Math 2.0

Global analysis, analysis on manifolds Problem

Hearing the shape of a drum

Builds on LaplacianDirichlet boundary conditionEigenvalue

Problem

Let Ω1\Omega_1 and Ω2\Omega_2 be bounded domains in ℝ2\R^2. Let Δ\Delta be the Laplacian. Suppose that −Δ-\Delta with the Dirichlet boundary condition has the same eigenvalues on Ω1\Omega_1 and on Ω2\Omega_2, with the same multiplicities. Must Ω1\Omega_1 and Ω2\Omega_2 be congruent?

Remarks

Mark Kac popularised the question in his 1966 article "Can one hear the shape of a drum?". The answer is no. In 1964 John Milnor had already found two flat tori of dimension 1616 that have the same spectrum but are not isometric. In 1992 Carolyn Gordon, David Webb and Scott Wolpert constructed two plane domains, both polygons, that have the same Dirichlet spectrum but are not congruent. Some features can be heard: by Weyl's law the spectrum determines the area of the domain, and for domains with smooth boundary it also determines the perimeter. Among plane domains with smooth boundary, the disc is therefore determined by its spectrum, by the equality case of the isoperimetric inequality.

Reading path

To understand this page from the ground up, read these 14 pages in order. The foundations come first.

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Dirichlet boundary condition, concept, no evidence levelDirichlet boundary co…conceptEigenvalue, concept, no evidence levelEigenvalueconceptEuclidean 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 levelLaplacianconceptHearing the shape of a drum, problem, no evidence levelHearing the shape of …problem
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.
  • no evidence level
  1. Dirichlet boundary condition concept
  2. Eigenvalue concept
  3. Euclidean space object
  4. Function 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. Hearing the shape of a drum problem

Further reading

  1. Web https://en.wikipedia.org/wiki/Hearing_the_shape_of_a_drum
About this page
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MSC58J53, 35P05, 58J50
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AboutLaplacian
UsesLaplacian, Dirichlet boundary condition, Eigenvalue
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Changed2 November 2026

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