Spectral Asymptotics for Variational Fractals

Abstract

In this paper a generalization of a classic result of H. Weyl concerning the asymptotics of the spectrum of the Laplace operator is proved for variational fractals. Physically we are studying the density of states for the diffusion trough a fractal media. A variational fractal is a couple $(K,\mathcal{E})$ where $K$ is a self-similar fractal and $\mathcal{E}$ is an energy form with some similarity properties connected with those of $K$. In this class we can find some of the most widely studied families of fractals as nested fractals, p.c.f. fractals, the Sierpinski carpet etc., as well as some regular self-similar Euclidean domains. We will see that if $r(x)$ is the number of eigenvalues associated with $\mathcal{E}$ smaller than $x$, then $r(x)x^{\nu /2}$, where $\nu$ is the intrinsic dimension of $(K,\mathcal{E})$.