Math 2.0

Partial differential equations Spectral theory and eigenvalue problems Conjecture

Schiffer's conjecture

Builds on LaplacianDirichlet eigenvalue

Statement

Let n≥2n \ge 2 and let Ω⊆ℝn\Omega \subseteq \R^n be a bounded, simply connected domain with smooth boundary ∂Ω\partial\Omega and outer unit normal ν\nu. Suppose that for some λ>0\lambda > 0 and some constant cc there is a nonzero function uu with Δu+λu=0 in Ω,∂u∂ν=0 on ∂Ω,u=c on ∂Ω,\Delta u + \lambda u = 0 \text{ in } \Omega, \qquad \frac{\partial u}{\partial \nu} = 0 \text{ on } \partial\Omega, \qquad u = c \text{ on } \partial\Omega, where Δ\Delta is the Laplacian. The conjecture states that Ω\Omega is then a ball.

Remarks

The boundary value problem is overdetermined: uu is a Neumann eigenfunction that is also constant on the boundary. Every ball admits such a solution: a radial Neumann eigenfunction is constant on each sphere centred at the centre of the ball. By a result of Williams (1976), for these domains the conjecture is equivalent to the Pompeiu problem, that only balls fail the Pompeiu property. Consider the analogous problem for a Dirichlet eigenfunction with constant normal derivative. By Serrin's theorem on overdetermined problems, when the Dirichlet eigenfunction is positive, the domain must be a ball.

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Dirichlet eigenvalue, concept, no evidence levelDirichlet eigenvalueconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptSchiffer's conjecture, conjecture, no evidence levelSchiffer's conjectureconjecture
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
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  1. Dirichlet eigenvalue concept
  2. Euclidean space object
  3. Function concept
  4. Total derivative concept
  5. Function of several variables concept
  6. Vector-valued function concept
  7. Differentiable function of several variables concept
  8. Partial derivative concept
  9. Vector field concept
  10. Gradient concept
  11. Divergence concept
  12. Laplacian concept
  13. Schiffer's conjecture conjecture

Further reading

  1. Web https://en.wikipedia.org/wiki/Pompeiu_problem
About this page
TagJ05VKG4XQB
URIm2://J05VKG4XQB
Kindconjecture
WritingW0 stub
MSC35N25, 35P05, 35R30
StatusThis copy of the site has no build data. Run m2 status J05VKG4XQB.
AboutPompeiu problem
UsesLaplacian, Dirichlet eigenvalue
Changed2 November 2026

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