Lesson plan
Dirichlet eigenvalue
In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of the drum can one deduce. Here a "drum" is thought of as an elastic membrane Ω, which is represented as a planar domain whose boundary is fixed. The Dirichlet eigenvalues are found by solving the following problem for an unknown function u ≠ 0 and eigenvalue λ Here Δ is the Laplacian, which is given in xy-coordinates by : \Delta u = \frac{\partial^2u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}. The boundary value problem ( ) is the Dirichlet problem for the Helmholtz equation, and so λ is known as a Dirichlet eigenvalue for Ω. Dirichlet eigenvalues are contrasted with Neumann eigenvalues: eigenvalues for the corresponding Neumann problem. The Laplace operator Δ appearing in ( ) is often known as the Dirichlet Laplacian when it is considered as accepting only functions u satisfying the Dirichlet boundary condition. More generally, in spectral geometry one considers ( ) on a manifold with boundary Ω. Then Δ is taken to be the Laplace–Beltrami operator, also with Dirichlet boundary conditions.…
Article summary
In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape. The problem of whether one can hear the shape of a drum is: given the Dirichlet eigenvalues, what features of the shape of the drum can one deduce. Here a "drum" is thought of as an elastic membrane Ω, which is represented as a planar domain whose boundary is fixed. The Dirichlet eigenvalues are found by solving the following problem for an unknown function u ≠ 0 and eigenvalue λ Here Δ is the Laplacian, which is given in xy-coordinates by : \Delta u = \frac{\partial^2u}{\partial x^2} + \frac{\partial^2 u}{\partial y^2}. The boundary value problem ( ) is the Dirichlet problem for the Helmholtz equation, and so λ is known as a Dirichlet eigenvalue for Ω. Dirichlet eigenvalues are contrasted with Neumann eigenvalues: eigenvalues for the corresponding Neumann problem. The Laplace operator Δ appearing in ( ) is often known as the Dirichlet Laplacian when it is considered as accepting only functions u satisfying the Dirichlet boundary condition. More generally, in spectral geometry one considers ( ) on a manifold with boundary Ω. Then Δ is taken to be the Laplace–Beltrami operator, also with Dirichlet boundary conditions.…
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Discussion questions
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