Payne-Philippin's overdetermined problems on compact surfaces

We investigate the overdetermined problem given by \begin{equation*} \Delta u=0 \text{ in } \Omega,\quad \frac{\partial u}{\partial\nu} =\sigma_1 u \text{ on } \partial \Omega, \quad |\nabla u|=\text{constant on } \partial \Omega, \end{equation*} where $\Omega$ is a connected compact Riemannian surface with smooth boundary $\partial \Omega$, and $\sigma_1$ is the first nonzero Steklov eigenvalue of $\Omega$. We prove that this overdetermined problem admits a nontrivial solution if and only if $\Omega$ is $\sigma$-homothetic to either the flat unit disk or a flat cylinder $[-T,T]\times S^1$ for some $T\ge T_1$. This gives a complete answer to the question raised by Payne and Philippin in [Z. Angew. Math. Phys. 42(6), 864-873, 1991] for $\sigma=\sigma_1$ and arbitrary surfaces. In particular, we completely characterize compact domains in 2-dimensional space forms for which the overdetermined problem is solvable.