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An isoperimetric inequality for the harmonic mean of the Steklov eigenvalues in hyperbolic space

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Abstract

In this article, we prove an isoperimetric inequality for the harmonic mean of the first \((n-1)\) nonzero Steklov eigenvalues on bounded domains in n-dimensional hyperbolic space. Our approach to prove this result also gives a similar inequality for the first n nonzero Steklov eigenvalues on bounded domains in n-dimensional Euclidean space.

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References

  1. Binoy, Santhanam, G.: Sharp upperbound and a comparison theorem for the first nonzero Steklov eigenvalue. J. Ramanujan Math. Soc. 29(2), 133–154 (2014)

  2. Brock, F.: An isoperimetric inequality for eigenvalues of the Stekloff problem. Z. Angew. Math. Mech. 81(1), 69–71 (2001)

  3. Colbois, B., Girouard, A., Gittins, K.: Steklov eigenvalues of submanifolds with prescribed boundary in Euclidean space. J. Geom. Anal. 29(2), 1811–1834 (2019)

    Article  MathSciNet  Google Scholar 

  4. Girouard, A., Laugesen, R.S., Siudeja, B.A.: Steklov eigenvalues and quasiconformal maps of simply connected planar domains. Arch. Ration. Mech. Anal. 219(2), 903–936 (2016)

    Article  MathSciNet  Google Scholar 

  5. Girouard, A., Polterovich, I.: Spectral geometry of the Steklov problem (survey article). J. Spect. Theory 7(2), 321–359 (2017)

    Article  MathSciNet  Google Scholar 

  6. Hersch, J., Payne, L.E.: Extremal principles and isoperimetric inequalities for some mixed problems of Stekloff’s type. Z. Angew. Math. Phys. 19(5), 802–817 (1968)

  7. Provenzano, L., Stubbe, J.: Weyl-type bounds for Steklov eigenvalues. J. Spect. Theory 9(1), 349–377 (2019)

    Article  MathSciNet  Google Scholar 

  8. Santhanam, G.: A sharp upper bound for the first eigenvalue of the Laplacian of compact hypersurfaces in rank-\(1\) symmetric spaces. Proc. Math. Sci. 117(3), 307–315 (2007)

    Article  MathSciNet  Google Scholar 

  9. Verma, S.: Bounds for the Steklov eigenvalues. Arch. Math. (Basel) 111(6), 657–668 (2018)

  10. Verma, S., Santhanam, G.: Sharp bounds for Steklov eigenvalues on star-shaped domain. Adv. Pure Appl. Math. 11(2), 47–56 (2020)

    Article  MathSciNet  Google Scholar 

  11. Stekloff, M.W.: Les problèmes fondamentaux de la physique mathématique. Ann. Sci. Éc. Norm. Supér. 19, 445–490 (1902)

  12. Weinstock, R.: Inequalities for a classical eigenvalue problem. J. Ration. Mech. Anal. 3, 745–753 (1954)

    MathSciNet  MATH  Google Scholar 

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  1. Tata Institute of Fundamental Research Centre for Applicable Mathematics, Bangalore, India

    Sheela Verma

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  1. Sheela Verma
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Correspondence to Sheela Verma.

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Verma, S. An isoperimetric inequality for the harmonic mean of the Steklov eigenvalues in hyperbolic space. Arch. Math. 116, 193–201 (2021). https://doi.org/10.1007/s00013-020-01549-x

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  • DOI: https://doi.org/10.1007/s00013-020-01549-x

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