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.
VANISHING VISCOSITY FOR FRACTAL SETS / Umberto, Mosco; Vivaldi, Maria Agostina. - In: DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS. - ISSN 1078-0947. - STAMPA. - 28:3(2010), pp. 1207-1235. [10.3934/dcds.2010.28.1207]
VANISHING VISCOSITY FOR FRACTAL SETS
VIVALDI, Maria Agostina
2010
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.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.