Quasiconformal Mappings and Neumann Eigenvalues of Divergent Elliptic Operators

We study spectral properties of divergence form elliptic operators $-\text{div}[A(z)\mathrm{\nabla }f(z)]$ 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.