The Behaviour of the Eigenvalues for a Class of Operators Related to some Self-Affine Fractals in ${\mathbb{R}}^2$

Abstract

The obtaining of sharp estimates for the asymptotic behaviour of the eigenvalues of the (semi-elliptic) operator acting in the anisotropic Sobolev space

generated by the quadratic form ${\int }_{\Omega }f(\gamma )g(\gamma )d\mu (\gamma )$ is investigated. Here $\mu$ is an appropriate self-affine fractal measure on the unit disc $\Omega \subset {\mathbb{R}}^2$.