In this article, we study Steklov eigenvalues and mixed Steklov Neumann eigenvalues on a bounded domain in \(\mathbb {R}^{n}\) , \(n \ge 2\) , with Lipschitz boundary, having a spherical hole. We focus on two main results related to Steklov eigenvalues. First, we obtain an explicit expression for the second nonzero Steklov eigenvalue on a concentric annular domain. Secondly, we derive a sharp upper bound of the first n nonzero Steklov eigenvalues on a domain \(\Omega \subset \mathbb {R}^{n}\) having symmetry of order 4 and a ball removed from its center. This bound is given in terms of the corresponding Steklov eigenvalues on a concentric annular domain of the volume same as \(\Omega \) . Next, we consider the mixed Steklov Neumann eigenvalue problem on doubly connected domains in \(\mathbb {R}^{n}\) having symmetry of order 4 and obtain an upper bound for the first n nonzero eigenvalues. We also provide some examples to illustrate that the symmetry assumption in our results is crucial. Finally, based on numerical evidence obtained by using FreeFEM++, we state some open problems about these eigenvalues.
