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. [ABSTRACT FROM AUTHOR]