Improved inequalities between Dirichlet and Neumann eigenvalues of the biharmonic operator

We prove that the $(k+d)$-th Neumann eigenvalue of the biharmonic operator on a bounded connected $d$-dimensional $(d\ge 2)$ Lipschitz domain is not larger than its $k$-th Dirichlet eigenvalue for all $k\in \mathbb{N}$. For a special class of domains with symmetries we obtain a stronger inequality. Namely, for this class of domains, we prove that the $(k+d+1)$-th Neumann eigenvalue of the biharmonic operator does not exceed its $k$-th Dirichlet eigenvalue for all $k\in \mathbb{N}$. In particular, in two dimensions, this special class consists of domains having an axis of symmetry.