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

We prove that among all doubly connected domains of R n of the form B 1 \ B 2 , where B 1 and B 2 are open balls of fixed radii such that B 2 ⊂ B 1 , the first non-trivial 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.