Conformal spectral stability estimates for the Neumann Laplacian

We study the eigenvalue problem for the Neumann-–Laplace operator in conformal regular planar domains $\Omega\subset\mathbb C$. Conformal regular domains support the Poincaré-–Sobolev inequality and this allows us to estimate the variation of the eigenvalues of the Neumann Laplacian upon domain perturbation via energy type integrals. Boundaries of such domains can have any Hausdorff dimension between one and two.