The overdetermined problem for harmonic u with Steklov boundary condition and constant |∇u| on the boundary admits nontrivial solutions if and only if the domain is σ-homothetic to the flat unit disk or a flat cylinder.
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Payne-Philippin's overdetermined problems on compact surfaces
The overdetermined problem for harmonic u with Steklov boundary condition and constant |∇u| on the boundary admits nontrivial solutions if and only if the domain is σ-homothetic to the flat unit disk or a flat cylinder.