Zhen-Guo Song

A sharp isoperimetric bound for the ratio of the fifth and fourth eigenvalues of Dirichlet Laplacian

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with $n\geq2$. The well-known Payne-P\'olya-Weinberger conjecture for consecutive eigenvalue ratios asserts that $$ \frac{\lambda_{m+1}(\Omega)}{\lambda_m(\Omega)}\leq\left(\frac{j_{n / 2,1}}{j_{n / 2-1,1}}\right)^2,\quad m=1,2,3,\cdots. $$ Here, $j_{p,k}$ denotes the...