Definition:Neumann Eigenvalue

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.

Let $N$ be the outward-pointing unit normal vector field on $\partial M$.

Suppose:

$\ds \nabla^2 u + \lambda u = 0$

$\bigvalueat {N u} {\partial M} = 0$

Then $\lambda$ is called a Neumann eigenvalue of $M$.

Source of Name

This entry was named for Carl Gottfried Neumann.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Problems