SYMMETRY AND NONSYMMETRY FOR THE OVERDETERMINED STEKLOFF EIGENVALUE PROBLEM

We consider the Stekloff eigenvalue problem (1.1)-(1.2); Payne and Philippin conjectured that if u is an eigenfunction which satisfies the overdetermined condition \($) over bar Vu\ = 1 on partial derivative Omega, then Omega should be a disk. In this paper we show that this conjecture holds if and only if the complex potential F associated to u vanishes only at one point. Then we show how to construct non-symmetric domains in the case where F vanishes at more than one point.