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

${\displaystyle {\begin{cases}\Delta u+\lambda u=0&{\rm {{in\ }\Omega }}\\u|_{\partial \Omega }=0.&\end{cases}}}$

(1)

Here Δ is the Laplacian, which is given in xy-coordinates by

${\displaystyle \Delta u={\frac {\partial ^{2}u}{\partial x^{2}}}+{\frac {\partial ^{2}u}{\partial y^{2}}}.}$

The boundary value problem (1) 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 (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:

${\displaystyle 0<\lambda _{1}\leq \lambda _{2}\leq \cdots ,\quad \lambda _{n}\to \infty ,}$

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 ${\displaystyle H_{0}^{2}(\Omega )}$ into ${\displaystyle L^{2}(\Omega )}$. 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,

${\displaystyle \lambda _{1}=\inf _{u\not =0}{\frac {\int _{\Omega }|\nabla u|^{2}}{\int _{\Omega }|u|^{2}}},}$

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 ${\displaystyle u\in H_{0}^{1}(\Omega )}$. Moreover, using results from the calculus of variations analogous to the Lax–Milgram theorem, one can show that a minimizer exists in ${\displaystyle H_{0}^{1}(\Omega )}$. More generally, one has

${\displaystyle \lambda _{k}=\sup \inf {\frac {\int _{\Omega }|\nabla u|^{2}}{\int _{\Omega }|u|^{2}}}}$

where the supremum is taken over all (k−1)-tuples ${\displaystyle \phi _{1},\dots ,\phi _{k-1}\in H_{0}^{1}(\Omega )}$ and the infimum over all u orthogonal to the ${\displaystyle \phi _{i}}$.