\begin{definition}[Definition:Dirichlet Eigenvalue of Riemannian Manifold]
Let $\struct {M, g}$ be a compact connected Riemannian manifold with non-empty boundary $\partial M$.
Let $u \in \map {\CC^\infty} M : M \to \R$ be a smooth mapping on $M$ not identically equal to zero.
Let $\Delta$ be the Laplacian.
Let $\lambda \in \R$ be a real number.
Suppose:
:$\ds \Delta u + \lambda u = 0$
:$\valueat u {\partial M} = 0$
Then $\lambda$ is called a '''Dirichlet eigenvalue of $M$'''.
{{Namedfor|Johann Peter Gustav Lejeune Dirichlet|cat = Dirichlet}}
\end{definition}