Vanishing viscosity for fractal sets
Umberto Mosco, Maria Agostina Vivaldi · Discrete and Continuous Dynamical Systems · 2010
We imbed an array of thin highly conductivefibers in a surrounding two-dimensional medium with smallviscosity. The resulting composite medium is described by a secondorder elliptic operator in divergence form with discontinuoussingular coefficients on an open domain of the plane. We study theasymptotic spectral behavior of the operator when, simultaneously,the viscosity vanishes and the fibers develop fractal geometry. Weprove that the spectral measure of the operator converges to thespectral measure of a self-adjoint operator associated with thelower-dimensional fractal limit of the thin fibers. The limitfiber is a compact set that disconnects the initial domain intoinfinitely many non-empty open components. Our approach is ofvariational nature and relies on Hilbert space convergence ofquadratic energy forms.