Some results on higher eigenvalue optimization

@article{Fraser2020SomeRO,
  title={Some results on higher eigenvalue optimization},
  author={Ailana M. Fraser and Richard M. Schoen},
  journal={Calculus of Variations and Partial Differential Equations},
  year={2020}
}
  • A. Fraser, R. Schoen
  • Published 8 October 2019
  • Mathematics
  • Calculus of Variations and Partial Differential Equations
In this paper we obtain several results concerning the optimization of higher Steklov eigenvalues both in two and higher dimensional cases. We first show that the normalized (by boundary length) $k$-th Steklov eigenvalue on the disk is not maximized for a smooth metric on the disk for $k\geq 3$. For $k=1$ the classical result of [W] shows that $\sigma_1$ is maximized by the standard metric on the round disk. For $k=2$ it was shown [GP1] that $\sigma_2$ is not maximized for a smooth metric. We… 
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