A sharp bound for the ratio of the first two eigenvalues of Dirichlet Laplacians and extensions
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.
scientific article; zbMATH DE number 91444