Abstract
We consider the overdetermined eigenvalue problem on a sufficiently regular connected open domain Ω on the 2-sphere \(\mathbb{S}^2\):
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.
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Souam, R. Schiffer’s Problem and an Isoperimetric Inequality for the First Buckling Eigenvalue of Domains on \(\mathbb{S}^{2}\). Ann Glob Anal Geom 27, 341–354 (2005). https://doi.org/10.1007/s10455-005-5219-9
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DOI: https://doi.org/10.1007/s10455-005-5219-9