An Isoperimetric Inequality for Eigenvalues of the Stekloff Problem

Let Ω be a bounded smooth domain in ℝn and let 0 = λ1 ≤ λ2 ≤ … denote the eigenvalues of the Stekloff problem: Δu = 0 in Ω and (∥u)/(∥ν) = λi on ∥Ω. We show that , where denotes the second eigenvalue of the Stekloff problem in a ball having the same measure as Ω. The proof is based on a weighted isoperimetric inequality.