4 problems

Let $\{\lambda_k\}_{k\ge 1}$ and $\{\mu_k\}_{k\ge 1}$ be the eigenvalues of ${\mathsf H}_{\rm D}$ and ${\mathsf H}_{\rm N}$, respectively, enumerated in non-decreasing order and re…

Let $\Omega\subset\mathbb{R}^n$ be a bounded domain with smooth boundary, and let $\Lambda_1(\Omega)$ be the first positive eigenvalue of the free-plate problem … with the natural…

Let $\Omega$ range over open sets of fixed measure, and consider the biharmonic operator with Dirichlet boundary conditions. Its fundamental tone is the first positive eigenvalue o…

Let $H_{\Omega}=H_{\Omega,\Delta^2}$ be the singular Green's function for the biharmonic operator $\Delta^2$ in a reasonable two-dimensional domain $\Omega$ under Dirichlet boundar…