The relation between the first and second natural frequencies of vibrations of a membrane (Q1803096)

The equation \(-\Delta u=\lambda u\) is considered in a domain \(D\subset\mathbb{R}^ 2\) with vanishing boundary condition. Let \(\lambda_ 1\leq\lambda_ 2\leq\dots\) be the corresponding eigenvalues. There are determined conditions under which the ratio \(\lambda_ 2/\lambda_ 1\) is locally extremal on \(D\) for all enough smooth one-parametric perturbations of \(\partial D\). The cases when \(\lambda_ 2\) is simple or double are investigated. These results confirm the conjecture that \(\lambda_ 2/\lambda_ 1\) is maximal if \(D\) is a circle. In the paper are used polar coordinates, previously obtained parametric expansions of eigenvalues and expressions for expansion coefficients in terms of Bessel functions as well as stationarity conditions.