Faber-Krahn Type Inequalities

The celebrated Faber-Krahn Theorem gives an important isoperimetric inequality concerning Dirichlet eigenvalues. It states that the ball has lowest first Dirichlet eigenvalue amongst all bounded domains of the same volume in R (with the standard Euclidean metric). It has been first conjectured by Rayleigh and proved independently by Faber [61] and Krahn [118] for the R; a proof of the generalized version can be found for example in [29]. The Faber-Krahn theorem can also be rephrased in the following way: for all drums with the same area and same tension the circular-shaped has the lowest tone.