×

A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions. (English) Zbl 0757.35052

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)

MSC:

35P15 Estimates of eigenvalues in context of PDEs
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
Full Text: DOI