Inequalities between Dirichlet and Neumann eigenvalues of the polyharmonic operators

Abstract:

We prove that $\mu _{k+m}^m <\lambda _k^m$, where $\mu _k^m$ ($\lambda _k^m$) are the eigenvalues of $(-\Delta )^m$ on $\Omega \subset \mathbb R^d$, $d\geq 2$, with Neumann (Dirichlet) boundary conditions.