Some sharp bounds for Steklov eigenvalues

This work is an extension of a result given by Kuttler and Sigillito (SIAM Rev \(10\):\(368-370\), \(1968\)) on a star-shaped bounded domain in \(\mathbb{R}^2\). Let \(\Omega\) be a star-shaped bounded domain in a hypersurface of revolution, having smooth boundary. In this article, we obtain a sharp lower bound for all Steklov eigenvalues on \(\Omega\) in terms of the Steklov eigenvalues of the largest geodesic ball contained in \(\Omega\) with the same center as \(\Omega\). We also obtain similar bounds for all Steklov eigenvalues on star-shaped bounded domain in paraboloid, \(P = \left\lbrace (x, y, z) \in \mathbb{R}^{3} : z = x^2 + y^2\right\rbrace\).