Symmetry and non-symmetry for the overdetermined Stekloff eigenvalue problem (Q1315363)

This paper deals with the Stekloff eigenvalue problem \(\Delta u= 0\) in \(D\), \(\partial u/\partial n= pu\) on \(\partial D\), where \(n\) denotes the exterior normal. There exist countably many eigenvalues \(0= p_ 1< p_ 2\leq p_ 3\leq\dots\). Payne and Philippin have shown that in a simply connected plane domain, if the second eigenfunction satisfies \(|\nabla u|= 1\) on \(\partial D\), then \(D\) must be a disk. It is shown that this is not anymore the case if \(u\) is any other eigenfunction. The proof is based on classical methods of conformal mappings.