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

In 1961 G. Polya published a paper about the eigenvalues of vibrating membranes. The “free vibrating membrane” corresponds to the Neumann–Laplace operator in bounded plane domains. In this paper we obtain estimates for the first non-trivial eigenvalue of this operator in a large class of domains tha...

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Bibliographic Details
Published in:Archive for rational mechanics and analysis 2016-08, Vol.221 (2), p.893-915
Main Authors: Gol’dshtein, V., Ukhlov, A.
Format: Article
Language:English
Subjects:
Archives
Classical Mechanics
Complex Systems
Eigenvalues
Estimates
Fluid- and Aerodynamics
Inequalities
Mathematical and Computational Physics
Membranes
Operators
Physics
Physics and Astronomy
Planes
Theoretical
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Summary:In 1961 G. Polya published a paper about the eigenvalues of vibrating membranes. The “free vibrating membrane” corresponds to the Neumann–Laplace operator in bounded plane domains. In this paper we obtain estimates for the first non-trivial eigenvalue of this operator in a large class of domains that we call conformal regular domains. This class includes convex domains, John domains etc. On the basis 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 which is based on weighted Poincaré–Sobolev inequalities, obtained by the authors recently.
ISSN:0003-9527
1432-0673
DOI:10.1007/s00205-016-0988-9