Abstract
Let \(\Omega \) be a star-shaped bounded domain in \((\mathbb {S}^{n}, ds^{2})\) with smooth boundary. In this article, we give a sharp lower bound for the first non-zero eigenvalue of the Steklov eigenvalue problem in \(\Omega .\) This result extends a result given by Kuttler and Sigillito (SIAM Rev 10:368–370, 1968) for a star-shaped bounded domain in \(\mathbb {R}^2\). Further we also obtain a two sided bound for the eigenvalues of the Steklov problem on a ball in \(\mathbb {R}^n\) with rotationally invariant metric and with bounded radial curvature.
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Acknowledgements
I would like to thank Prof. G. Santhanam for the discussions and valuable comments that were helpful in carrying out this work. The author wishes to thank the referee for his important suggestions which led to the improvement of results in the original manuscript.
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The author is supported by University Grants Commission, India.
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Verma, S. Bounds for the Steklov eigenvalues. Arch. Math. 111, 657–668 (2018). https://doi.org/10.1007/s00013-018-1238-1
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DOI: https://doi.org/10.1007/s00013-018-1238-1