Upper bound estimation for the ratio of the first two eigenvalues of Robin Laplacian

The celebrated conjecture by Payne, P\'{o}lya and Weinberger (1956) states that for the fixed membrane problem, the ratio of the first two eigenvalues, $\lambda_2/\lambda_1$, is maximized by a disk. A more general dimensional version of this conjecture was later resolved by Ashbaugh and Benguria in the 1990s. For the Robin Laplacian, Payne and Schaefer (2001) formulated an analogous conjecture, positing that the ratio $\mu_2/\mu_1$ is also maximized by a disk for a range of the boundary parameter $\sigma$. This was later restated by Henrot in 2003. In this work, under some suitable conditions, we affirm this conjecture for all dimensions $N\geq2$ and for all $\sigma>0$. Furthermore, we prove that the maximum value of $\mu_2/\mu_1$ is strictly decreasing in $\sigma$ over the entire interval $(0,+\infty)$. Our result provides a positive answer to a variant of Yau's Problem 77: by measuring the ratio of the first two eigenfrequencies, one can determine whether an elastically supported drum is circular.