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The Journal of Geometric Analysis

, Volume 29, Issue 2, pp 1811–1834 | Cite as

Steklov Eigenvalues of Submanifolds with Prescribed Boundary in Euclidean Space

  • Bruno Colbois
  • Alexandre GirouardEmail author
  • Katie Gittins
Article

Abstract

We obtain upper and lower bounds for Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. A very general upper bound is proved, which depends only on the geometry of the fixed boundary and on the measure of the interior. Sharp lower bounds are given for hypersurfaces of revolution with connected boundary: We prove that each eigenvalue is uniquely minimized by the ball. We also observe that each surface of revolution with connected boundary is Steklov isospectral to the disk.

Keywords

Steklov problem Euclidean space Prescribed boundary manifolds Hypersurfaces of revolution 

Mathematics Subject Classification

35P15 58C40 

Notes

Acknowledgements

We thank one of the anonymous referees for suggesting an elegant proof to Proposition 1.10. We thank another referee for carefully reading the paper and suggesting several improvements, in particular regarding Theorem 1.11. While a postdoctoral student at the Université de Neuchâtel, KG was supported by the Swiss National Science Foundation Grant No. 200021_163228 entitled Geometric Spectral Theory. KG also acknowledges support from the Max Planck Institute for Mathematics, Bonn. AG acknowledges support from the NSERC Discovery Grants Program.

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Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  • Bruno Colbois
    • 1
  • Alexandre Girouard
    • 2
    Email author
  • Katie Gittins
    • 3
  1. 1.Institut de MathématiquesUniversité de NeuchâtelNeuchâtelSwitzerland
  2. 2.Département de mathématiques et de statistique, Pavillon Alexandre-VachonUniversité LavalQuébecCanada
  3. 3.Max Planck Institute for MathematicsBonnGermany

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