the Dirichlet eigenvalue problem
/ DEER-ish-lay /
Strike a drum and it rings in a definite set of pitches — a lowest fundamental tone and a ladder of higher overtones, fixed entirely by the shape of the drumhead. The Dirichlet eigenvalue problem is the mathematics of exactly this: find the special numbers lambda and the special shapes u for which - Laplacian u = lambda u inside a region, with u held at zero on the rigid boundary. The lambda are the squared frequencies, the eigenfunctions u are the modes of vibration, and the clamped edge is the Dirichlet boundary condition.
Formally: on a bounded domain, find nonzero u with - Laplacian u = lambda u in the interior and u = 0 on the boundary. Because the Laplacian here is self-adjoint and its inverse is compact (Rellich-Kondrachov again), the spectral theorem applies and the structure is beautifully rigid. The eigenvalues form a discrete sequence 0 < lambda_1 < lambda_2 is not greater than lambda_3 is not greater than ... marching off to infinity, with no accumulation point except at infinity and each value of finite multiplicity. The eigenfunctions can be chosen orthonormal in L^2 and they form a complete basis — every reasonable function on the domain expands in them, the higher-dimensional analogue of a Fourier series. On a rectangle the eigenfunctions are products of sines and the eigenvalues are sums of squared integers times pi^2; on a disk they are Bessel functions times angular sines and cosines.
This spectrum is the bridge between geometry and analysis. It controls how fast solutions of the heat equation decay (each Fourier mode dies like e^(-lambda_n t), so the smallest eigenvalue sets the slowest decay), the natural frequencies of the wave equation, and the ground-state energy in quantum mechanics. Kac's famous question 'can you hear the shape of a drum?' — does the spectrum determine the domain? — turned out to have the answer no in general (there exist distinct shapes with identical spectra), but the eigenvalues do reveal a great deal of geometry, including the area and perimeter through Weyl's law.
On a square of side pi with u = 0 on all four edges, separation of variables gives eigenfunctions sin(m x) sin(n y) for positive integers m, n, with eigenvalues lambda_(m,n) = m^2 + n^2. The fundamental is m = n = 1 with lambda_1 = 2; the next is m,n = 1,2 or 2,1 with lambda = 5 (a doubly degenerate overtone). These are exactly the resonant pitches of a square drum.
Each eigenvalue is a squared frequency, each eigenfunction a vibration mode — degeneracies are repeated tones.
The boundary condition is part of the problem, not a detail: the Dirichlet spectrum (u = 0 on the edge) differs from the Neumann spectrum (normal derivative zero), which famously starts at lambda = 0 with a constant eigenfunction. Change the boundary condition and you change every eigenvalue. Also, eigenfunctions for distinct eigenvalues are automatically orthogonal, but within a degenerate eigenspace you must choose an orthogonal basis by hand.