Discrete and Continuous Dynamical Systems - Series A (DCDS-A)

Vanishing viscosity for fractal sets

Pages: 1207 - 1235, Volume 28, Issue 3, November 2010      doi:10.3934/dcds.2010.28.1207

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Umberto Mosco - Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280, United States (email)
Maria Agostina Vivaldi - Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Universit√† degli Studi di Roma "La Sapienza‚ÄĚ, Via A. Scarpa 16, 00161 Roma, Italy (email)

Abstract: We imbed an array of thin highly conductive fibers in a surrounding two-dimensional medium with small viscosity. The resulting composite medium is described by a second order elliptic operator in divergence form with discontinuous singular coefficients on an open domain of the plane. We study the asymptotic spectral behavior of the operator when, simultaneously, the viscosity vanishes and the fibers develop fractal geometry. We prove that the spectral measure of the operator converges to the spectral measure of a self-adjoint operator associated with the lower-dimensional fractal limit of the thin fibers. The limit fiber is a compact set that disconnects the initial domain into infinitely many non-empty open components. Our approach is of variational nature and relies on Hilbert space convergence of quadratic energy forms.

Keywords:  Elliptic operators, weigths, singular homogenization, fractals, spectral asymptotics.
Mathematics Subject Classification:  35J20, 35J70; 35B20, 35B27, 74K05, 74K15.

Received: April 2010;      Available Online: April 2010.