Remarks on a Stekloff Eigenvalue Problem

Abstract

Consider the Stekloff eigenvalue problem \[\begin{gathered} \Delta ^2 u = 0\quad {\text{in }}R, \hfill \\ u = \Delta u - q\tfrac{{\partial u}}{{\partial n}} = 0\quad {\text{on }}\partial R \hfill. \\ \end{gathered} \] It will be shown that the first eigenvalue is simple, and its eigenfunction does not change sign. Bounds for the first eigenvalue are discussed, particularly for the square, and a conjecture is disproved.