhäuser, 1996. | Zbl | MR

[27] R. Rammal - G. Toulouse, Random walks on fractal structures and percolation clusters, J. Physique Lettres 44 (1983), L13-L22.

[28] L. Saloff-Coste, A note on Poincaré, Sobolev and Harnack inequalities, Intern. Math. Res. Notices (1992), 27-38. | Zbl | MR

[29] E.M. Stein, Harmonic analysis, Princeton Univ. Series, 1994.

[30] N. Th. Varopoulos, Sobolev inequalities on Lie groups and symmetric spaces, J. Funct. Anal. 86 (1989), 19-40. | Zbl | MR