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July  2010, 28(3): 1207-1235. doi: 10.3934/dcds.2010.28.1207

Vanishing viscosity for fractal sets

Umberto Mosco 1, and  Maria Agostina Vivaldi 2,

1. 

Worcester Polytechnic Institute, 100 Institute Road, Worcester, MA 01609-2280, United States
2. 

Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Università degli Studi di Roma "La Sapienza”, Via A. Scarpa 16, 00161 Roma, Italy

Received  April 2010 Published  April 2010

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: singular homogenization, fractals, spectral asymptotics., Elliptic operators, weigths.
Mathematics Subject Classification: 35J20, 35J70; 35B20, 35B27, 74K05, 74K1.
Citation: Umberto Mosco, Maria Agostina Vivaldi. Vanishing viscosity for fractal sets. Discrete & Continuous Dynamical Systems, 2010, 28 (3) : 1207-1235. doi: 10.3934/dcds.2010.28.1207
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