Eigenvalue distribution of some fractal semi-elliptic differential operators: Combinatorial approach

We obtain the sharp order of growth of the eigenvalue distribution function for the operator in the anisotropic Sobolev space\(H^{t_1 ,t_2 } (Q)\), generated by the quadratic form ∫ Q ∣u∣2dμ, whereQ⊂ℝ2 is the unit square and μ is a probability self-affine fractal measure onQ. The geometry of Supp μ should be in a certain way consistent with the parameterst 1 ,t 2 .