Misplaced Pages

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 ${\displaystyle \mathbb {R} ^{n}}$, ${\displaystyle n\geq 2}$. 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 ${\displaystyle n=2}$, 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.

See also

• Hearing the shape of a drum

References

1. Benguria, Rafael D. (2001) , "Rayleigh–Faber–Krahn inequality", Encyclopedia of Mathematics, SpringerLink, retrieved 6 November 2011
2. Chavel, Isaac (1984). Eigenvalues in Riemannian geometry. OCLC 1106800772.

v t e Linear algebra
Outline Glossary
Basic concepts	Scalar Vector Vector space Scalar multiplication Vector projection Linear span Linear map Linear projection Linear independence Linear combination Multilinear map Basis Change of basis Row and column vectors Row and column spaces Kernel Eigenvalues and eigenvectors Transpose Linear equations
Matrices	Block Decomposition Invertible Minor Multiplication Rank Transformation Cramer's rule Gaussian elimination Productive matrix
Bilinear	Orthogonality Dot product Hadamard product Inner product space Outer product Kronecker product Gram–Schmidt process
Multilinear algebra	Determinant Cross product Triple product Seven-dimensional cross product Geometric algebra Exterior algebra Bivector Multivector Tensor Outermorphism
Vector space constructions	Dual Direct sum Function space Quotient Subspace Tensor product
Numerical	Floating-point Numerical stability Basic Linear Algebra Subprograms Sparse matrix Comparison of linear algebra libraries
Category

This mathematical analysis–related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

This Riemannian geometry-related article is a stub. You can help Misplaced Pages by expanding it.

• v
• t
• e

• Elliptic partial differential equations
• Riemannian geometry
• Spectral theory
• Mathematical analysis stubs
• Riemannian geometry stubs