High Quality Content by WIKIPEDIA articles! 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 ?, which is represented as a planar domain whose boundary is fixed.The boundary value problem (1) is, of course, the Dirichlet problem for the Helmholtz equation, and so ? is known as a Dirichlet eigenvalue for ?. Dirichlet eigenvalues are contrasted with Neumann eigenvalues: eigenvalues for the corresponding Neumann problem. The Laplace operator ? appearing in (1) 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 (1) on a manifold with boundary ?. Then ? is taken to be the Laplace-Beltrami operator, also with Dirichlet boundary conditions.

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