Spectral Stability Estimates of Dirichlet Divergence Form Elliptic Operators

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