Inequalities between Dirichlet and Neumann eigenvalues of the Polyharmonic operators

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