Dirichlet eigenvalue

In mathematics, the Dirichlet eigenvalues are the fundamental modes of vibration of an idealized drum with a given shape.

Here a "drum" is thought of as an elastic membrane Ω, which is represented as a planar domain whose boundary is fixed.

More generally, in spectral geometry one considers (1) on a manifold with boundary Ω.

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.

In fact, the Dirichlet Laplacian has a continuous extension to an operator from the Sobolev space

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.

To wit, the infimum is taken over all u of compact support that do not vanish identically in Ω.

Moreover, using results from the calculus of variations analogous to the Lax–Milgram theorem, one can show that a minimizer exists in

[3] The modes of pump should not avoid the active core used in double-clad fiber amplifiers.

The spiral-shaped domain happens to be especially efficient for such an application due to the boundary behavior of modes of Dirichlet laplacian.

Similarly, all the modes of the Dirichlet Laplacian have non-zero values in vicinity of the chunk.

As the mode is steady-state solution of the propagation equation (with trivial dependence of the longitudinal coordinate), the total force should be zero.