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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 , . 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.

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  • レイリー＝フェイバー＝クラーンの不等式 (ja)
  • Rayleigh–Faber–Krahn inequality (en)
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  • 数学のスペクトル幾何学の分野において、レイリー＝フェイバー＝クラーンの不等式（レイリー＝フェイバー＝クラーンのふとうしき、英: Rayleigh–Faber–Krahn inequality）は、その成立を予想したレイリー卿と、それをそれぞれ独自に証明したとの名にちなむ、, 内のある有界領域上のラプラス作用素の最小のディリクレ固有値に関する不等式である。この不等式では、その第一ディリクレ固有値は、同じ体積のユークリッド球の対応するディリクレ固有値より小さくはならないことが示される。さらに、その第一ディリクレ固有値が対応する球のそれと等しいなら、その領域は実際に球でなければならない意味で、その不等式は rigid である。 より一般に、フェイバー＝クラーンの不等式は、が成立する任意のリーマン多様体において成立する。 (ja)
  • 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 , . 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. (en)
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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 , . 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. In particular, according to Cartan–Hadamard conjecture, it should hold in all simply connected manifolds of nonpositive curvature. (en)
  • 数学のスペクトル幾何学の分野において、レイリー＝フェイバー＝クラーンの不等式（レイリー＝フェイバー＝クラーンのふとうしき、英: Rayleigh–Faber–Krahn inequality）は、その成立を予想したレイリー卿と、それをそれぞれ独自に証明したとの名にちなむ、, 内のある有界領域上のラプラス作用素の最小のディリクレ固有値に関する不等式である。この不等式では、その第一ディリクレ固有値は、同じ体積のユークリッド球の対応するディリクレ固有値より小さくはならないことが示される。さらに、その第一ディリクレ固有値が対応する球のそれと等しいなら、その領域は実際に球でなければならない意味で、その不等式は rigid である。 より一般に、フェイバー＝クラーンの不等式は、が成立する任意のリーマン多様体において成立する。 (ja)
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