On the First Eigenvalues of Free Vibrating Membrane in Conformal Regular Domains

In 1961 G.Polya published a paper about the eigenvalues of vibrating membrane. The "free vibrating membrane"' corresponds to the Neumann-Laplace operator in bounded plane domains. In this paper we obtain estimates for the first eigenvalue of this operator in a large class of domains that we call as conformal regular domains, that includes convex domains, John domains etc... On the base of our estimates we conjecture that the eigenvalues of the Neumann-Laplace operator depend on the hyperbolic metrics of plane domains. We propose a new method for the estimates that is based on weighted Poincar\'e-Sobolev inequalities obtained by the authors recently.