Definition:Dirichlet Eigenvalue
(Redirected from Definition:Dirichlet Eigenvalue of Riemannian Manifold)
Jump to navigation
Jump to search
Definition
Let $\struct {M, g}$ be a compact connected Riemannian manifold with non-empty boundary $\partial M$.
Let $u \in \map {C^\infty} M : M \to \R$ be a smooth mapping on $M$ not identically equal to zero.
Let $\nabla^2$ be the Laplace-Beltrami operator.
Let $\lambda \in \R$ be a real number.
Suppose:
- $\ds \nabla^2 u + \lambda u = 0$
- $\valueat u {\partial M} = 0$
Then $\lambda$ is called a Dirichlet eigenvalue of $M$.
Source of Name
This entry was named for Johann Peter Gustav Lejeune Dirichlet.
Sources
- 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems