Payne-Philippin's overdetermined problems on compact surfaces

Abstract

We investigate the overdetermined problem given by \begin{equation*} Δu=0 \text{ in } Ω,\quad \frac{\partial u}{\partialν} =σ_1 u \text{ on } \partial Ω, \quad |\nabla u|=\text{constant on } \partial Ω, \end{equation*} where $\hat{I}\text{c◯}$ is a connected compact Riemannian surface with smooth boundary $\partial \hat{I}\text{c◯}$, and $\ddot{I}{ƒ}_1$ is the first nonzero Steklov eigenvalue of $\hat{I}\text{c◯}$. We prove that this overdetermined problem admits a nontrivial solution if and only if $\hat{I}\text{c◯}$ is $\ddot{I}ƒ$-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 $\ddot{I}ƒ=\ddot{I}{ƒ}_1$ and arbitrary surfaces. In particular, we completely characterize compact domains in 2-dimensional space forms for which the overdetermined problem is solvable.

16 pages