Fractal Reinforcement of Elastic Membranes

We describe a homogenization model of an elastic membrane reinforced by the inclusion of a fractal string. We follow a variational approach consisting in proving the convergence of certain energy functionals. This leads to the spectral convergence of a sequence of weighted second-order elliptic partial differential operators to a singular elliptic operator with a fractal term.

[1]  M. Fukushima,et al.  Dirichlet forms and symmetric Markov processes , 1994 .

[2]  Jun Kigami,et al.  Harmonic calculus on p.c.f. self-similar sets , 1993 .

[3]  L. Rogers.  Stochastic differential equations and diffusion processes: Nobuyuki Ikeda and Shinzo Watanabe North-Holland, Amsterdam, 1981, xiv + 464 pages, Dfl.175.00 , 1982 .

[4]  Tadashi Shima,et al.  On a spectral analysis for the Sierpinski gasket , 1992 .

[5]  Y. Oshima,et al.  Dirichlet forms, diffusion processes and spectral dimensions for nested fractals , 2010 .

[6]  Martin T. Barlow,et al.  Brownian motion on the Sierpinski gasket , 1988 .

[7]  Jaak Peetre,et al.  Function spaces on subsets of Rn , 1984 .

[8]  S. Goldstein.  Random Walks and Diffusions on Fractals , 1987 .

[9]  T. Lindstrøm.  Brownian Motion On Nested Fractals , 1990 .

[10]  Alf Jonsson,et al.  Dirichlet Forms and Brownian Motion Penetrating Fractals , 2000 .

[11]  Umberto Mosco,et al.  An Example of Fractal Singular Homogenization , 2007 .

[12]  Hans Triebel,et al.  Fractals and Spectra: Related to Fourier Analysis and Function Spaces , 1997 .

[13]  Maria Rosaria Lancia,et al.  A Transmission Problem with a Fractal Interface , 2002 .

[14]  Tom Lindstrøm.  BROWNIAN MOTION PENETRATING THE SIERPINSKI GASKET , 1990 .

[15]  U. Mosco.  Convergence of convex sets and of solutions of variational inequalities , 1969 .

[16]  D. Gilbarg,et al.  Elliptic Partial Differential Equa-tions of Second Order , 1977 .

[17]  W. Hackbusch.  Elliptic Differential Equations , 1992 .

[18]  H. Attouch.  Variational convergence for functions and operators , 1984 .

[19]  Takashi Kumagai,et al.  Brownian Motion Penetrating Fractals: An Application of the Trace Theorem of Besov Spaces , 2000 .

[20]  A. Jonsson.  Brownian motion on fractals and function spaces , 1996 .

[21]  Tosio Kato.  Perturbation theory for linear operators , 1966 .

[22]  M. Fukushima.  Functional Analysis in Markov Processes , 1982 .

[23]  Dirichlet forms associated with direct product diffusion processes , 1982 .

[24]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[25]  H. Triebel.  Fractals and spectra , 1997 .

[26]  Gunter H. Meyer,et al.  On Diffusion in a Fractured Medium , 1971 .

[27]  Harry Kesten,et al.  Percolation Theory and Ergodic Theory of Infinite Particle Systems , 1987 .

[28]  Alf Jonsson,et al.  A trace theorem for the Dirichlet form on the Sierpinski gasket , 2005 .

[29]  Umberto Mosco,et al.  Composite media and asymptotic dirichlet forms , 1994 .

[30]  S. Kusuoka,et al.  Lecture on diffusion processes on nested fractals , 1993 .

[31]  L. C. G. Rogers,et al.  STOCHASTIC DIFFERENTIAL EQUATIONS AND DIFFUSION PROCESSES (North‐Holland Mathematical Library, 24) , 1982 .

[32]  Harmonization and homogenization on fractals , 1993 .

[33]  E. Sanchez-Palencia,et al.  Phénomènes de transmission à travers des couches minces de conductivitéélevée , 1974 .

[34]  Carl M. O'Brien,et al.  Probabilistic Methods in Mathematical Physics , 1989 .