  In this paper we consider conformal spectral estimates of the
Dirichlet-Laplace operator in conformal regular domains $\Omega \subset \mathbb
R^2$. This study is based on the geometric theory of composition operators on
Sobolev spaces that permits us to estimate constants of the Poincar\'e-Sobolev
inequalities. On this base we obtain lower estimates of the first eigenvalue of
the Dirichlet-Laplace operator in a class of conformal regular domains. As a
consequence we obtain conformal estimates of the ground state energy of quantum
billiards.
