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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>{{cite web|last=Benguria|first=Rafael D.|title=Rayleigh–Faber–Krahn inequality|url=http://eom.springer.de/r/r130040.htm|work=Encyclopaedia of Mathematics|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.

More generally, the Faber–Krahn inequality holds in any [[Riemannian manifold]] in which the [[isoperimetric inequality]] holds.

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

==References==
<references />

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