Payne-Philippin's overdetermined problems on compact surfaces

TL;DR

This paper characterizes all compact surfaces where a specific overdetermined harmonic problem with boundary conditions involving the first Steklov eigenvalue admits solutions, identifying only flat disks and cylinders as solutions.

Contribution

It provides a complete classification of compact surfaces supporting solutions to the overdetermined problem, answering a longstanding question by Payne and Philippin.

Findings

Solutions exist only for flat disks and cylinders.

Characterizes domains in space forms where the problem is solvable.

Completes the classification for 2D compact surfaces.

Abstract

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.