  We study spectral estimates of the divergence form uniform elliptic operators
$-\textrm{div}[A(z) \nabla f(z)]$ with the Dirichlet boundary condition in
bounded non-Lipschitz simply connected domains $\Omega \subset \mathbb C$. The
suggested method is based on the quasiconformal composition operators on
Sobolev spaces with applications to the weighted Poincar\'e-Sobolev
inequalities.
