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Variational fractals. (English) Zbl 1016.28010

A variational fractal is a triple \((K,\mu,E)\), where \(K\) is a self-similar set in the Euclidean space satisfying the open-set condition and a certain “boundary” condition, \(\mu\) is the normalized \(d_f\)-dimensional Hausdorff measure restricted to \(K\), where \(d_f\) denotes the Hausdorff dimension of \(K\), and \(E\) is a symmetric Dirichlet form on \(L^2(K,\mu)\) satisfying five somewhat technical conditions. Examples of variational fractals include nested fractals with the “standard” form \(E\) constructed by T. Lindstrøm [Mem. Am. Math. Soc. 420 (1990; Zbl 0688.60065)] and M. Fukushima [in: Ideas and methods in mathematical analysis, stochastics, and applications (Oslo, 1988), 151-161 (1992; Zbl 0764.60081)]. For each \(\delta> 0\), the author equips \(K\) with the quasi-metric \(d(x,y)=|x-y|^\delta\) for \(x,y\in K\), and studies various Poincaré-type inequalities associated with \(d\) and the Dirichlet form \(E\). The main result in the paper states that if \(\delta\) equals the intrinsic dimension of \(K\), then the following scaled Poincaré inequality holds: There exist \(q\geq 1\) and \(C> 0\) such that \[ \int_{B(x,r)}|u-\overline u_{B(x,r)}|^2\mu(dy)\leq C\Biggl({r\over\text{diam }K}\Biggr)^2 \int_{B(x,qr)}\gamma[u] (dy) \] for \(u\) in the domain of \(E\), \(x\in K\) and \(0< r<\text{diam }K\), where \(B(x,r)\) denotes the ball in \(K\) with respect to the quasi-metric \(d\), \(\gamma[u]\) denotes the so-called local energy measure associated with \(E(u,u)\), and \(\overline u_{B(x,r)}= \int_{B(x,r)}\overline u\mu(dy)\), where \(\overline u\) denotes the quasi-continuous representative of \(u\). Finally, the author considers some examples.

MSC:

28A80 Fractals
26D10 Inequalities involving derivatives and differential and integral operators
31C25 Dirichlet forms
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
49J45 Methods involving semicontinuity and convergence; relaxation

References:

[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 . Numdam | MR 1023950 | Zbl 0691.60070 · Zbl 0691.60070
[3] M.T. Barlow - E.A. Perkins , Brownian motion on the Sierpinski gasket , Prob. Theo. Rel. Fields 79 ( 1988 ), 543 - 624 . MR 966175 | Zbl 0635.60090 · Zbl 0635.60090 · doi:10.1007/BF00318785
[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 . MR 1133491 | Zbl 0760.49004 · Zbl 0760.49004
[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 . MR 1378473 | Zbl 0851.31008 · Zbl 0851.31008 · doi:10.1007/BF01759352
[6] M. Biroli - U. Mosco , Sobolev and isoperimetric inequalities for Dirichlet forms on homogeneous spaces , Rend. Mat. Acc. Lincei 9 , 6 ( 1995 ), 37 - 44 . MR 1340280 | Zbl 0837.31006 · Zbl 0837.31006
[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 . MR 1260455 | Zbl 0833.46020 · Zbl 0833.46020 · doi:10.1007/BF01053449
[8] A. Bunde - S. Havlin , Fractals and Disordered Systems , Springer-Verlag , Berlin - Heidelberg , 1991 . MR 1130617 | Zbl 0746.58009 · Zbl 0746.58009
[9] E.A. Carlen - S. Kusuoka - D.W. Stroock , Upper bounds for symmetric Markov transition functions , Ann. Inst. H. Poincaré 2 ( 1987 ), 245 - 287 . Numdam | MR 898496 | Zbl 0634.60066 · Zbl 0634.60066
[10] R.R. Coifman - G. Weiss , Analyse harmonique sur certaines éspaces homogenes , Lect. Notes in Math. 242 , Springer V. , Berlin - Heidelberg - New York , 1971 . MR 499948 | Zbl 0224.43006 · Zbl 0224.43006 · doi:10.1007/BFb0058946
[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 . MR 1190496 | Zbl 0764.60081 · Zbl 0764.60081
[12] M. Fukushima - Y. Oshima - M. Takeda , Dirichlet forms and Symmetric Markov Processes , Walter De Gruyter Co. , 1995 . MR 1303354 | Zbl 0838.31001 · Zbl 0838.31001
[13] M. Fukushima - T. Shima , On a spectral analysis for the Sierpinski gasket , Potential Anal. 1 ( 1992 ), 1 - 35 . MR 1245223 | Zbl 1081.31501 · Zbl 1081.31501 · doi:10.1007/BF00249784
[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 . MR 894545 | Zbl 0621.60073 · Zbl 0621.60073
[15] J.E. Hutchinson , Fractals and selfsimilarity , Indiana Univ. Math. J. 30 ( 1981 ), 713 - 747 . MR 625600 | Zbl 0598.28011 · Zbl 0598.28011 · doi:10.1512/iumj.1981.30.30055
[16] J. Kigami , A harmonic calculus on the Sierpinski spaces , Japan J. Appl. Math. 6 ( 1989 ), 259 - 290 . MR 1001286 | Zbl 0686.31003 · Zbl 0686.31003 · doi:10.1007/BF03167882
[17] S.M. Kozlov , Harmonization and homogenization on fractals , Commun. Math. Phys. 153 ( 1993 ), 159 - 339 . Article | MR 1218305 | Zbl 0767.58033 · Zbl 0767.58033 · doi:10.1007/BF02096647
[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 . MR 933827 | Zbl 0645.60081 · Zbl 0645.60081
[19] S. Kusuoka , Diffusion processes in nested fractals, in ”Statistical Mechanics and Fractals” , Lect. Notes in Math. 1567 , Springer V ., 1993 . · Zbl 0787.60119
[20] S. Kusuoka - X.Y. Zhou , Dirichlet forms on fractals: Poincaré constant and resistance , Prob. Th. Rel. Fields 93 ( 1992 ) 169 - 196 . MR 1176724 | Zbl 0767.60076 · Zbl 0767.60076 · doi:10.1007/BF01195228
[21] T. Lindstrøm , Brownian motion on nested fractals , Memoirs AMS , N. 420 , 83 ( 1990 ). MR 988082 | Zbl 0688.60065 · Zbl 0688.60065
[22] U. Mosco , Composite media and asymptotic Dirichlet forms , J. Funct. Anal. 123 , No. 2 ( 1994 ), 368 - 421 . MR 1283033 | Zbl 0808.46042 · Zbl 0808.46042 · doi:10.1006/jfan.1994.1093
[23] U. Mosco , Variational metrics on self-similar fractals , C. R. Acad. Sci. Paris , t. 321 , Série I ( 1995 ), 715 - 720 . MR 1354712 | Zbl 0898.28003 · Zbl 0898.28003
[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 . MR 1453892 | Zbl 0884.58001 · Zbl 0884.58001
[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 . MR 1492702 | Zbl 0898.58021 · Zbl 0898.58021
[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 . MR 1414498 | Zbl 0877.46028 · Zbl 0877.46028
[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 . MR 1150597 | Zbl 0769.58054 · Zbl 0769.58054 · doi:10.1155/S1073792892000047
[29] E.M. Stein , Harmonic analysis , Princeton Univ. Series , 1994 . · Zbl 0811.68038
[30] N. Th. Varopoulos , Sobolev inequalities on Lie groups and symmetric spaces , J. Funct. Anal. 86 ( 1989 ), 19 - 40 . MR 1013932 | Zbl 0697.22013 · Zbl 0697.22013 · doi:10.1016/0022-1236(89)90063-3
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