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

@article{Goldshtein2016OnTF,
  title={On the First Eigenvalues of Free Vibrating Membranes in Conformal Regular Domains},
  author={Vladimir Gol'dshtein and A. Ukhlov},
  journal={Archive for Rational Mechanics and Analysis},
  year={2016},
  volume={221},
  pages={893-915}
}
AbstractIn 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… 

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