Schiffer’s Problem and an Isoperimetric Inequality for the First Buckling Eigenvalue of Domains on $$\mathbb{S}^{2}$$

Abstract

We consider the overdetermined eigenvalue problem on a sufficiently regular connected open domain Ω on the 2-sphere \(\mathbb{S}^2\): $$ \begin{array}{@{}l@{}} \Delta u+\alpha u = 0\quad {\rm in}\quad \Omega,\\ u = {\rm constant},\ \displaystyle\frac{\partial u}{\partial \nu} = {\rm constant\quad on}\quad \partial \Omega, \end{array}$$ where α ≠ 0. We show that if α = 2 and Ω is simply connected then the problem admits a (nonzero) solution if and only if Ω is a geodesic disk. We furthermore extend to domains on \(\mathbb{S}^{2}\) the isoperimetric inequality of Payne–Weinberger for the first buckling eigenvalue of compact planar domains. As a corollary we prove that Ω is a geodesic disk if the above overdetermined eigenvalue problem admits a (nonzero) solution with ∂u/∂ν = 0 on ∂Ω and α = λ2 the second eigenvalue of the Laplacian with Dirichlet boundary condition. This extends a result proved in the case of the Euclidean plane by C. Berenstein.