Differential operators | Partial differential equations | Spectral theory
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 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 : 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. 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 has where the supremum is taken over all (k−1)-tuples and the infimum over all u orthogonal to the . (Wikipedia).
