Spectral Estimates of the Dirichlet-Laplace Operator in Conformal Regular Domains

We consider conformal spectral estimates of the Dirichlet–Laplace operator in conformal regular domains Ω ⊂ ℝ², based on the geometric theory of composition operators on Sobolev spaces, which permits us to estimate constants in the Poincaré-Sobolev inequalities. We obtain lower estimates for the first eigenvalue of the Dirichlet–Laplace operator in the class of conformal regular domains and conformal estimates for the ground state energy of quantum billiards.