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

TL;DR

This paper provides new estimates for the first eigenvalue of the Neumann-Laplace operator in conformal regular domains, connecting eigenvalues to hyperbolic metrics and introducing a novel estimation method based on weighted inequalities.

Contribution

It introduces a new estimation technique for eigenvalues in conformal regular domains and conjectures a link between eigenvalues and hyperbolic metrics.

Findings

Derived eigenvalue estimates for conformal regular domains.

Proposed a conjecture relating eigenvalues to hyperbolic metrics.

Developed a new method based on weighted Poincaré-Sobolev inequalities.

Abstract

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.