Range of the first two eigenvalues of the laplacian

@article{Wolf1994RangeOT,
  title={Range of the first two eigenvalues of the laplacian},
  author={S. A. Wolf and Joseph B. Keller},
  journal={Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences},
  year={1994},
  volume={447},
  pages={397 - 412}
}
For each planar domain D of unit area, the first two Dirichlet eigenvalues of —∆ on D determine a point (λ 1 ( D ), λ 2 ( D ) in the (λ 1 , λ 2 ) plane. As D varies over all such domains, this point varies over a set R which we determine. Its boundary consists of two semi-infinite straight lines and a curve connecting their endpoints. This curve is found numerially. We also show how to minimize the n th eigenvalue when the minimizing domain is diconnected. For n = 3 we show that the minimizing… CONTINUE READING

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