Math 2.0

Partial differential equations Spectral theory and eigenvalue problems Problem

Steklov eigenvalue problem

Builds on LaplacianSobolev spaceDirichlet-to-Neumann operator

Problem

Let Ω⊆ℝn\Omega \subseteq \R^n be a bounded domain with Lipschitz boundary ∂Ω\partial\Omega and outer unit normal ν\nu. Find all σ∈ℝ\sigma \in \R for which some nonzero u∈H1(Ω)u \in H^1(\Omega) solves Δu=0 in Ω,∂u∂ν=σu on ∂Ω,\Delta u = 0 \text{ in } \Omega, \qquad \frac{\partial u}{\partial \nu} = \sigma u \text{ on } \partial\Omega, in the weak sense. Here Δ\Delta is the Laplacian and H1(Ω)H^1(\Omega) is a Sobolev space. Such numbers σ\sigma are the Steklov eigenvalues of Ω\Omega, and one asks how they depend on the geometry of Ω\Omega.

Remarks

The Steklov eigenvalues are the eigenvalues of the Dirichlet-to-Neumann operator of Ω\Omega, which maps boundary values ff to the normal derivative of the harmonic extension of ff. They form a sequence 0=σ0<σ1≤σ2≤⋯→∞0 = \sigma_0 < \sigma_1 \le \sigma_2 \le \dots \to \infty, and σ0=0\sigma_0 = 0 belongs to the constant functions. Example: for the unit disc, the functions rkcos⁡(kθ)r^k \cos(k\theta) and rksin⁡(kθ)r^k \sin(k\theta) are harmonic with ∂ru=ku\partial_r u = k u on the circle, so the eigenvalues are 0,1,1,2,2,3,3,…0, 1, 1, 2, 2, 3, 3, \dots. Weinstock proved that among simply connected planar domains of given perimeter, the disc maximises σ1\sigma_1.

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  1. Dirichlet-to-Neumann operator concept
  2. Euclidean space object
  3. Function concept
  4. Sobolev space concept
  5. Total derivative concept
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Differentiable function of several variables concept
  9. Partial derivative concept
  10. Vector field concept
  11. Gradient concept
  12. Divergence concept
  13. Laplacian concept
  14. Steklov eigenvalue problem problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Poincar%C3%A9%E2%80%93Steklov_operator
About this page
TagT6WEFSKYHZ
URIm2://T6WEFSKYHZ
Kindproblem
WritingW0 stub
MSC35P15, 35P05, 35J25
StatusThis copy of the site has no build data. Run m2 status T6WEFSKYHZ.
AboutDirichlet-to-Neumann operator
UsesLaplacian, Sobolev space, Dirichlet-to-Neumann operator
Changed2 November 2026

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