Definition:Neumann Eigenvalue of Riemannian Manifold

Definition
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.

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

Suppose:


 * $\ds \Delta u + \lambda u = 0$


 * $\valueat {N u} {\partial M} = 0$

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