An annular sector-like domain admitting an overdetermined Helmholtz solution in a space form must be a spherical sector and the solution is radially symmetric.
Federer.Geometric Measure Theory, volume 153 ofDie Grundlehren der mathematischen Wissenschaften
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Overdetermined elliptic problems on model Riemannian manifolds
An annular sector-like domain admitting an overdetermined Helmholtz solution in a space form must be a spherical sector and the solution is radially symmetric.