Dirichlet eigenvalue

Remarks

The Dirichlet eigenvalues form a sequence $0 < \lambda_1(\Omega) < \lambda_2(\Omega) \le \lambda_3(\Omega) \le \dots \to \infty$, listed with multiplicity, when $\Omega$ is connected. They are the eigenvalues of the Dirichlet Laplacian, a self-adjoint operator with compact inverse. The first one is the minimum of the Rayleigh quotient: $\lambda_1(\Omega) = \min \set{ \int_\Omega \abs{\nabla u}^2 \, dx / \int_\Omega u^2 \, dx : u \in H^1_0(\Omega), u \ne 0 }$.

Example: for $\Omega = (0, \pi)$, the function $u(x) = \sin(kx)$ with $k \ge 1$ an integer vanishes at $0$ and $\pi$ and satisfies $-u'' = k^2 u$. So $k^2$ is a Dirichlet eigenvalue, and these are all of them.

Non-example: on $(0, \pi)$, the number $0$ is not a Dirichlet eigenvalue. The solutions of $-u'' = 0$ are $u = a + bx$, and the boundary conditions force $a = b = 0$. With the Neumann condition $u' = 0$ instead, the constant function shows that $0$ is a Neumann eigenvalue.