Definition:Dirichlet Eigenvalue

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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