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

We prove that the first non-trivial Steklov eigenvalue of a spherical shell is maximal 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.; We prove that the first non-trivial Steklov eigenvalue of a spherical shell is maximal when the balls are concentric. We also show that the ideas used may apply to a mixed boundary conditions eigenvalue problem found in literature.