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Overdetermined Steklov eigenvalue problems on compact surfaces

Hang Chen, Bohan Wu

TL;DR

This work provides a complete σ1-based classification of overdetermined Steklov problems on compact surfaces: a domain is solvable if and only if it is σ-homothetic to either the flat unit disk or a flat cylinder A_T with T≥T1, where T1 is the unique positive root of $1/x= anh x$. The authors develop a geometric-analytic framework using the auxiliary function $f= abla u|/|u|$, Gauss-Bonnet, and Weinstock inequalities in simply connected cases to force disk-like behavior, and they leverage separation of variables on annuli to classify multiply connected domains, extending the classification to space forms (Euclidean, spherical, hyperbolic). They also discuss weaker overdetermined conditions, higher eigenvalues, and a Serrin-type result for mean-zero eigenfunctions, linking boundary geometry, conformal structure, and Steklov spectrum through the notion of σ-homothety. Overall, the paper resolves, in the σ1 setting, Payne and Philippin’s question for arbitrary surfaces and provides a complete picture in 2D space forms. The results illuminate the geometric rigidity induced by overdetermined Steklov conditions and connect spectral, conformal, and topological data in a unified classification.

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 $Ω$ 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.

Overdetermined Steklov eigenvalue problems on compact surfaces

TL;DR

This work provides a complete σ1-based classification of overdetermined Steklov problems on compact surfaces: a domain is solvable if and only if it is σ-homothetic to either the flat unit disk or a flat cylinder A_T with T≥T1, where T1 is the unique positive root of . The authors develop a geometric-analytic framework using the auxiliary function , Gauss-Bonnet, and Weinstock inequalities in simply connected cases to force disk-like behavior, and they leverage separation of variables on annuli to classify multiply connected domains, extending the classification to space forms (Euclidean, spherical, hyperbolic). They also discuss weaker overdetermined conditions, higher eigenvalues, and a Serrin-type result for mean-zero eigenfunctions, linking boundary geometry, conformal structure, and Steklov spectrum through the notion of σ-homothety. Overall, the paper resolves, in the σ1 setting, Payne and Philippin’s question for arbitrary surfaces and provides a complete picture in 2D space forms. The results illuminate the geometric rigidity induced by overdetermined Steklov conditions and connect spectral, conformal, and topological data in a unified classification.

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 is a connected compact Riemannian surface with smooth boundary , and is the first nonzero Steklov eigenvalue of . We prove that this overdetermined problem admits a nontrivial solution if and only if is -homothetic to either the flat unit disk or a flat cylinder for some . 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 and arbitrary surfaces. In particular, we completely characterize compact domains in 2-dimensional space forms for which the overdetermined problem is solvable.

Paper Structure

This paper contains 12 sections, 25 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Normal exponential maps
  4. Basic properties of Steklov eigenfunctions
  5. The case of simply connected surfaces
  6. The case of multiply connected surfaces
  7. Riemannian surfaces
  8. Domains in space forms
  9. Further discussion
  10. A weaker overdetermined condition
  11. Solvability for higher eigenvalues
  12. Another overdetermined condition

Key Result

Theorem 1.2

Let $\Omega\subset \mathbb{R}^2$ be a simply connected domain with smooth boundary $\partial \Omega$. Then the overdetermined problem eq-s1, eq-s2 and eq-s3 with $\sigma=\sigma_1$ is solvable if and only if $\Omega$ is a round disk.

Theorems & Definitions (47)

  • Theorem 1.2: PP91*Theorem 1
  • Theorem 1.3: LS25*Theorem 2.3 and Corollary 2.4
  • Definition 1.4
  • Definition 1.5: cf. FS16*Definition 2.1
  • Theorem 1.6
  • Remark 1.7
  • Theorem 1.8
  • Theorem 1.9
  • Corollary 1.10
  • Lemma 2.1
  • ...and 37 more