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

V. Gol’dshtein, A. Ukhlov

Research output: Contribution to journalArticlepeer-review

19 Scopus citations

Abstract

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 firstnon-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.

Original languageEnglish
Pages (from-to)893-915
Number of pages23
JournalArchive for Rational Mechanics and Analysis
Volume221
Issue number2
DOIs
StatePublished - 1 Aug 2016

Fingerprint

Dive into the research topics of 'On the First Eigenvalues of Free Vibrating Membranes in Conformal Regular Domains'. Together they form a unique fingerprint.

Cite this