An isoperimetric inequality for eigenvalues of the Stekloff problem

Let Ohm be a bounded smooth domain in R-n and let 0 = lambda (1) less than or equal to lambda (2) less than or equal to ... denote the eigenvalues of the Stekloff problem: Deltau = 0 in Ohm and (partial derivativeu)/(partial derivativev) = lambdau on partial derivative Ohm. We show that Sigma (n+1)( i=2) lambda (-1)(i) greater than or equal to n/lambda (2)*, where lambda (2 )* denotes the second eigenvalue of the Stekloff problem in a ball having t he same measure as Ohm. The proof is based on a weighted isoperimetric ineq uality. MSC (2000): 35P15, 26D15.