  We study spectral stability estimates of elliptic operators in divergence
form $-\textrm{div} [A(w) \nabla g(w)]$ with the Dirichlet boundary condition
in non-Lipschitz domains $\widetilde{\Omega} \subset \mathbb C$. The suggested
method is based on connections of planar quasiconformal mappings with Sobolev
spaces and its applications to the Poincar\'e inequalities.
