“Rayleigh–Faber–Krahn inequality”的意思、由来-开放百科全书

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Rayleigh–Faber–Krahn inequality

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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 concerning the lowest Dirichlet eigenvalue of the Laplace operator on a bounded domain in , .^[1] 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 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 , 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.

References

1. ^{{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}}

{{Linear algebra}}{{Areas of mathematics |collapsed}}{{DEFAULTSORT:Rayleigh-Faber-Krahn inequality}}{{mathanalysis-stub}}

3 : Elliptic partial differential equations|Riemannian geometry|Spectral theory

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