Symmetry and non-symmetry for the overdetermined Stekloff eigenvalue problem

Abstract

We consider the Stekloff eigenvalue problem (1.1)–(1.2); Payne and Philippin conjectured that ifu is an eigenfunction which satisfies the overdetermined condition ▽u=1 on ∂Ω, then Ω should be a disk. In this paper we show that this conjecture holds if and only if the complex potentialF associated tou vanishes only at one point. Then we show how to construct non-symmetric domains in the case whereF vanishes at more than one point.