  In this paper we obtain lower estimates of the first non-trivial eigenvalues
of the degenerate $p$-Laplace operator, $p>2$, in a large class of non-convex
domains. This study is based on applications of the geometric theory of
composition operators on Sobolev spaces that permits us to estimates constants
of Poincar\'e-Sobolev inequalities and as an application to derive lower
estimates of the first non-trivial eigenvalues for the Alhfors domains (i.e. to
quasidiscs). This class of domains includes some snowflakes type domains with
fractal boundaries.
