Where to place a spherical obstacle so as to maximize the first nonzero Steklov eigenvalue

We prove that among all doubly connected domains of ${\mathbb{R}}^n$ of the form $B_1\mathrm{\backslash }\overline{B_2}$, where $B_1$ and $B_2$ are open balls of fixed radii such that $\overline{B_2}\subset B_1$, the first nonzero Steklov eigenvalue achieves its maximal value uniquely when the balls are concentric. Furthermore, we show that the ideas of our proof also apply to a mixed boundary conditions eigenvalue problem found in literature.