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- Dirichlet_eigenvalue abstract "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</dl>Here Δ is the Laplacian, which is given in xy-coordinates byThe boundary value problem (1) is, of course, 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 (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 Ω. Then Δ is taken to be the Laplace-Beltrami operator, also with Dirichlet boundary conditions.It can be shown, using the spectral theorem for compact self-adjoint operators that the eigenspaces are finite-dimensional and that the Dirichlet eigenvalues λ are real, positive, and have no limit point. Thus they can be arranged in increasing order:where each eigenvalue is counted according to its geometric multiplicity. The eigenspaces are orthogonal in the space of square-integrable functions, and consist of smooth functions. In fact, the Dirichlet Laplacian has a continuous extension to an operator from the Sobolev space into . This operator is invertible, and its inverse is compact and self-adjoint so that the usual spectral theorem can be applied to obtain the eigenspaces of Δ and the reciprocals 1/λ of its eigenvalues.One of the primary tools in the study of the Dirichlet eigenvalues is the max-min principle: the first eigenvalue λ1 minimizes the Dirichlet energy. To wit,the infimum is taken over all u of compact support that do not vanish identically in Ω. By a density argument, this infimum agrees with that taken over nonzero . Moreover, using results from the calculus of variations analogous to the Lax–Milgram theorem, one can show that a minimizer exists in . More generally, one haswhere the supremum is taken over all (k−1)-tuples and the infimum over all u orthogonal to the φi.".
- Dirichlet_eigenvalue thumbnail SpiralCladding.png?width=300.
- Dirichlet_eigenvalue wikiPageID "22860539".
- Dirichlet_eigenvalue wikiPageRevisionID "602343609".
- Dirichlet_eigenvalue first "Rafael D.".
- Dirichlet_eigenvalue hasPhotoCollection Dirichlet_eigenvalue.
- Dirichlet_eigenvalue id "d/d130170".
- Dirichlet_eigenvalue last "Benguria".
- Dirichlet_eigenvalue title "Dirichlet eigenvalues".
- Dirichlet_eigenvalue year "2001".
- Dirichlet_eigenvalue subject Category:Differential_operators.
- Dirichlet_eigenvalue subject Category:Partial_differential_equations.
- Dirichlet_eigenvalue subject Category:Spectral_theory.
- Dirichlet_eigenvalue type Abstraction100002137.
- Dirichlet_eigenvalue type Communication100033020.
- Dirichlet_eigenvalue type DifferentialEquation106670521.
- Dirichlet_eigenvalue type DifferentialOperators.
- Dirichlet_eigenvalue type Equation106669864.
- Dirichlet_eigenvalue type Function113783816.
- Dirichlet_eigenvalue type MathematicalRelation113783581.
- Dirichlet_eigenvalue type MathematicalStatement106732169.
- Dirichlet_eigenvalue type Message106598915.
- Dirichlet_eigenvalue type Operator113786413.
- Dirichlet_eigenvalue type PartialDifferentialEquation106670866.
- Dirichlet_eigenvalue type PartialDifferentialEquations.
- Dirichlet_eigenvalue type Relation100031921.
- Dirichlet_eigenvalue type Statement106722453.
- Dirichlet_eigenvalue comment "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.".
- Dirichlet_eigenvalue label "Dirichlet eigenvalue".
- Dirichlet_eigenvalue sameAs m.0642dlt.
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- Dirichlet_eigenvalue sameAs Q5280763.
- Dirichlet_eigenvalue sameAs Dirichlet_eigenvalue.
- Dirichlet_eigenvalue wasDerivedFrom Dirichlet_eigenvalue?oldid=602343609.
- Dirichlet_eigenvalue depiction SpiralCladding.png.
- Dirichlet_eigenvalue isPrimaryTopicOf Dirichlet_eigenvalue.