View source for Rayleigh–Faber–Krahn inequality

{{Short description|Geometric Wave Phenomenon}}
In [[spectral geometry]], the '''Rayleigh–Faber–Krahn inequality''', named after its conjecturer, [[Lord Rayleigh]], and two individuals who independently proved the conjecture, [[G. Faber]] and [[Edgar Krahn]], is an [[Inequality (mathematics)|inequality]] concerning the lowest [[Dirichlet eigenvalue]] of the [[Laplace operator]] on a bounded domain in <math>\mathbb{R}^n</math>, <math>n \ge 2</math>.<ref name=springer>{{SpringerEOM|last=Benguria|first=Rafael D.|title=Rayleigh–Faber–Krahn inequality|publisher=SpringerLink|accessdate=6 November 2011}}</ref>  It states that the first Dirichlet eigenvalue is no less than the corresponding Dirichlet eigenvalue of a Euclidean ball having the same volume.  Furthermore, the inequality is [[Rigidity (mathematics)|rigid]] in the sense that if the first Dirichlet eigenvalue is equal to that of the corresponding ball, then the domain must actually be a ball. In the case of <math>n=2</math>, the inequality essentially states that among all drums of equal area, the circular drum (uniquely) has the lowest voice.

More generally, the Faber–Krahn inequality holds in any [[Riemannian manifold]] in which the [[isoperimetric inequality]] holds.<ref>{{Cite book|url=http://worldcat.org/oclc/1106800772|title=Eigenvalues in Riemannian geometry|last=Chavel|first= Isaac|date=1984|oclc=1106800772}}</ref> In particular, according to [[Cartan–Hadamard conjecture]], it should hold in all simply connected manifolds of nonpositive curvature.

==See also==
* [[Hearing the shape of a drum]]

== References ==
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{{Linear algebra}}
{{Functional analysis}}
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[[Category:Elliptic partial differential equations]]
[[Category:Riemannian geometry]]
[[Category:Spectral theory]]


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