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

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 Poincaré-Sobolev inequalities obtained by the authors recently.

21 pages

References in corpus (4)

• Brennan's conjecture for composition operators on Sobolev spaces
• Conformal spectral stability for the Dirichlet-Laplace operator
• V.A. Steklov and the Problem of Sharp (Exact) Constants in Inequalities of Mathematical Physics
• Poincaré inequalities in quasihyperbolic boundary condition domains

Cited by in corpus (6)

• Spectral Estimates of the $p$-Laplace Neumann operator and Brennan's Conjecture
• Spectral Properties of the Neumann-Laplace operator in Quasiconformal Regular Domains
• Space Quasiconformal Mappings and Neumann Eigenvalues in Fractal Domains
• On the hyperbolic distance of $n$-times punctured spheres
• On growth monotonicity estimates of the principal Dirichlet-Laplacian eigenvalue
• Sharp Poincaré inequalities in a class of non-convex sets