Spectral Properties of the Neumann-Laplace operator in Quasiconformal Regular Domains

In this paper we study spectral properties of the Neumann-Laplace operator in planar quasiconformal regular domains $\mathrm{\Omega }\subset {\mathbb{R}}^2$. 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.