Ashbaugh, Mark S.; Benguria, Rafael D. A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions. (English) Zbl 0757.35052 Ann. Math. (2) 135, No. 3, 601-628 (1992). The authors present a proof for the Payne-Polya-Weinberger conjecture. It says that the ratio \(\lambda_ 2/\lambda_ 1\) of the membrane eigenvalues \[ \Delta\varphi+\lambda\varphi=0\quad\text{in }D, \qquad \varphi|_{\partial D}=0 \] is maximal for the ball. The proof relies on the Rayleigh principle and uses rearrangement techniques together with specific properties of the Bessel functions and their zeros.Extensions to problems with more general operators are given. Reviewer: C.Bandle (Basel) Cited in 4 ReviewsCited in 75 Documents MSC: 35P15 Estimates of eigenvalues in context of PDEs 35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation Keywords:Payne-Polya-Weinberger conjecture; membrane eigenvalues; Rayleigh principle × Cite Format Result Cite Review PDF Full Text: DOI