Overdetermined Steklov eigenvalue problems on compact surfaces

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 $Ω$ is a connected compact Riemannian surface with smooth boundary $\partial Ω$, and $σ_1$ is the first nonzero Steklov eigenvalue of $Ω$. We prove that this overdetermined problem admits a nontrivial solution if and only if $Ω$ is $σ$-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. \textbf{42}(6), 864–873, 1991] for $σ=σ_1$ and arbitrary surfaces. In particular, we completely characterize compact domains in 2-dimensional space forms for which the overdetermined problem is solvable.

💡 Research Summary

The paper studies an overdetermined boundary value problem for a compact, connected Riemannian surface Ω with smooth boundary. The problem consists of three conditions: (i) u is harmonic in Ω (Δu=0), (ii) u satisfies the first non‑trivial Steklov boundary condition ∂νu=σ₁u on ∂Ω, where σ₁ is the first positive Steklov eigenvalue, and (iii) the magnitude of the gradient is constant on the boundary, |∇u|=c. The authors ask for which domains Ω a non‑trivial solution exists and what the geometry of such domains must be.

A key notion introduced is σ‑homothety (or σ‑homothetic equivalence): two surfaces are σ‑homothetic if there exists a conformal diffeomorphism whose conformal factor is constant along the boundary. This relation preserves the Steklov spectrum up to a constant factor, so solvability of the overdetermined problem is invariant under σ‑homothety.

The analysis proceeds by selecting a first Steklov eigenfunction u and defining the auxiliary function f=log|∇u|. Lemma 3.2 shows that f satisfies Δf=K, where K is the Gaussian curvature of Ω; this identity is special to two dimensions. Proposition 3.3, derived via careful local coordinate calculations, proves that on the boundary the normal derivative of f obeys ∂νf=σ₁−κ, where κ denotes the geodesic curvature of ∂Ω. Consequently κ must be constant, forcing the boundary to be a curve of constant curvature.

For simply connected surfaces, the Gauss–Bonnet theorem yields ∫ΩK dA=2π, and the Weinsteck inequality together with the constancy of κ forces the boundary length to be fixed. The only σ‑homothetic model satisfying these constraints is the flat unit disk D in the Euclidean plane. Hence any simply connected solution domain is σ‑homothetic to D, and the solution space coincides with the first Steklov eigenspace spanned by the coordinate functions.

In the multiply connected case, the authors first exhibit annular examples A_T=