A Rayleigh-Faber-Krahn Inequality and Some Monotonicity Properties for Eigenvalue Problems with Mixed Boundary Conditions
Catherine Bandle · International series of numerical mathematics · 2008
An eigenvalue problem is considered whose eigenvalues appear in the interior and on the boundary. It has been shown in [ 1 ] that there exists an infinite sequence of positive and an infinite sequence of negative eigenvalues. The lowest positive and the largest negative eigenvalue λ 1 , resp. λ −1 can be characterised by means of a Rayleigh principle. It turns out that among all domains of given volume the ball has the smallest λ 1 . A partial result in this direction is established for λ −1 . The proof uses the isoperimetric inequality of Krahn-Bossel-Daners. Some monotonicity properties similar to those for the elastically supported membrane are included.