  We study the eigenvalue problem for the Neumann-Laplace operator in conformal
regular planar domains $\Omega\subset\mathbb{C}$. Conformal regular domains
support the Poincar\'e 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.
