Payne-Philippin's overdetermined problems on compact surfaces

2025/12/07 by Chen, Hang, Wu, Bohan
Mathematics · Computer Science · #Nonlinear Partial Differential Equations #Advanced Mathematical Modeling in Engineering #Numerical methods in inverse problems

paper · doi:10.48550/arxiv.2512.06740

Abstract

We investigate the overdetermined problem given by Δu=0 in Ω, (∂ u)/(∂ν) =σ₁ u on ∂ Ω, |∇ u|=constant on ∂ Ω, where Ω is a connected compact Riemannian surface with smooth boundary ∂ Ω, and σ₁ 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]× S¹ for some T≥ T₁. This gives a complete answer to the question raised by Payne and Philippin in [Z. Angew. Math. Phys. 42(6), 864-873, 1991] for σ=σ₁ and arbitrary surfaces. In particular, we completely characterize compact domains in 2-dimensional space forms for which the overdetermined problem is solvable.

Citations

• Weinstock inequality in hyperbolic space
• Some recent developments on the Steklov eigenvalue problem
• On the spectra of three Steklov eigenvalue problems on warped product manifolds
• A partially overdetermined problem in a half ball
• Comparison of Steklov eigenvalues on a domain and Laplacian eigenvalues on its boundary in Riemannian manifolds
• Spectral geometry of the Steklov problem
• Sharp upper bound and a comparison theorem for the first nonzero Steklov eigenvalue
• The first Steklov eigenvalue, conformal geometry, and minimal surfaces
• Serrin’s result for hyperbolic space and sphere
• A symmetry problem in potential theory

Related

• Sharp lower bounds for the first eigenvalue of Steklov-type eigenvalue problems on a compact surface
• Some recent developments on the Steklov eigenvalue problem
• Bounds for the first non-zero Steklov eigenvalue
• The first Steklov eigenvalue, conformal geometry, and minimal surfaces
• Isoperimetric inequalities involving Steklov eigenvalues on surfaces