  We study spectral properties of divergence form elliptic operators
$-\textrm{div} [A(z) \nabla f(z)]$ with the Neumann boundary condition in
planar domains (including some fractal type domains), that satisfy to the
quasihyperbolic boundary conditions. Our method is based on an interplay
between quasiconformal mappings, elliptic operators and composition operators
on Sobolev spaces.
