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  and Krahn  for the R; a proof of the generalized version can be found for example in . 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.
KeywordsConnected Graph Normal Derivative Boundary Edge Small Root Degree Sequence
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