  In this paper we study spectral properties of the Neumann-Laplace operator in
planar quasiconformal regular domains $\Omega\subset\mathbb R^2$. This study is
based on the quasiconformal theory of composition operators on Sobolev spaces.
Using the composition operators theory we obtain estimates of constants in
Poincar\'e-Sobolev inequalities and as a consequence lower estimates of the
first non-trivial eigenvalue of the Neumann-Laplace operator in planar
quasiconformal regular domains.
