Isoperimetric inequalities for eigenvalues of the Laplacian

Abstract

Dedicated to Professor Barry Simon on his sixtieth birthday.Abstract. We give a survey of universal inequalities for low-lying eigenvalues of the laplacian.2000 Mathematics Subject Classification. Primary 35P15; Secondary 35J05, 49R50.

Isoperimetric inequalities for eigenvalues of the Laplacian and the Schrödinger operator

“…Take to be the rectangle of sides a and b. Its Dirichlet eigenvalues are given by 15) with n, m = 1, 2, . .…”

“…There are many interesting results that we have left out of this review. For different perspectives, selections and emphasis, please refer for example to the reviews [3][4][5]15,60], among many others. …”

The Saint-Venant Inequality for the Laplace Operator with Robin Boundary Conditions

“…Here λ 1 (Ω) is defined as the lowest value for which the problem −∆u = λ 1 (Ω)u in Ω, u = 0 on ∂Ω, admits a non trivial solution. Following the review paper [21] (see also [3]), Lord Rayleigh was motivated in his conjecture by the study of the principal frequency of vibration of a plane elastic membrane fixed at its boundary, stating that the circular shape has the lowest mode of vibration (and giving some evidence of it).…”

Krahn’s proof of the Rayleigh conjecture revisited

“…A classic is also Bandle's book [4]. More recent surveys include [3,21]. The old classic is the book [33] by Pólya and Szegö.…”