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 Ω⊂R2. 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 Poincaré-Sobolev inequalities and as a consequence lower estimates of the first non-trivial eigenvalue of the Neumann-Laplace operator in planar quasiconformal regular domains