Sharp bounds and geometric properties of the first non trivial Steklov Neumann Eigenvalue

In this article, we study the mixed Steklov–Neumann eigenvalue problem on doubly connected domains. First, we show that among all doubly connected domains in ${\mathbb{R}}^n$ of the form $B_{R_2}\setminus \overline{B_{R_1}}$, where $B_{R_1}$ and $B_{R_2}$ are open balls of fixed radii satisfying $\overline{B_{R_1}}\subset B_{R_2}$, the first non-zero Steklov–Neumann eigenvalue attains its maximal value when the balls are concentric. Next, we establish bounds for the first non-zero Steklov–Neumann eigenvalue on a doubly connected star-shaped domain contained in a hypersurface equipped with a revolution-type metric. We also derive the asymptotic behavior of the first non-zero Steklov–Neumann eigenvalue on a bounded domain with a spherical hole in ${\mathbb{R}}^n$ as the radius of the hole approaches zero. Finally, we study the number of nodal domains of the eigenfunction corresponding to the first non zero Steklov–Neumann eigenvalue on a bounded domain in ${\mathbb{R}}^n$ having a spherical hole.